Combinatorial Coding Theory
Gennian Ge
Zhejiang University
2011 June
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
1 / 297
Part I
1 Constant-Weight Codes
2 Constant-Composition Codes
3 Permutation Codes
Part II
4 Optical Orthogonal Codes
5 Conflict-Avoiding Codes
6 Impulse Radio Sequences
Part III
7 Comma-Free Codes
8 Frequency-Hopping Sequences
Part IV
9 Deletion-Correcting Codes
10 Codebook
11 Cover-Free Family
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
2 / 297
Part I
Constant-Weight Codes
Constant-Composition Codes
Permutation Array
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Combinatorial Coding Theory
2011 June
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Constant Weight Codes
1
Constant Weight Codes
Introduction
CWC and Designs
Binary Constant Weight Codes
q-Ary Constant Weight Codes with q ≥ 3
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
4 / 297
Constant Weight Codes
1
Introduction
Constant Weight Codes
Introduction
CWC and Designs
Binary Constant Weight Codes
q-Ary Constant Weight Codes with q ≥ 3
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
5 / 297
Constant Weight Codes
Introduction
Preliminaries
A q-ary code of length n is a set C ⊆ ZX
q for some X of size n. The
elements of C are called codewords. The Hamming norm or the
Hamming weight of a vector u ∈ ZX
q is defined as
kuk = |{x ∈ X : ux 6= 0}|. The distance induced by this norm is called the
Hamming distance, denoted dH , so that dH (u, v) = ku − vk, for
u, v ∈ ZX
q.
For any two vectors u, v ∈ ZX
q , define their support as
supp(u, v) = {x ∈ X : ux 6= vx }. We write supp(u) instead of supp(u, 0)
and also call supp(u) the support of u.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
6 / 297
Constant Weight Codes
Introduction
Constant Weight Codes
A code C is said to have minimum distance d if dH (u, v) ≥ d for all
distinct u, v ∈ C. If kuk = w for every codeword u ∈ C, then C is said to
be of (constant) weight w . A q-ary constant-weight code of length n,
distance d, and weight w is denoted as an (n, d, w )q code.
The number of codewords in an (n, d, w )q code is called the size of the
code. The maximum size of an (n, d, w )q code is denoted Aq (n, d, w ) and
the (n, d, w )q codes achieving this size are said to be optimal.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
7 / 297
Constant Weight Codes
Introduction
Background
Constant-weight codes have been recently introduced in a number of
engineering applications, including
code-division multiple-access (CDMA) systems for optical fibers,
protocol design for the collision channel without feedback,
automatic-repeat-request error-control systems, and
parallel asynchronous communication.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
8 / 297
Constant Weight Codes
Introduction
In addition, they often serve as building blocks in
design of spherical codes and
DC-free constrained codes.
Further applications have been reported in
frequency-hopping spread-spectrum systems,
radar and sonar signal design,
mobile radio and synchronization.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
9 / 297
Constant Weight Codes
Introduction
Upper Bounds
Theorem
If w 2 − nw + nδ > 0, then A(n, 2δ, w ) ≤
j
nδ
w 2 −nw +nδ
k
.
Lemma
n(q − 1)
Aq (n, d, w ) ≤
Aq (n − 1, d, w − 1) .
w
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
10 / 297
Constant Weight Codes
Introduction
Johnson Bound1
Theorem
If w ≥ δ, then
n
A(n, 2δ, w ) ≤
w
n−1
n−w +δ
...
.
w −1
δ
(1)
1
S. M. Johnson. A new upper bound for error-correcting codes. IRE Trans.,
IT-8:203–207, 1962.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
11 / 297
Constant Weight Codes
1
CWC and Designs
Constant Weight Codes
Introduction
CWC and Designs
Binary Constant Weight Codes
q-Ary Constant Weight Codes with q ≥ 3
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
12 / 297
Constant Weight Codes
CWC and Designs
The incidence vectors of the blocks of a t-(v , k, λ) design with maximum
block intersection number s form a constant weight code of weight w = k,
length n = v , and minimum distance d = 2(k − s).
In particular, the incidence vectors of a t-(v , k, 1) packing form a constant
weight code of weight w = k, length n = v , and minimum distance
d = 2(k − t + 1).
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
13 / 297
Constant Weight Codes
CWC and Designs
The rows of the point by block incidence matrix of a 2-design with
parameters v , k, λ, r , b form a constant weight code with length n = b,
weight w = r , and minimum distance d = 2(r − λ).
The existence of a Steiner system or a 2-design implies the existence of an
optimal constant weight code. Therefore, any table for such designs gives
a table of values for A(n, 2δ, w ).
Example
For example, the existence of an S(5, 6, 12) implies that A(12, 4, 6) = 132.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Constant Weight Codes
CWC and Designs
When d is large, there are not much results and general method. But
there are still some paper dealing with the case when n is small.
The tables on web page2 give the best known lower (and upper) bounds,
which are focus on the codes wight minimum distance
d ∈ {4, 6, 8, 10, 12, 14, 16, 18}.
2
http://www.win.tue.nl/˜aeb/codes/Andw.html
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
15 / 297
Constant Weight Codes
CWC and Designs
Group Divisible Design
Let K and G be sets of positive integers and let λ be a positive integer. A
group divisible design of index λ and order v ((K , λ)-GDD) is a triple
(V , G, B), where V is a finite set of cardinality v , G is a partition of V into
parts (groups) whose sizes lie in G , and B is a family of subsets (blocks)
of V that satisfy
if B ∈ B then |B| ∈ K ,
every pair of distinct elements of V occurs in exactly λ blocks or one
group, but not both, and
|G| > 1.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
16 / 297
Constant Weight Codes
CWC and Designs
If v = a1 g1 + a2 g2 + · · · + as gs , and if there are a i groups of size gi ,
i = 1, 2, . . . , s, then the (K , λ)-GDD is of type g1a1 g2a2 . . . gsas . This is the
exponential notation for the group type.
If K = {k}, then the (K , λ)-GDD is a (k, λ)-GDD. If λ = 1, the GDD is a
K -GDD. Furthermore, a ({k}, 1)-GDD is a k-GDD.
A GDD is uniform if all groups have the same size, that is, if it is of type
mu for some positive integer u.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
17 / 297
Constant Weight Codes
CWC and Designs
GDD to CWC
An optimal (v , k, d)g +1 code can be constructed from a k-GDD of type
gv.
Let Im denote the set {1, 2, . . . , m}. Suppose that
(Iv × Ig , {{i} × Ig |i ∈ Iv }, B) is a k-GDD of type g v . For each block
{(i1 , a1 ), (i2 , a2 ), . . . , (ik , ak )} ∈ B, we construct a codeword of length v by
putting aj in position ij , 1 ≤ j ≤ k, and zeros elsewhere.
Then k − 1 ≤ d ≤ 2(k − 2) + 1.
If d = 2(k − 2) + 1, then we call such a GDD a generalized Steiner
system GS(2, k, v , g ).
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
18 / 297
Constant Weight Codes
CWC and Designs
Known of GS
For k = 3
The sufficient existence condition of a GS (2, 3, n, g ) for g = 2, 3, 4, 5 and
9 is also sufficient with exception (g , n) = (2, 6).
For k = 4
There exists a GS(2, 4, v , 2) for any integer v > 7 and v ≡ 1 (mod 3) with
one exception for v = 7 and three possible exceptions for v ∈ {13, 52, 58}.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
19 / 297
Constant Weight Codes
CWC and Designs
Holey Packing
Let k, g and n ≥ k be positive integers.
Definition
A holey packing, k-HP of type g n , is an ordered triple (X , G, B) where X
is a gn-set (of points), G = {G1 , G2 , . . . , Gn } is a partition of X into n
holes (or groups) of g points, and B is a collection of k-subsets (called
blocks) of X such that:
any pair of points from distinct groups occurs in at most one of
blocks, and
no block contains two distinct points of any group (this is the ‘hole’).
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
20 / 297
Constant Weight Codes
CWC and Designs
Maximum Distance Holey Packing
Definition
A k-HP of type g n with g > 1 is called a maximum distance holey
packing, denoted by MDHP(2, k, n, g ), if
it has maximum number of blocks; and
its derived code has minimum Hamming distance d = 2(k − 2) + 1.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
21 / 297
Constant Weight Codes
CWC and Designs
MDHP and CWC
Using this definition, we can state the fundamental relation between
CWCs and HPs as the following theorem.
Theorem
For any integer g > 1, an MDHP(2, k, n, g ) is equivalent to an optimal
(n, k, 2(k − 2) + 1)g +1 code.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
22 / 297
Constant Weight Codes
CWC and Designs
Some Results
For all integers n satisfying n = 2 (mod 3) and n ≥ 8, an
MDPH(2, 3, n, 2) exists; there does not exist an MDHP(2, 3, 5, 2).
(
c if q ≤ n,
b (q−1)n
2
Aq (n, 3, 2) =
n
if q > n.
2
j
n−1 k n
Aq (n, 4, 3) = min{ (q−1)n
, 3 } for all n and q;
3
2
Aq (n, 2w − 1, w ) = (q − 1)n/w for all sufficiently large n satisfying
w |(q − 1)n.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
23 / 297
Constant Composition Codes
2
Constant Composition Codes
Introduction
Group Divisible Codes
Results on Small d and q
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
24 / 297
Constant Composition Codes
2
Introduction
Constant Composition Codes
Introduction
Group Divisible Codes
Results on Small d and q
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
25 / 297
Constant Composition Codes
Introduction
Definition
The composition of a vector u ∈ ZX
q is the tuple w = [w1 , . . . , wq−1 ],
where wj = |{x ∈ X : ux = j}|. A code C is said to be of constant
composition if every codeword in C has composition w .
A q-ary code of length n, distance d, and constant composition w is
referred to as an (n, d, w )q -code. The maximum size of an
(n, d, w )q -code is denoted Aq (n, d, w ) and the (n, d, w )q -codes achieving
this size are called optimal.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
26 / 297
Constant Composition Codes
Introduction
Background
Communication over an electric power line.
Frequency hopping.
Indeed, whenever a different cost is associated with each symbol in
the underlying alphabet, uniform cost of codewords leads to constant
composition.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
27 / 297
Constant Composition Codes
Introduction
Upper Bound
Lemma

Pq−1 wi n
i=1
Pq−1


 i=1 wi w1 ,...,wq−1
n
Aq (n, d, [w1 , . . . , wq−1 ]) = b Pq−1 c
wi


 i=1
1
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
if d ≤ 2,
P
if d = 2 q−1
i=1 wi ,
Pq−1
if d ≥ 2 i=1 wi + 1.
2011 June
28 / 297
Constant Composition Codes
Introduction
Johnson Bound3
Theorem
Aq (n, d, [w1 , . . . , wq−1 ]) ≤
n
Aq (n − 1, d, [w1,i , . . . , wq−1,i ])
wi
for any i = 1, . . . , q − 1, where
wj,i
(
wj − 1
=
wj
if j = i,
if j =
6 i.
3
M. Svanström, P. R. J. Östergård, and G. T. Bogdanova. Bounds and constructions
for ternary constant-composition codes. IEEE Trans. Inform. Theory, 48(1):101–111,
2002.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
29 / 297
Constant Composition Codes
2
Group Divisible Codes
Constant Composition Codes
Introduction
Group Divisible Codes
Results on Small d and q
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
30 / 297
Constant Composition Codes
Group Divisible Codes
Group Divisible Codes
A group divisible code (GDC) of distance d is a tripe (X , G, C), where
G = {G1 , . . . , Gt } is a partition of X with cardinality |X | = n and C ⊆ ZqX
is a q-ary code of length n, such that dH (u, v) ≥ d for all distinct u, v ∈ C,
and ku |Gi k ≤ 1 for all u ∈ C, 1 ≤ i ≤ t.
Elements of G are called groups. The type of a GDC (X , G, C) is the
multiset *|G | : G ∈ G+. The size of a GDC (X , G, C) is |C|.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Constant Composition Codes
Group Divisible Codes
We denote a GDC (X , G, C) of distance d as w -GDC(d) if C is of constant
weight w . If we want to emphasize the composition of the codewords, we
denote the GDC as w -GDC(d) when every u ∈ C has composition w .
Note that an (n, d, w )q -code of size s is equivalent to a w -GDC(d) of type
1n with size s.
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Combinatorial Coding Theory
2011 June
32 / 297
Constant Composition Codes
Group Divisible Codes
Constructions
Filling in Groups
Let d ≤ 2(w − 1). Suppose there exists
a w -GDC(d) (X , G, C) of type g1t1 · · · gsts with size a,
a (gi , d, w )q -code Ci of size bi for each i, 1 ≤ i ≤ s,
P
P
then there exists a ( si=1 ti gi , d, w )q -code C 0 of size a + si=1 ti bi .
In particular, if C and Ci , 1 ≤ i ≤ s, are of constant composition w , then
C 0 is also of constant composition w .
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
33 / 297
Constant Composition Codes
Group Divisible Codes
Adjoining y Points
Let y ∈ Z≥0 . Suppose there exists a (master) w -GDC(d) of type
g1t1 · · · gsts with size a, and suppose the following (ingredients) also exist:
a (g1 + y , d, w )q -code of size b.
a w -GDC(d) of type 1gi y 1 with size ci for 2 ≤ i ≤ s,
a w -GDC(d) of type 1g1 y 1 with size c1 if t1 ≥ 2.
P
Then, there exists a (y + si=1 ti gi , d, w )q -code of size
a + b + (t1 − 1)c1 +
s
X
ti ci .
i=2
Furthermore, if the master and ingredient codes are of constant
composition, then so is the resulting code.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
34 / 297
Constant Composition Codes
Group Divisible Codes
Inflation Construction
Suppose there exists a w -GDC(d) (X , G, C) of type g1t1 · · · gsts with size a.
Suppose further that there exists a TD(w , m), then there exists a
w -GDC(d) of type (mg1 )t1 · · · (mgs )ts with size am2 .
If the original GDC is of constant composition w , then so is the derived
GDC.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
35 / 297
Constant Composition Codes
Group Divisible Codes
Theorem (Fundamental Construction)
Let d ≤ 2(w − 1), D = (X , G, A) be a (master) GDD, and ω : X → Z≥0
be a weight function. Suppose that for each A ∈ A, there exists an
(ingredient) w -GDC(d) of
Ptype *ω(a) : a ∈ A+. Then there exists a
?
w -GDC(d) D of type * x∈G ω(x) : G ∈ G+.
Furthermore, if the ingredient GDCs are of constant composition w , then
D? is also of constant composition w .
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
36 / 297
Constant Composition Codes
2
Results on Small d and q
Constant Composition Codes
Introduction
Group Divisible Codes
Results on Small d and q
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
37 / 297
Constant Composition Codes
Results on Small d and q
For all integers n ≥ 3
A3 (n, 4, [2, 1]) =
Gennian Ge (Zhejiang University)
 n(n−2)

 4
n(n−1)
4

 (n−1)2
4
+
n−3 12
Combinatorial Coding Theory
if n ≡ 0
if n ≡ 0
(mod 2)
(mod 2)
if n ≡ 0
(mod 2)
2011 June
38 / 297
Constant Composition Codes
Results on Small d and q
For all integers n ≥ 3


if n = 3
3
A4 (n, 3, [1, 1, 1]) = n(n − 1) − 2 if n ∈ {5, 6}


n(n − 1)
if n ∈
6 {3, 5, 6}
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
39 / 297
Constant Composition Codes
Results on Small d and q
For all integers n ≥ 3


1





6



11
A4 (n, 4, [1, 1, 1]) =
16




23



n n−1 2
if
if
if
if
if
if
n=3
n=5
n=6
n=7
n=8
n 6∈ {3, 5, 6, 7, 8}
except possibly for n ∈ {9, 13, 15, 17}.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
40 / 297
Constant Composition Codes
Results on Small d and q
For all integers n ≥ 3


1



2
A4 (n, 5, [1, 1, 1]) =

4



n
Gennian Ge (Zhejiang University)
if
if
if
if
Combinatorial Coding Theory
n≤4
n=5
n=6
n≥7
2011 June
41 / 297
Constant Composition Codes
Results on Small d and q
For all integers n ≥ 3


1




3

A3 (n, 5, [3, 1]) = 10


n n−1


[ 3 [ 2 ]] − 1

 n n−1
[ 3 [ 2 ]]
Gennian Ge (Zhejiang University)
if n = 4
if n = 6
if n = 9
if n ≡ 5 (mod 6)
otherwise
Combinatorial Coding Theory
2011 June
42 / 297
Constant Composition Codes
Results on Small d and q
For all integers n ≥ 3


1




2



3
A3 (n, 5, [2, 2]) =

7





[ n2 [ n−1

2 ]] − 1


[ n [ n−1 ]]
2 2
Gennian Ge (Zhejiang University)
if n = 4
if n = 5
if n = 6
if n = 7
if n ≥ 8 and n ≡ 3
otherwise
Combinatorial Coding Theory
(mod 4)
2011 June
43 / 297
Permutation Codes
3
Permutation Codes
Introduction
Equidistant Permutation Arrays
Equivalent Combinatorial Designs
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Combinatorial Coding Theory
2011 June
44 / 297
Permutation Codes
Gennian Ge (Zhejiang University)
Introduction
Combinatorial Coding Theory
2011 June
45 / 297
Permutation Codes
3
Introduction
Permutation Codes
Introduction
Equidistant Permutation Arrays
Equivalent Combinatorial Designs
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
45 / 297
Permutation Codes
Introduction
Definition
Let X be a nonempty set of cardinality n. A permutation code (or
permutation array) of length n and minimum distance d, denoted by
PA(n, d) or simply PA, is an m × n array in which
every row contains every symbol of X exactly once, and
any two distinct rows differ in at least d positions.
The size of such a PA is the number of rows m. The maximum size of a
PA(n, d) is denoted M(n, d).
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Combinatorial Coding Theory
2011 June
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Permutation Codes
Introduction
Remarks
The ordering of rows of a PA(n, d) is immaterial; a PA(n, d) is often
presented as a set of words of length n forming a code of Hamming
distance at least d.
M(n, d) ≥ max {M(n − 1, d), M(n, d + 1)}.
By taking all possible permutations as rows, M(n, 1) = M(n, 2) = n!.
From a Latin square of order n, M(n, n) = n.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
47 / 297
Permutation Codes
Introduction
Theorem
M(n, d) ≤ nM(n − 1, d). Therefore, M(n, d) ≤ n!/(d − 1)!4. In addition,
M(n, d) is not equal to n!/(d − 1)! − 1.
Theorem
Distinct permutations in the alternating group An have minimum distance
3. Therefore, M(n, 3) = n!/2.
Theorem (Colbourn, Kløve, and Ling4 )
Suppose there exist k MOLS of order n. Then M(n, n − 1) ≥ kn.
4
C. J. Colbourn, T. Kløve, and A. C. H. Ling. Permutation arrays for power-line
communication and mutually orthogonal Latin squares. IEEE Trans. Inform. Theory,
50(6):1289–1291, 2004.
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Combinatorial Coding Theory
2011 June
48 / 297
Permutation Codes
Introduction
Theorem (Chu et. al.5 )
Suppose there are disjoint PA(n1 , 4) of sizes s1 , . . . , sp and disjoint
PA(n2 , 4) of sizes t1 , . . . , tp . If c is the size of a binary code of length
n = n1 +P
n2 , distance 4, and constant weight n1 , then there is a PA(n, 4)
of size c pj=1 sj tj .
Theorem
Let Dk be the number of derangements of order k. Then
n!
Pd−1 n
k=0 k
n!
≤ M(n, d) ≤ P d−1
.
b
Dk
2 c n
k=0
k Dk
5
W. Chu, C. J. Colbourn, and P. Dukes. Constructions for permutation codes in
power-line communications. Des. Codes Cryptogr., 32(1-3):51–64, 2004.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
49 / 297
Permutation Codes
3
Equidistant Permutation Arrays
Permutation Codes
Introduction
Equidistant Permutation Arrays
Equivalent Combinatorial Designs
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Combinatorial Coding Theory
2011 June
50 / 297
Permutation Codes
Equidistant Permutation Arrays
Equidistant Permutation Arrays
A PA(n, d) in which any two distinct rows differ in exactly d positions is
an equidistant permutation array (EPA). The maximum size of an
EPA(n, d) is denoted M 0 (n, d).
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Combinatorial Coding Theory
2011 June
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Permutation Codes
Equidistant Permutation Arrays
Theorem
M 0 (n, d) ≤ M 0 (n + 1, d), M(n, d);
M 0 (n, n) = n, M 0 (n, 1) = 1, M 0 (n, 2) = 2;
For n ≥ 4, M 0 (n, 3) = n − 1. For n ≥ 10, M 0 (n, 4) =
For 6 ≤ n ≤ 23, M 0 (n, 5) = max 10, n−1
.
2
n
2
.
Theorem
If there exist k + 2 MOLS(n), then M 0 (n + k + 2, n + 2) ≥ (k + 1)n + 1.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Permutation Codes
Equidistant Permutation Arrays
Theorem (Mathon and Vanstone6 )
If q is a prime power, then M 0 (q 2 + q + 1, q 2 + q) ≥ q 3 + q 2 .
Theorem (van Rees and Vanstone7 )
(d + 2)2 2
(n − d)
M (n, d) ≤ max M (d + 1, d),
, d − 5d + 7, 2 +
.
2
dd/3e
0
0
6
R. Mathon and S. A. Vanstone. On the existence of doubly resolvable Kirkman
systems and equidistant permutation arrays. Discrete Math., 30(2):157–172, 1980.
7
G. H. J. Van Rees and S. A. Vanstone. Equidistant permutation arrays: a bound. J.
Austral. Math. Soc. Ser. A, 33(2):262–274, 1982.
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Combinatorial Coding Theory
2011 June
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Permutation Codes
3
Equivalent Combinatorial Designs
Permutation Codes
Introduction
Equidistant Permutation Arrays
Equivalent Combinatorial Designs
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Combinatorial Coding Theory
2011 June
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Permutation Codes
Equivalent Combinatorial Designs
Definition
Let X be a set of cardinality v . A generalized Room square (packing)
of side n and index λ defined on X is an n × n array F having the
following properties:
every cell of F contains a subset (possibly empty) of X ,
each symbol of X occurs once in each row and column of F , and
any two distinct symbols of X occur together in exactly (at most) λ
cells of F .
Denote such an array by GRS(n, λ; v ) (GRSP(n, λ; v )).
Example
Room squares are GRSs where λ = 1 and every nonempty cell contains
exactly two elements.
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Combinatorial Coding Theory
2011 June
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Permutation Codes
Equivalent Combinatorial Designs
Theorem
There exists a GRS(n, λ; v ) if and only if there exists an
EPA(n, n − λ) of size v .
There exists a GRSP(n, λ; v ) if and only if there exists a PA(n, n − λ)
of size v .
An (r , λ)-design ((r , λ)-packing) on v points that admits two orthogonal
resolutions is equivalent to an EPA(r , r − λ) (PA(r , r − λ)) of size v .
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Combinatorial Coding Theory
2011 June
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Part II
Optical Orthogonal Codes
Conflict-Avoiding Codes
Impulse Radio Sequences
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Combinatorial Coding Theory
2011 June
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Optical Orthogonal Codes
4
Optical Orthogonal Codes
Introduction
Basic Results
Cyclic Designs
Preliminary
Constructions via Skew Starters
Constructions via Weil’s Theorem
Recursive Constructions
Two-Dimensional OOC
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Combinatorial Coding Theory
2011 June
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Optical Orthogonal Codes
Gennian Ge (Zhejiang University)
Introduction
Combinatorial Coding Theory
2011 June
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Optical Orthogonal Codes
4
Introduction
Optical Orthogonal Codes
Introduction
Basic Results
Cyclic Designs
Preliminary
Constructions via Skew Starters
Constructions via Weil’s Theorem
Recursive Constructions
Two-Dimensional OOC
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Combinatorial Coding Theory
2011 June
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Optical Orthogonal Codes
Introduction
Correlation Function
Let x = (x0 , x1 , . . . , xv −1 ) and y = (y0 , y1 , . . . , yv −1 ) be two
(0, 1)-sequences of length v . Define the correlation function of x and y to
be
v −1
X
λx,y (t) =
xi yi+t
(2)
i=0
where the addition in subscripts is taken modulo v .
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Combinatorial Coding Theory
2011 June
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Optical Orthogonal Codes
Introduction
Optical Orthogonal Codes
A (v , k, λa , λc ) optical orthogonal code (OOC) is a family C of (0, 1)
sequences of length v with constant Hamming weight k satisfying the
following two properties:
(The Auto-Correlation Property) For any x ∈ C and any integer r 6≡ 0
(mod v )
λx (r ) ≤ λa ,
(The Cross-Correlation Property) For any two distinct codewords x
and y ∈ C and any integer r
λx,y (r ) ≤ λc .
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Optical Orthogonal Codes
Introduction
When λa = λc = λ, a (v , k, λa , λc ) OOC is simply refer to a (v , k, λ)
OOC.
The numbers λa and λc are called the auto- and cross-correlation
constraints.
The (0, 1) sequences of an optical orthogonal code are called its
codewords.
The size of an optical orthogonal code, denoted by |C|, is the number
of codewords in it.
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2011 June
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Optical Orthogonal Codes
Introduction
Background
Auto- and cross-correlations are of interest in optical communication
systems where the 1’s and 0’s in a sequence correspond to the
presence or absence of pulses of transmitted light.
Sequences with low Hamming cross-correlation are also of interest in
frequency-hopping multiple-access communication systems.
Here each user’s signal hops over the entire transmission bandwidth in
a pseudo-random fashion, and it is desired that collisions,
simultaneous transmissions over the same frequency sub-band
between the signals of different users, are minimized.
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2011 June
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Optical Orthogonal Codes
Introduction
Set-Theoretical Perspective
The set-theoretical perspective offers a convenient notation for OOCs
when k is much smaller than v . Let X = supp(x) for each codeword x.
Then the correlation properties for codewords X and Y , X 6= Y , can be
reformulated as follows:
(The Auto-Correlation Property) |X ∩ (r ⊕ X )| ≤ λa , r 6≡ 0 (mod v );
(The Cross-Correlation Property) |X ∩ (r ⊕ Y )| ≤ λc ,
where r ⊕ X := {r ⊕ x : x ∈ X } and all integers under consideration are
taken in Zv (i.e. modulo v ).
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2011 June
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Optical Orthogonal Codes
Introduction
From a practical point of view, a code with a large size is required. To find
the best possible codes, we need to determine an upper bound on the size
of an OOC with the given parameters.
Let M(v , k, λa , λc ) be the largest possible size of a (v , k, λa , λc ) OOC. An
OOC achieving this maximum size is said to be optimal. The
determination of the exact values of M(v , k, λa , λc ) and the specific
construction of optimal codes are of interest.
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Combinatorial Coding Theory
2011 June
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Optical Orthogonal Codes
Introduction
Johnson Bound
An upper bound of M(v , k, λ) can be derived from the size of
constant-weight codes.
Theorem
M(v , k, λ) ≤
1
1 v −1 v −2
v −λ
A(v , 2k − 2λ, k) ≤ b b
b
b· · · b
c · · · cccc.
v
k k −1 k −2
k −λ
The codes achieving this bound are optimal.
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2011 June
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Optical Orthogonal Codes
Introduction
Proof.
For any (v , k, λ) optical orthogonal code C, let C 0 be the error-correcting
code consisting of all cyclic shifts of codewords of C. Since for every
codeword of C, its v cyclic shifts are all distinct, we have |C 0 | = v |C|.
Every v -tuple in C 0 has Hamming weight k. Furthermore, for any two
members of C 0 , there are at most λ bit positions where they both have a
1. Therefore, C 0 has minimum distance at least 2k − 2λ, and we have
|C 0 | ≤ A(v , 2k − 2λ, k).
Since |C 0 | = v |C|, this implies the theorem.
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Combinatorial Coding Theory
2011 June
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Optical Orthogonal Codes
Introduction
Some Facts
There are also some other bounds.
Theorem
n j
ko
−λ
M(n, w , λ) ≤ max 1, ww2 −nλ
, w 2 > nλ.
Lemma
Given an (v , k, λa , λc ) code C, we can use it as an (v , k, λ0a , λ0c ) code with
λ0a ≥ λa and λ0c ≥ λc .
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2011 June
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Optical Orthogonal Codes
Introduction
Lemma
Given an (v , k, λa , λc ) code C, we can construct a (tv , tk, tk, tλc ) code C 0
with the same number of codewords.
Proof.
For each codeword x of C, construct a codeword z of C 0 by concatenating
t copies of x.
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Combinatorial Coding Theory
2011 June
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Optical Orthogonal Codes
Introduction
Lemma
Given an (v , k, λa , λc ) code C with M codewords, there
is an
(v , 2k − 2λc , 2λa + 2λc , k + 3λc ) code C 0 with M
codewords.
2
Proof.
C 0 = {x ∨ y : x, y ∈ C, x 6= y}, where ∨ is the bit-wise XOR operation.
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Combinatorial Coding Theory
2011 June
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Optical Orthogonal Codes
4
Basic Results
Optical Orthogonal Codes
Introduction
Basic Results
Cyclic Designs
Preliminary
Constructions via Skew Starters
Constructions via Weil’s Theorem
Recursive Constructions
Two-Dimensional OOC
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Optical Orthogonal Codes
Basic Results
Research on OOCs has mainly concentrated on the case where
λa = λc = λ in the literature.
Theorem (Brickell and Wei8 )
An optimal (v , 3, 1)-OOC exists if and only if v 6= 6t + 2 with t ≡ 2 or 3
(mod 4).
8
E. F. Brickell and V. K. Wei. Optical orthogonal codes and cyclic block designs. In
Eighteenth Southeastern International Conference on Combinatorics, Graph Theory, and
Computing (Boca Raton, Fla., 1987), volume 58, pages 175–192, 1987.
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2011 June
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Optical Orthogonal Codes
Basic Results
k ≥4
When k ≥ 4, although there are some partial results, the existence
problem for an optimal (v , 4, 1)-OOC is far from settled.
Let n = p1a1 . . . psas be the prime factorization of n. We use the notation
n ∈ Pr (m) if pi ≡ r (mod m) for i, 1 ≤ i ≤ s. If r = 1, we simply write
P(m).
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2011 June
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Optical Orthogonal Codes
Basic Results
Results on (v , 4, 1)-OOCs
There exists an optimal (kn, 4, 1)-OOC for
k ∈ {12} and n ∈ P(4);
k ∈ {2, 22, 34, 46, 50, 58} and n ∈ P(6);
k ∈ {3, 21, 33, 51, 57, 63, 69} and n ∈ P5 (8);
k ∈ {1, 4, 5, 7, 11, 20, 28, 40, 44, 52, 64, 280} and n ∈ P(12);
k ∈ {6, 30, 54, 78, 174, 222} and n ∈ P7 (12) or n ∈ P11 (12);
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Optical Orthogonal Codes
Basic Results
The only complete congruence classes of v for which the existence of an
optimal (v , 4, 1)-OOC was solved are v ≡ 6 (mod 12) and v ≡ 0
(mod 24).
Lemma (Ge and Yin9 )
There exists an optimal (v , 4, 1)-OOC for all positive integers v ≡ 6
(mod 12) or v ≡ 24 (mod 48).
Lemma (Chang, Fuji-Hara, and Miao10 )
There exists an optimal (v , 4, 1)-OOC for every positive integer v ≡ 0
(mod 24).
9
G. Ge and J. Yin. Constructions for optimal (v, 4, 1) optical orthogonal codes.
IEEE Trans. Inform. Theory, 47(7):2998–3004, 2001.
10
Y. Chang, R. Fuji-Hara, and Y. Miao. Combinatorial constructions of optimal
optical orthogonal codes with weight 4. IEEE Trans. Inform. Theory, 49 (5):1283–1292,
2003.
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2011 June
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Optical Orthogonal Codes
Basic Results
Results on (v , 5, 1)-OOCs
Let p be a prime. There exists an optimal (gp, 5, 1)-OOC for
g = 1 and p ≡ 1 (mod 20);
g = 1 and p = 10m + 1 where m > 1 is a prime;
g = 4 and p ≡ 1 (mod 10);
g = 5 and p ≡ 1 (mod 4);
g ∈ {8, 12} and p ≡ 11 (mod 20);
g ∈ {60, 80, 100, 120, 140, 160, 180} and v is a product of primes
greater than 5.
See Ma and Chang11 .
11
S. Ma and Y. Chang. Constructions of optimal optical orthogonal codes with weight
five. J. Combin. Des., 13(1):54–69, 2005.
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2011 June
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Optical Orthogonal Codes
Basic Results
Result on (v , 4, 2)-OOCs
There exists an optimal (v , 4, 2)-OOC for
v = 2n for any integer n ≥ 3;
not any v ≡ 0 (mod 24);
v = st for s ∈ {4n − 1 : n ≥ 1} ∪ {1, 27, 33, 39, 51, 87} and t is an
integer taken from set P = {p ≡ 7
(mod 12) : p is a prime} ∪ {2n − 1 : n ≡ 1
(mod 2), n > 1} ∪ {25, 37, 61, 73}, or a product of such integers;
v = 5a n for any integer a ≥ 0 and n ∈ {14, 18, 20, 42};
v = 2a+3 5b 7c or 2a+3 5b 11c for any a, b, c ≥ 0;
v = 4n for n = p1r1 . . . psrs where pi = 13 or pi ≡ 5 (mod 12) and
pi < 1500000.
See Feng, Chang, and Ji.12
12
T. Feng, Y. Chang, and L. Ji. Constructions for strictly cyclic 3-designs and
applications to optimal OOCs with λ = 2. J. Combin. Theory Ser. A., 115:1527–1551,
2008.
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Combinatorial Coding Theory
2011 June
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Optical Orthogonal Codes
4
Cyclic Designs
Optical Orthogonal Codes
Introduction
Basic Results
Cyclic Designs
Preliminary
Constructions via Skew Starters
Constructions via Weil’s Theorem
Recursive Constructions
Two-Dimensional OOC
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2011 June
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Optical Orthogonal Codes
Cyclic Designs
There are several methods for constructing OOCs that can be categorized
into two cases:
direct methods which use the mathematical structures such as
projective geometry, finite field theory and design theory;
search methods which use the computer algorithms such as greedy
and accelerated greedy algorithms and outer-product matrix
algorithm.
Optimal orthogonal codes are closely related to combinatorial
configurations.
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2011 June
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Optical Orthogonal Codes
Cyclic Designs
Packing Designs
Definition
A (v , k, λ) packing, denoted by P(k, λ; v ), is a pair (X , B) where X is a
v -set (of points), and B is a collection of k-subsets (called blocks) of X
such that any pair of distinct points from X occurs in at most λ blocks.
For some values of v the blocks of such a packing may contain every
2-subset of X exactly λ times, then it is just a BIBD(v , k, λ).
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Optical Orthogonal Codes
Cyclic Designs
Automorphism
Consider a packing P(k, 1; v ) (X , B). Let h be a permutation on X . For
any block B = {b1 , . . . , bk }, define h(B) = {h(b1 ), . . . , h(bk )}. If
h(B) = {h(B) : B ∈ B} = B, then h is called an automorphism of the
packing P(k, 1; v ).
Any automorphism h partitions B into equivalence classes called the orbits
of B under h. An arbitrary set of representatives for these orbits of B is
called the base blocks of the packing.
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Optical Orthogonal Codes
Cyclic Designs
Cyclic Packing
A P(k, 1; v ) is said to be cyclic if it admits an automorphism consisting of
a single cycle of length v .
We use CP(k, 1; v ) to denote a cyclic P(k, 1; v ) in which each of its block
orbits under the automorphism contains exactly v distinct blocks.
When the packing is a BIBD, it is also denoted by CB.
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Optical Orthogonal Codes
Cyclic Designs
As the terminology suggests, if (X , B) is a CP(k, 1; v ), then the point set
X can be identified with Zv , the residue ring of integers modulo v . The
cyclic automorphism is then just the bijection mapping i to i + 1 modulo v .
A CP(k, 1; v ) is uniquely determined by its base blocks. Given an arbitrary
set of base blocks of a CP(k, 1; v ), one can obtain the packing by
successively adding 1 to each base block modulo v .
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Optical Orthogonal Codes
Cyclic Designs
Difference Family
A convenient way of viewing a CP(k, 1; v ) is from the difference family
perspective.
CP(k, 1; v ) can be defined equivalently as a family B = {B1 , B2 , . . . , Bt }
of t k-subsets (called base blocks) of Zv such that the differences
resulting from B
∆B = {a − b : a, b ∈ B, a 6= b, B ∈ B}
cover each element in Zv \ {0} at most once.
k
j
v −1
v −1
. When t = b k(k−1)
c, the CP(k, 1; v ) is
Obviously, we have t ≤ k(k−1)
called optimal.
The difference leave of B, denoted by DL(B), is defined to be the set of all
nonzero integers in Zv which are not covered by differences in B.
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Optical Orthogonal Codes
Cyclic Designs
Equivalence
Theorem (Yin13 )
An optimal (v , k, 1) OOC is equivalent to an optimal CP(k, 1; v ).
Proof
Given an optimal CP(k, 1; v ), (Zv , B), we can take an arbitrary set of base
blocks and then construct a (0, 1)-sequence of length v from each base
block whose nonzero bit positions are exactly indexed by the block. It is
easy to see that the derived (0, 1)-sequences constitute an optimal
(v , k, 1) OOC.
13
J. Yin. Some combinatorial constructions for optical orthogonal codes. Discrete
Math., 185 (1-3):201–219, 1998.
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2011 June
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Optical Orthogonal Codes
Cyclic Designs
Proof (cont.)
Conversely, let C be an optimal (v , k, 1) OOC. For each of binary
sequences in C, we construct a k-subset of Zv , by taking
j thekindex set of
v −1
k-subsets of
its nonzero bit positions. This creates a family F of k(k−1)
Zv . The correlation properties of the optical orthogonal code guarantee
that |(r + X ) ∩ Y | ≤ 1 for any integer r in Z, and any X , Y in F except
when X = Y and r ≡ 0 (mod v ). Thus, we can take F as a set of base
blocks to form an optimal CP(k, 1; v ) with the automorphism i → i + 1
(mod v ).
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2011 June
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Optical Orthogonal Codes
Cyclic Designs
Regular Cyclic Packing
Definition
A CP(k, 1; v ) is termed g -regular if the subset Zv \ ∆B forms an additive
subgroup of Zv with order g .
It is important to observe that a g -regular CP(k, 1; v ) is not necessarily
optimal by definition. The cyclic constraint of a g -regular CP(k, 1; v ) gives
rise to the following two results.
Lemma
A necessary condition for the existence of a g -regular CP(k, 1; v ) is
v ≡ g (mod k(k − 1)).
If 1 ≤ g ≤ k(k − 1), then a g -regular CP(k, 1; v ) is optimal.
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2011 June
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Optical Orthogonal Codes
Cyclic Designs
Generalization
Let X be a v -set (points) and B be some k-subsets of X (blocks). We
call (X , B) a t-(v , k, λ) packing if every t-subsets of X is contained in
almost λ blocks of B.
It is called cyclic if it admits a cyclic automorphism group of order v that
is spanned by a v -cycle. If a block orbit has v distinct blocks then this
block orbit is full, otherwise it is short.
A cyclic t-(v , k, λ) packing (Zv , B) having only full block orbits is denoted
by t-CP(k, λ; v ). If t = 2, the parameter t is often omitted.
A cyclic packing with maximum possible number of blocks is optimal.
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2011 June
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Optical Orthogonal Codes
Cyclic Designs
Theorem (Fuji-Hara and Miao14 )
If λ < k, a cyclic (λ + 1)-(v , k, 1) packing of b full orbits is equivalent to a
(v , k, λ)-OOC of size b.
We will overview some constructions of cyclic packings.
14
R. Fuji-Hara and Y. Miao. Optical orthogonal codes: their bounds and new optimal
constructions. IEEE Trans. Inform. Theory, 46(7):2396–2406, 2000.
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Combinatorial Coding Theory
2011 June
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Optical Orthogonal Codes
Cyclic Designs
Constructions via Skew Starter
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2011 June
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Optical Orthogonal Codes
Cyclic Designs
Skew Starter
Let G be an additive group of order v > 1. A skew starter in G is a set
of unordered pairs S = {{xi , yi } : 1 ≤ i ≤ (v − 1)/2} which satisfies the
following three properties:
{xi : 1 ≤ i ≤ (v − 1)/2} ∪ {yi : 1 ≤ i ≤ (v − 1)/2} = G \{0}.
{±(xi − yi ) : 1 ≤ i ≤ (v − 1)/2} = G \{0}.
{±(xi + yi ) : 1 ≤ i ≤ (v − 1)/2} = G \{0}.
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Optical Orthogonal Codes
Cyclic Designs
According to the definition, a skew starter in G can exists only if v is odd.
Furthermore, if we write X = {xi : 1 ≤ i ≤ (v − 1)/2} and
Y = {yi : 1 ≤ i ≤ (v − 1)/2}, then we may assume that X = −Y and
hence we have X ∪ (−X ) = Y ∪ (−Y ) = X ∪ Y = G \{0}.
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2011 June
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Optical Orthogonal Codes
Cyclic Designs
Results on Skew Starters
Theorem (Chen, Ge, and Zhu15 )
There exists a skew starter in Zv for all v such that gcd(v , 150) = 1 or 25.
There does not exist any skew starter in Zv of v ≡ 0 (mod 3).
15
K. Chen, G. Ge, and L. Zhu. Starters and related codes. J. Statist. Plann.
Inference, 86:379395, 2000.
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Optical Orthogonal Codes
Cyclic Designs
Lemma
Let v be an odd positive integer and gcd(v , 3) = 1. If a skew starter in Zv
exists, then so does an 18-regular CP(4, 1; 18v ).
Proof
For the stated value of v , where v = 2u + 1. By assumption, let S be a
skew starter in Zv . Now let B consist of the following 3u 4-subsets of
Zv × Z18 :
{(x, 0), (y , 0), (x + y , 7), (0, 8)},
{(0, 0), (x + y , 1), (x, 13), (y , 15)},
{(0, 0), (−x − y , 9), (−x, 3), (−y , 5)},
where {x, y } run over all pairs in S.
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2011 June
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Optical Orthogonal Codes
Cyclic Designs
Proof (details I)
Since (v , 6) = 1, the residue ring of integers modulo 18v is isomorphic to
Zv × Z18 . It is readily calculated in the additive group of Zv × Z18 that
the differences in B, namely ∆B, can be partitioned into the following ten
classes depending on the value in the second coordinate:
(±(x − y ), 0),
{x, y } ∈ S;
(±(x + y ), 9),
{x, y } ∈ S;
(±(x + y ), ±1),
{x, y } ∈ S;
(±(x − y ), ±2),
{x, y } ∈ S;
(−x, 3), (−y , 3), (x, −3), (y , −3),
(x, ±4), (−x, ±4),
{x, y } ∈ S;
(−x, 5), (−y , 5), (x, −5), (y , −5),
(y , ±6), (−y , ±6),
{x, y } ∈ S;
(x, 7), (y , 7), (−x, −7), (−y , −7),
(−x, 8), (−y , 8), (x, −8), (y , −8),
Gennian Ge (Zhejiang University)
{x, y } ∈ S;
{x, y } ∈ S;
{x, y } ∈ S;
{x, y } ∈ S.
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Proof (details II)
Noticing that S is a skew starter in Zv , one can easily see that each
element of Zv × Z18 occurs in ∆B at most once. Further, we have
Zv × Z18 \∆B = {0} × Z18 , which forms a subgroup of the additive group
of Zv × Z18 . Therefore, B is a 18-regular CP(4, 1; 18v ).
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2011 June
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Optical Orthogonal Codes
Cyclic Designs
Lemma
Let v be an odd positive integer and gcd(v , 3) = 1. Then so does a
24-regular CP(4, 1; 24v ) exists provided that a skew starter in Zv exists.
Proof.
The construction is similar to that in the former Lemma. We make use of
ring Zv × Z24 which is isomorphic to Z24v , since gcd(24, v ) = 1 under the
assumptions. The required 2(v − 1) base blocks over Zv × Z24 are:
{(x + y , 0), (x, 3), (0, 4), (y , 9)},
{(x + y , 0), (x, 21), (0, 16), (y , 15)},
{(x, 0), (y , 0), (0, 7), (x + y , 11)},
{(−x, 0), (y , 12), (x, 2), (−y , 10)},
where {x, y } run over all pairs of a skew starter in Zv .
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Lemma
Let v be an odd positive integer and gcd(v , 3) = 1. Then so does a
72-regular CP(4, 1; 72v ) exists if a skew starter in Zv exists.
Proof.
This design is obtained by taking the following 6(v − 1) base blocks over
Zv × Z72 are:
{(−x, 0), (y , 36), (x, 10), (−y , 26)},
{(0, 5), (x, 0), (y , 0), (x + y , 21)},
{(δx, 0), (−δx, 1), (δy , 3), (−δy , 7)},
{(δx, 0), (−δx, 8), (δy , 22), (−δy , 53)},
{(δx, 0), (−δx, 9), (δy , 32), (−δy , 43)},
{(δx, 0), (−δx, 12), (δy , 30), (−δy , 47)},
{(δx, 0), (−δx, 13), (δy , 28), (−δy , 52)},
where δ ∈ {1, −1} and {x, y } run over all pairs of a skew starter in Zv .
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Constructions via Weil’s Theorem
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Definitions
A multiplicative character of a finite field GF(q) is a homomorphism
from the multiplicative group of GF(q) into the multiplicative group of
complex numbers of absolute value 1, i.e., a map
χ : GF(q)∗ → {z ∈ C : |z| = 1}
with the property that χ(xy ) = χ(x)χ(y ) for any (x, y ) ∈ GF(q) × GF(q).
In addition, we define χ(0) = 0 for any non-trivial multiplicative character
χ of GF(q).
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Weil’s Theorem
Theorem
Let ψ be a multiplicative character of GF(q) of order m > 1 and let
f ∈ GF (q)[x] be a monic polynomial of positive degree that is not an m-th
power of a polynomial. Let d be the number of distinct roots of f in its
splitting field over GF(q), then for every a ∈ GF (q), we have
X
√
ψ(a · f (c)) ≤ (d − 1) q
c∈GF (q)
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Fixed a prime p ≡ 1 (mod n) and a primitive element w ∈ Zp , C0n will
denote the multiplicative subgroup {w in : 0 ≤ i < (p − 1)/n} of the n-th
powers modulo p, while Cjn will denote the coset of C0n in Z∗p (= C01 )
represented by w j , i.e., Cjn = w j · C0n .
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As an application of Weil’s theorem, we obtain
Lemma
Let p be a prime ≡ 1 (mod q) with
√
p − [4(q − 1)5 + 15(q − 1)4 + 20(q − 1)3 + 10(q − 1)2 ] p − 5q 4 > 0.
Then, for any given 5-tuple (j1 , j2 , j3 , j4 , j5 ) ∈ {0, 1, . . . , q − 1}5 and any
given 5-tuple (c1 , c2 , c3 , c4 , c5 ) of pairwise distinct elements of Zp , there
exists an element x ∈ Zp such that x + ci ∈ Cjqi for each i.
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Lemma
Let p ≡ 1 (mod 10) be a prime, and 2 ∈ Ci50 . If there exists a pair (x, y )
satisfying the following properties:
x, x − 1 ∈ C15 , x − 2 ∈ C25 , and (x − 3)/2, 2x − 3 ∈ C35 ;
5 , y , y + x, y − (x − 3)/2, and y + 3(x − 1)/2 belong
y + x/2 ∈ C4−i
0
5 , C 5 , C 5 , C 5 }, where all subscripts
to distinct classes of {C2−i
3−i0
4−i0
5−i0
0
are taken modulo 5,
then there exists a 4-regular CP(5, 1; 4p).
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Proof.
Note that −1 ∈ C05 . Hence, {1, −1} is a subgroup of C05 . By the
hypothesis, there exists a pair (x, y ) satisfying the properties (1)-(2). The
desired 4-regular CP(5, 1; 4p) can be constructed by taking the following
(p − 1)/5 base blocks based on the additive group of Zp × Z4 ,
{00 , 10 , (x − 1)0 , (x/2)1 , (1/2)3 )} · r
{00 , ((x − 3)/2)0 , y2 , (−x − y )2 , (−x/2)1 )} · r
where r runs over all representative of C05 /{1, −1}. It is readily checked
that the list of differences arising from these base blocks does cover each
element of (Zp × Z4 )\({0} × Z4 ) exactly once, while any element of the
additive subgroup {0} × Z4 is not covered.
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Lemma
Let p be a prime p ≡ 1 (mod 10) and p > 8.7916 × 107 , then there exists
a pair (x, y ) satisfying the two conditions in the previous lemma.
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Proof.
Since 0, −1, −2, −3/2, −3 are distinct elements in Zp (if p ≥ 11), by
Lemma 36 with q = 5, an element x satisfying the first condition (1)
always exists in Zp for any prime p ≡ 1 (mod 10) and p ≥ 8.7916 × 107 .
Obviously, x 6= 0, 1, 2, 3/2, 3. Once the element x ∈ Zp has been
determined, we can again apply Lemma 36 to obtain the required element
y satisfying condition (2) for any prime p ≡ 1 (mod 10) and
p ≥ 8.7916 × 107 .
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Combining the previous two lemmas, we get the following result.
Theorem
A 4-regular CP(5, 1; 4p) exists for any prime p ≡ 1 (mod 10) and
p > 8.7916 × 107 .
Remark
For primes p ≡ 1 (mod 10) and p < 8.7916 × 107 , a computer search
gives a 4-regular CP(5, 1; 4p) except for p = 11.
Actually, the existence of 4-regular CP(5, 1; 4p) is guaranteed by a
certain element in a finite field of order p.
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Recursive Constructions
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Construction
Suppose that both a g -regular CP(k, 1; v ) and an optimal CP(k, 1; g )
exist. Then an optimal CP(k, 1; v ) also exists. Moreover, if the given
CP(k, 1; g ) is r -regular, then so is the derived CP(k, 1; v ).
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Proof.
Let F be the family of base blocks of the given g -regular CP(k, 1; v ). We
then have DL(F) ∪ {0} = {0, u, 2u, . . . , (g − 1)u} where u = v /g .
To complete an optimal CP(k, 1; v ), we assume that B is the family of
base blocks of an optimal CP(k, 1; g ). For each
B = {b0 , b1 , . . . , bk−1 } ∈ B, we take one base block
uB = {ub0 , ub1 , . . . , ubk−1 } (mod v ).
Since the existence of a g -regular CP(k, 1; v ) implies v ≡ g
(mod k(k − 1)), the difference leaves of an optimal CP(k, 1; v ) and an
optimal CP(k, 1; g ) have the same cardinality. So, the family
F ∪ {uB : B ∈ B} forms the desired optimal CP(k, 1; v ). The second part
is straightforward according to the above construction. The proof then is
complete.
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Difference Matrix
A difference matrix (DM) (k, m)-DM is a k × m matrix D = (dij )
(0 ≤ i ≤ k − 1, 0 ≤ j ≤ m − 1) whose each entry is from an abelian group
G such that for any two distinct rows s and t, the list
{dsj · dtj−1 : j = 0, 1, . . . , m − 1}
contains each element of G exactly once.
If G = Zm , such a difference matrix is said to be cyclic. We denote it
(k, m)-CDM.
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Difference matrices have been investigated extensively.
Lemma (Colbourn and Colbourn16 )
Let v and k be positive integers such that gcd(v , (k − 1)!) = 1. Let
dij = ij (mod v ) for i = 0, 1, . . . , k − 1 and j = 0, 1, . . . , v − 1. Then
D = (dij ) is a (k, v )-CDM. In particular, if v is an odd prime number, then
there exists a (k, v )-CDM for any integer k (≤ v ).
Lemma (Evans17 )
There is a (5, m)-CDM if and only if v is odd and v 6= 3, 9, except possibly
v = 9p where p is an prime and p 6∈ {5, 7, 11, 13, 17, 23, 29, 31, 109}.
16
M. J. Colbourn and C. J. Colbourn. Recursive constructions for cyclic block designs.
J. Statist. Plan. Inference, 10:97–103, 1984.
17
A. Evans. On orthogonal orthomorphisms of cyclic and non-abelian groups. II. J.
Combin. Des. 15(3):195–209, 2008.
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Construction
Suppose that there exist:
a g -regular CP(k, 1; v );
a (k, m)-DM; and
an optimal CP(k, 1; gm).
Then there exists an optimal CP(k, 1; mv ).
Moreover, if the given CP(k, 1; gm) is r -regular, then so is the derived
CP(k, 1; mv ).
Proof
It suffices to construct a gm-regular CP(k, 1; mv ) which proceeds as
follows.
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Proof. (cont.)
Suppose as before F is the family of base blocks of the given g -regular
CP(k, 1; v ) whose difference leave plus zero consists of the additive
subgroup {0, u, 2u, . . . , (g − 1)u} of Zv , where u = v /g . Let D = (dij ) be
a (k, m)-DM where dij ∈ Zm for 0 ≤ i ≤ k − 1 and 0 ≤ j ≤ m − 1.
Now the desired gm-regular CP(k, 1; mv ) will be based on Zmv , whose
difference leave plus zero forms the subgroup
H = {0, u, 2u, . . . , (g − 1)u, . . . , (mg − 1)u} of Zmv . The required base
blocks can be constructed in the following way.
For each base block B = {b0 , b1 , . . . , bk−1 } ∈ F, we take m base blocks
Bj = {bi + vdij : 0 ≤ i ≤ k − 1} for j = 0, 1, . . . , m − 1, where the additive
operation is performed in Zmv . We need only to verify that the differences
arising from these base blocks cover each integer in Zmv \H exactly once.
It is a routine matter of checking and omitted.
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Group Divisible Design
A group divisible design (GDD) is a triple (X , G, B) which satisfies the
following properties:
G is a partition of a set X (of points) into subsets called groups;
B is a set of subsets of X (called blocks) such that a group and a
block contain at most one common point;
every pair of points from distinct groups occurs in a unique block.
We use GD(k, 1, n; nv ) to denote a k-GDD of type nv .
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Given positive integers n and v , let Iv = {0, 1, 2, . . . , v − 1} and
X = Iv × Zn . The points of X are denoted by (a, b).
Definition
A GD(k, 1, n; nv ) based on point set X having group set
G = {{i} × Zn : i ∈ Iv } and block set B is said to be n-cyclic, if for any
B ∈ B, adding 1 successively to the second component of each point of B
modulo n always gives n distinct blocks of B.
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According to the definition, it is easy to see that an n-regular CP(k, 1; nv )
is also an n-cyclic GD(k, 1, n; nv ).
Lemma
Suppose that both a g -regular CP(h, 1; v ) and an n-cyclic GD(k, 1, n; nh)
exist. Then there exists an ng -regular CP(k, 1; nv ). Moreover, if an
optimal CP(k, 1; ng ) exists, then an optimal CP(k, 1; nv ) also exists.
Lemma
Suppose that both a CB(h, 1; v ) with v ≡ 1 (mod h(h − 1)) and an
n-cyclic GD(k, 1, n; nh) exist. Then there exists an n-regular CP(k, 1; nv ).
Moreover, if an optimal CP(k, 1; n) exists, then an optimal CP(k, 1; nv )
also exists.
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4
Two-Dimensional OOC
Optical Orthogonal Codes
Introduction
Basic Results
Cyclic Designs
Preliminary
Constructions via Skew Starters
Constructions via Weil’s Theorem
Recursive Constructions
Two-Dimensional OOC
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Two-Dimensional OOC
Definition
A two-dimensional (u × v , k, λa , λc ) optical orthogonal code C is a family
of u × v (0, 1)-matrices (called codewords) of Hamming weight k
satisfying:
(The auto-correlation property) for any matrix A = (aij )u×v ∈ C and
any integer r 6≡ 0 (mod v ),
u−1 X
v −1
X
aij ai,j⊕r ≤ λa ;
i=0 j=0
(The cross-correlation property) for any two distince matrices
A = (aij )u×v and B = (bij )u×v ∈ C and any integer r ,
u−1 X
v −1
X
aij bi,j⊕r ≤ λc .
i=0 j=0
Where ⊕ denotes the addition modulo v .
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Two-Dimensional OOC
A (u × v , k, λa , λc ) two-dimensional optical orthogonal code is
denoted by 2-D (u × v , k, λa , λc ) OOC.
When λa = λc = λ, a 2-D (u × v , k, λa , λc ) OOC is simply refer to a
2-D (u × v , k, λ) OOC.
Obviously a 2-D (1 × v , k, λa , λc ) OOC is just a (v , k, λa , λc ) OOC
(also called a 1-D (v , k, λa , λc ) OOC).
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The merit of 2-D OOCs
when the number of users or the weight of codes is increased, the
length of the sequences increases rapidly. A large code length causes
the chip rate of the O-CDMA system to exceed the maximum chip
rate currently attainable in practice.
By spreading in both wavelength and time, the chip rate can be
reduced considerably.
Technologies such as wavelength-division-multiplexing (WDM) and
dense-WDM have made it possible to spread codes in both
wavelength and time.
These codes are referred to wavelength-time hopping codes,
multiple-wavelength codes, two-dimensional optical orthogonal codes,
etc., which tend to require smaller code length and hence lower chip
rate.
Here we shall refer to these codes as two-dimensional optical orthogonal
codes.
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Two-Dimensional OOC
Strictly Cyclic Packing
A t-(u × v , k, λ) packing is said to be v -cyclic if it admits an
automorphism consisting of u cycles of length v .
Let G be an automorphism group of a t-(u × v , k, λ) packing. For any
block B of the packing design, the subgroup {π ∈ G : B π = B} is called
the stabilizer of B in G .
If the stabilizer of any block of a v -cyclic t-(u × v , k, 1) packing is trivial
in Zv , then the packing is called strictly v -cyclic, denoted as a
t-(u × v , k, λ)-SCP.
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Two-Dimensional OOC
The set X of a t-(u × v , k, λ)-SCP can be denoted as Iu × Zv , where Iu is
a set of size u and Zv is a cyclic group of size v . In fact, each {i} × Zv ,
where i ∈ Iu , is an orbit under the specific automorphism.
In a t-(u × v , k, λ)-SCP, we can always find a subset B0 ⊂ B, where
B0 = v1 |B|, such that B can be generated by using Zv on B0 . Blocks in B0
are called base blocks (or starter blocks).
We shall use the notation (−, +1 mod v ) to denote that each element
(x, y ) ∈ Iu × Zv of a base block are generated to (x, y + i mod v )
i = 1, 2, . . . , v − 1. In this way, we can obtain all the blocks from base
blocks B0 .
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Two-Dimensional OOC
Lemma (Cao and Wei18 )
An optimal 2-D (u × v , k, t − 1) OOC is equivalent to an optimal
t-(u × v , k, 1)-SCP, provided that t ≤ k holds.
Proof I
Conversely, if we have a (u × v , k, t − 1) OOC, then for each u × v
(0, 1)-matrix Xi whose rows are indexed by Iu and columns are indexed by
Zv , we can construct a base block Bi such that (n, m) ∈ Bi if and only if
Xi ’s (n, m) cell equals to 1. It is readily checked that these base blocks
form a t-(u × v , k, 1)-SCP.
18
H. Cao and R. Wei. Combinatorial constructions for optimal two-dimensional
optical orthogonal codes. IEEE Trans. Inform. Theory, 55:1387–1394, 2009.
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Proof II
Suppose (X , B) is a t-(u × v , k, 1)-SCP, where X = Iu × Zv . For each base
block B = {(a1 , b1 ), (a2 , b2 ), . . . , (ak , bk )}, construct a u × v (0, 1)-matrix
XB such that its (n, m) cell equals to 1 if and only if there is an i,
1 ≤ i ≤ k, such that (n, m) = (ai , bi ). Since any two blocks intersect at
most t − 1 points and all the blocks can be developed cyclically from the
base blocks, it follows that these matrices form a (u × v , k, t − 1) OOC.
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Two-Dimensional OOC
Results
Theorem (Wang, Shan, and Yin19 )
Let u be any positive integer and v be any odd integer. There exists an
optimal 2-D (u × v , 3, 1) OOC.
Let v ∈ {2, 4, 5, 8, 9}. There is an optimal 2-D (u × v , 4, 2) OOC for
any u ≡ 2, 4 (mod 6).
There exists an optimal 2-D (u × 3, 4, 2)-OOC for any u ≡ 0 (mod 2).
Let p ≡ 7 (mod 12) be a prime or
p ∈ {37, 61, 73, 109, 157, 181, 229, 277}. There exist an optimal 2-D
(u × p, 4, 2)-OOC for any u ≡ 2, 4 (mod 6) or u ≡ 0 (mod 12).
19
J. Wang, X. Shan, and J. Yin. On constructions for optimal two-dimensional optical
orthogonal codes. Des. Codes and Cryptogr., 54:43–60, 2010.
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Conflict-Avoiding Codes
5
Conflict-Avoiding Codes
Introduction
Results
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Conflict-Avoiding Codes
5
Introduction
Conflict-Avoiding Codes
Introduction
Results
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Introduction
Background
In a multiple-access channel without feedback model, the time axis is
partitioned into slots whose duration corresponds to the transmission time
for one packet and all users are supposed to have slot synchronization, but
no other synchronization is assumed.
If more than one user is sending packets in a particular slot simultaneously,
then there is a conflict and the channel output in that slot is the
unreadable collision symbol, called an erasure.
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Introduction
Protocol Sequences
If the binary protocol sequence xi = (xi,0 , xi,1 , · · · , xi,n−1 ) has Hamming
weight k, then user i sends k packets in each frame of n slots, where his or
her protocol sequence appears.
When a user i is sending a message by using a protocol sequence xi , a
different message from the other user may be sent by a protocol sequence
xj or its cyclic shift since only slot synchronization is assumed.
The set C = {x1 , · · · , xN } of N binary sequences is called an (N, u, n, σ)
protocol sequence set if any xi ∈ C is of length n and has the property
that at least σ successful packet transmissions in a frame are guaranteed
for each active user, provided that at most u users out of N potential
users are active.
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Introduction
On the assumption that the number of collisions of any two distinct
sequences is at most λ, in order to guarantee each user that at least σ
packets in a frame survive from collision, the weight k of the (N, u, n, σ)
protocol sequence set satisfies k ≥ σ + λ(u − 1).
If there is at least one packet that survives from collision, then there may
be a chance to use an inner code for erasure correction. Let l be the bit
length of each slot. An (n0 = kl, k 0 = σl, d 0 = kl − σl + 1) shortened
Reed-Solomon (RS) code over GF (q) can be used as a code for each user
to decode his or her σ survives packets into k transmitted packets.
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Conflict-Avoiding Codes
Introduction
Conflict-Avoiding Codes
In order to use an inner code, every protocol sequence of C should have
constant weight k. Such an (N, u, n, σ) protocol sequence set is also called
a conflict-avoiding code (CAC) of length n with weight k (CAC(n, k)).
Hence CAC can be viewed as an OOC without auto-correlation property.
The maximum size of some code in CAC(n, k) is denoted by M(n, k). A
code C ∈ CAC (n, k) is said to be optimal if |C | = M(n, k). The
advantage of using an optimal CAC is that it enables the largest number
of potential users to transmit packets efficiently and reliably in such a
multiple-access channel model.
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Introduction
Set Notations
Let P(n, k) denote the set of all k-subsets of Z = Z/nZ. Each element
x ∈ P(n, k) can be identified with a binary sequence in {0, 1}n of
Hamming weight k representing the indices of the nonzero positions.
Given a k-subset x ∈ P(n, k), we define the difference set of x by
∆(x) = {j − i : i, j ∈ x, i 6= j}.
Note that ∆(x) contains at most k(k − 1) differences. Furthermore,
i ∈ ∆(x) implies (n − i) ∈ ∆(x), i.e., ∆(x) is symmetric with respect to
n/2.
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Introduction
In a set notation, for the case of λ = 1 a CAC of length n with weight k is
a subset C ⊂ P(n, k) satisfying the condition
∆(x) ∩ ∆(y ) = ∅
for any x, y ∈ C with x 6= y .
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Introduction
A codeword x is called equi-difference if the elements in x form an
arithmetic progression, i.e.,
x = {a, a + δ, a + 2δ, · · · , a + (k − 1)δ}
for some a and δ in Z.
A CAC is said to be equi-difference if all codewords are
equi-difference.
We use the symbol CACe (n, k) for an equi-difference CAC of length n
and weight k.
We let M e (n, k) be the maximal number of codewords among such
CACs.
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5
Results
Conflict-Avoiding Codes
Introduction
Results
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Results
The spectrum of the M(n, k) for even length n and weight k = 3 is
completely settled. Unfortunately, the case of odd code lengths and weight
three is still very far from being solved.
Theorem (Levenshtein and Tonchev20 )
If n ≡ 2 (mod 4), then M(n, 3) =
n−2
4
20
V. I. Levenshtein and V. D. Tonchev. Optimal conflict-avoiding codes for three
active users. In Proc. IEEE Int. Symp. Inform. Theory, pages 535–537, Adelaide,
Australia, 2005.
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2011 June
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Conflict-Avoiding Codes
Results
Theorem (Mishima, Fu, and Uruno21 )
Let n = 16m, then


if m ≡ 0 (mod 2)
7n/32
M(n, 3) = (7n − 16)/32 if m ≡ 1, 5 (mod 6)


(7n + 16)/32 if m ≡ 3 (mod 6)
with the exceptions M(48, 3) = 10 and M(64, 3) = 13.
21
M. Mishima, H.-L. Fu, and S. Uruno. Optimal conflict- avoiding codes of length
n ≡ 0 (mod 16) and weight 3. Des. Codes Cryptogr., 52(3):275–291, 2009.
Gennian Ge (Zhejiang University)
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2011 June
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Conflict-Avoiding Codes
Results
Theorem (Fu, Lin, and Mishima22 )
Let n = 8m + 4. Then


(7n + 4)/32




(7n + 12)/32



(7n − 12)/32
M(n, 3) =

(7n − 4)/32





(7n − 20)/32



(7n + 20)/32
if
if
if
if
if
if
m ≡ 0 (mod 4)
m ≡ 1 (mod 12)
m ≡ 2, 6 (mod 12)
m ≡ 3 (mod 4)
m ≡ 5, 9 (mod 12)
m ≡ 10 (mod 12)
22
H. L. Fu, Y. H. Lin, and Mishima. Optimal conflict-avoiding codes of even length
and weight 3. IEEE Trans. Inform. Theory, 56(11):5747–5756, 2010.
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2011 June
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Conflict-Avoiding Codes
Results
Theorem
For all w ≥ 2, we have
lim sup
n→∞
Gennian Ge (Zhejiang University)
M e (n, w )
1
M(n, w )
= lim sup
=
n
n
2w
−2
n→∞
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2011 June
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Impulse Radio Sequences
6
Impulse Radio Sequences
Introduction
Bounds
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Combinatorial Coding Theory
2011 June
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Impulse Radio Sequences
6
Introduction
Impulse Radio Sequences
Introduction
Bounds
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Combinatorial Coding Theory
2011 June
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Impulse Radio Sequences
Introduction
Background
Impulse radio sequences (IRSs) were first introduced by Chu and
Colbourn. This class of sequences was formulated to study sequence or
signal design for ultrawideband (UWB) radio with unmodulated
time-hopping. UWB systems for wireless communication have recently
become an important area of research.
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Combinatorial Coding Theory
2011 June
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Impulse Radio Sequences
Introduction
Definition
An (m, k, λa , λc ) impulse radio sequence (or (m, k, λa , λc )-IRS in short) C
is a family of (0, 1) sequences of length km and weight k satisfying the
following three properties.
The pulse position property: For any x = {xi }km−1
i=0 ∈ C, the support
of x can be expressed as
supp(x) = {ai + im : i ∈ Zk , ai ∈ Zm }.
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Combinatorial Coding Theory
2011 June
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Impulse Radio Sequences
Introduction
The auto-correlation property: For every integer t, t 6≡ 0 (mod km)
λx (t) ≤ λa .
The cross-correlation property: For every integer t
λx,y (t) ≤ λc .
The subscripts are reduced modulo km whenever necessary. When
λa = λc = λ, the notation (m, k, λ)-IRS is used.
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Combinatorial Coding Theory
2011 June
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Impulse Radio Sequences
Introduction
Relation with OOC
The definition of impulse radio sequences shows its close relationship with
optical orthogonal codes.
Lemma
An (m, k, λa , λc )-IRS is a (km, k, λa , λc ) OOC.
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Combinatorial Coding Theory
2011 June
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Impulse Radio Sequences
6
Bounds
Impulse Radio Sequences
Introduction
Bounds
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Combinatorial Coding Theory
2011 June
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Impulse Radio Sequences
Bounds
Let Φ(m, k, λa , λc ) denote the maximal possible number of sequences in
an (m, k, λa , λc )-IRS. An IRS achieving that size is called optimal.
Since it is difficult to decide the exact value of Φ(m, k, λa , λc ), the upper
bounds are of interest.
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Combinatorial Coding Theory
2011 June
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Impulse Radio Sequences
Bounds
Johnson Bound
In the case λa = λc = λ, Φ(m, k, λa , λc ) is bounded by the Johnson
bound for constant-weight error-correcting codes.
Theorem
Φ(m, k, λ) ≤
Gennian Ge (Zhejiang University)
1 km − 1
km − λ
...
...
k k −1
k −λ
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2011 June
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Impulse Radio Sequences
Bounds
Other Bounds23
j
k
√
 1 (m + 3m2 − 4m + 2) ,
6
k
Φ(m, 3, 1) ≤ j
√
 1 (m − 1 + 3m2 − 6m + 1)
6
if m is odd
if m is even.
If k is even, then
k 2 m − 2km − 2k + 4m
Φ(m, k, 1) ≤
.
k 3 − 2k 2 + 2k
23
J. Gao and Y. Chang. New upper bounds for impulse radio sequences. IEEE Trans.
Inform. Theory, 52(5):2255–2260, 2006.
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Combinatorial Coding Theory
2011 June
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Impulse Radio Sequences
Bounds
If k and m are odd and k ≥ 5, then
$
%
p
(k − 1)m + (k 2 − 2k + 9)m2 + 16m + 8
Φ(m, k, 1) ≤
.
2k 2 − 2k
If k ≥ 5 is odd and m is even, then
$
√ %
km − m − 2 + ∆
Φ(m, k, 1) ≤
2k 2 − 2k
where ∆ = (k 2 − 2k + 9)m2 − (4k − 20)m + 4.
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2011 June
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Impulse Radio Sequences
Bounds
A General Bound24
Theorem
Let m, k, and λ be positive integers such that k > λ ≥ 2. Then
√
(k 2 + 2k − 6)m2 − (10k − 9)m + 9 + t
c
Φ(m, k, λ) ≤ bλ3
3k(k − 1)(k − 2)
where t = (4k 4 − 8k 3 − 8k 2 + 24k − 12)m4 + (4k 3 − 4k 2 + 12k −
12)m3 − (11k 2 + 18k − 21)m2 − (6k − 30)m + 9 and
λ3 = b
km − λ
km − 3 km − 4
b
b. . . b
c . . . ccc
k −1 k −4
k −λ
for λ ≥ 3, or λ3 = 1 if λ = 2.
24
H. Cao and R. Wei. Combinatorial constructions for optimal two-dimensional
optical orthogonal codes. IEEE Trans. Inform. Theory, 55(3):1387–1394, 2009.
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2011 June
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Impulse Radio Sequences
Bounds
Corollary
Let m and k ≥ 3 be positive integers. Then
(k 2 + 2k − 6)m2 − (10k − 9)m + 9 +
Φ(m, k, 2) ≤ b
3k(k − 1)(k − 2)
√
t
c.
where t is the same as above.
Corollary
Let m be positive integer. Then
9m2 − 21m + 9 +
Φ(m, 3, 2) ≤ b
18
√
t
c.
where t = 96m4 + 96m3 − 132m2 + 12m + 9.
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Combinatorial Coding Theory
2011 June
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Impulse Radio Sequences
Bounds
Remark
Use c(m, k, λ) to denote the bound in theorem, then for any given positive
integer k and λ where k > λ, we have
lim (J(m, k, λ) − c(m, k, λ)) = +∞.
m→∞
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Combinatorial Coding Theory
2011 June
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Part III
Comma-Free Codes
Frequency-Hopping Sequences
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Combinatorial Coding Theory
2011 June
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Comma-Free Codes
7
Comma-Free Codes
Introduction
Comma-Free Index
Difference System of Sets
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Combinatorial Coding Theory
2011 June
157 / 297
Comma-Free Codes
Gennian Ge (Zhejiang University)
Introduction
Combinatorial Coding Theory
2011 June
158 / 297
Comma-Free Codes
7
Introduction
Comma-Free Codes
Introduction
Comma-Free Index
Difference System of Sets
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Combinatorial Coding Theory
2011 June
158 / 297
Comma-Free Codes
Introduction
Background
We consider the process of transmitting data over a channel, where the
data being sent can be thought as a stream of symbols from a finite
alphabet F (e.g., F = {0, 1}):
. . . 10110110100111100 . . .
The data stream consists of consecutive messages, each message being a
sequence of n consecutive symbols:
. . . x1 . . . xn y1 . . . yn . . .
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Combinatorial Coding Theory
2011 June
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Comma-Free Codes
Introduction
The synchronization problem that arises at the receiving end is the task
to partition correctly the data stream into messages of length n. (As
opposed to conceiving incorrectly a sequence of n symbols being the
concatenation of the end of one message with the beginning of another
message as a single message:)
. . . xi xi+1 . . . xn y1 . . . yi yi+1 . . .
One way to resolve the synchronization problem is by requiring that the
collection of admissible messages has the property that no single message
coincides with a concatenation of the others.
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2011 June
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Comma-Free Codes
Introduction
Definitions
Let Fq be the set of words (sequences) of length n over the alphabet
Fq = {0, 1, . . . , q − 1}. For any x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ Fqn ,
and i = 1, . . . , n − 1, we set
Ti (x, y ) = xi+1 . . . xn y1 . . . yi
and call Ti (x, y ) a splice of x and y . In particular, Ti (x, x) is a cyclic shift
of x.
A code C ⊂ Fqn is called comma-free, if any splice of two codewords is
not a codeword. Let M(n, q) be the maximum size of a comma-free code
C ⊂ Fqn .
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2011 June
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Comma-Free Codes
Introduction
Theorem (Golomb, Gordon, and Welch25 )
1X
µ(d)q n/d
n
where the sum is taken over all divisors d of n and µ is the Möbius
function.
M(n, q) ≤ B(n, q) =
The proof is based on the fact that B(n, q) is the number of words x ∈ Fqn
of period n which are not pairwise equivalent with respect to their cyclic
shifts. By definition, a comma-free code can contain only one
representative of the equivalence class.
25
S. W. Golomb, B. Gordon, and L. R. Welch. Comma-free codes. Canad. J. Math.,
10:202–209, 1958.
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2011 June
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Comma-Free Codes
Introduction
Theorem (Eastman26 )
For any q ≥ 2 and any odd n, there is a comma-free code Eqn with size
B(n, q).
However, the bound B(n, q) is generally not tight for even n. Golomb,
Gordon, and Welch proved that M(2, q) = q 2 /3 whereas
B(2, q) = q(q − 1)/2. In fact, they showed that for any even n = 2l ,
there exists a minimal number m(l) such that M(n, q) < B(n, q) if
q > m(l), and that m(l) ≤ 3l + l. Jiggs proved that m(l) ≤ 2l + l.
26
W. L. Eastman. On the construction of comma-free codes. IEEE Trans. Inform.
Theory, IT-11:263–267, 1965.
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2011 June
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Comma-Free Codes
Introduction
The combinatorial problem that helped to prove the previous inequality
was refined allowing for better upper bounds on m(l).
n
We use the extended alphabet Fq = {0, 1, . . . , q − 1, ∗} and denote by Fq
the set of words (sequences) of length n over the alphabet Fq . We say that
a binary word x = x1 . . . xn ∈ F2n covers y = y1 . . . yn ∈ F2n if xi ≥ yi for
all i = 1, . . . , n, and if there exists a position where this inequality is strict.
n
This definition can be extended to words from F2 if the condition xi ≥ yi
is only applied to the positions i where xi 6= ∗ and yi =
6 ∗.
n
We call two words from F2n or F2 comparable if one of the words covers
the other.
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2011 June
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Comma-Free Codes
Introduction
It is easily seen that the maximum number of pairwise comparable words
of F2n equals n + 1; for instance, {000, 001, 011, 111} forms a maximal set
for n = 3.
We denote by t(n) the maximum size of a set of pairwise comparable
n
words of F2 . In particular, {000, 0 ∗ 1, ∗10, 10∗, 111} forms a maximal set
n
of pairwise comparable words of F2 and t(3) = 5.
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Combinatorial Coding Theory
2011 June
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Comma-Free Codes
Introduction
The following statement in fact shows that m(l) ≤ t(l) + l.
Theorem
For even n = 2l, M(n, q) < B(n, q) if q > t(l) + l.
Known results on t(l) are
t(l) < l
t(h2
log2 l
2
for l ≥ 8, and
+ h + 1) ≥ h(h2 + h + 1) + 2 for any h = 1, 2, . . . .
Problem
Whether there exists a constant c, c ≥ 3/2, such that t(l) = O(l c ) as
l → ∞.
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2011 June
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Comma-Free Codes
7
Comma-Free Index
Comma-Free Codes
Introduction
Comma-Free Index
Difference System of Sets
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Combinatorial Coding Theory
2011 June
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Comma-Free Codes
Comma-Free Index
Definitions
The notion of comma-free codes gives natural rise to a parameter ρ(C ) of
a code C ⊂ Fqn which is called the comma-free index or code
separation and is defined as
ρ(C ) = min d(Ti (x, y ), z)
where Ti (x, y ) is a splice. The minimum is taken over all x, y , z ∈ C , and
i = 1, . . . , n − 1.
If the comma-free index ρ(C ) is positive, it is possible to distinguish a
codeword from a concatenation of two codewords even in case that up to
b(ρ(C ) − 1)/2c errors have occurred in the given codeword.
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2011 June
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Comma-Free Codes
Comma-Free Index
n
We extend this definition to codes C ⊂ Fq by assuming that in finding
the Hamming distance between two words one does not take into account
the positions where letters ∗ occur.
n
If a word z ∈ Fq forms a code (of size 1) with a comma-free index of at
least ρ then we call z a comma-free sequence of index ρ or simply a
comma-free sequence if ρ = 1.
Let M(n, q, ρ) be the maximum cardinality of a code C ⊂ Fqn with
ρ(C ) = ρ.
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2011 June
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Comma-Free Codes
Comma-Free Index
Since a comma-free code is defined as a code C with ρ(C ) = 1, we have
M(n, q, 1) = M(n, q).
Theorem
For any fixed ρ ≥ 1,
M(n, 2, ρ) %
1 2n
c(ρ)e n
as n → ∞, where c(ρ) is a constant.
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Combinatorial Coding Theory
2011 June
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Comma-Free Codes
Comma-Free Index
Theorem
If q is a prime power and a coset of a linear [n, k]-code C has comma-free
index ρ, then
√
r = n − k ≥ ρn.
(3)
For constructing cosets of linear [n, k]-codes C ⊂ Fqn whose redundancy is
close to the bound, Levenshtein introduced and investigated the following
combinatorial notion.
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Combinatorial Coding Theory
2011 June
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Comma-Free Codes
7
Difference System of Sets
Comma-Free Codes
Introduction
Comma-Free Index
Difference System of Sets
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Combinatorial Coding Theory
2011 June
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Comma-Free Codes
Difference System of Sets
Definition
A collection Q of q disjoint subsets Qi of Nn = {0, 1, . . . , n − 1},
i = 0, 1, . . . , q − 1, is called a difference system of sets (DSS) of index ρ
if for each number s, s = 1, . . . , n − 1, the equation
x −y =s
(mod n)
(4)
has at least ρ solutions for x ∈ Qi , y ∈ Qj , i, j = 0, . . . , q − 1, i 6= j. We
use the notation DSS(n, τ0 , τ1 , . . . , τq−1 , ρ) for it, where τi = |Qi |.
It is worth to underline that x and y should belong to different subsets.
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2011 June
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Comma-Free Codes
Difference System of Sets
DSS and Comma-Free Sequence
For a collection Q of disjoint subsets Qi of Nn = {0, . . . , n − 1},
i = 0, . . . , q − 1 (in particular, for a DSS) consider a sequence
z(Q) = z0 z1 . . . zn−1 ∈ Fq
n
where
zi = ∗, if i 6∈
Sq−1
i=0
Qi , and
zi = j, if i ∈ Qj (i = 0, . . . , n − 1).
It is easily seen that such a collection Q is a DSS of index ρ if and only if
z(Q) is a comma-free sequence of index ρ.
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2011 June
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Comma-Free Codes
Difference System of Sets
Example
The collection Q of subsets Q0 = {0, 9}, Q1 = {1, 18}, Q2 = {3, 14} of
N25 forms a DSS of index 1. The corresponding sequence
z(Q) = 01 ∗ 2 ∗ ∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ ∗2 ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗ ∈ F3
25
is a comma-free sequence.
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2011 June
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Comma-Free Codes
Difference System of Sets
Optimal DSS
For a DSS Q of index ρ, consider a code C (Q) ⊂ Fqn of redundancy
| ∪q−1
i=0 Qi | whose information positions are in the places, where z(Q) has
∗, and the remaining positions equal 0.
From the definition of a DSS of index ρ, the shift of C (Q) ⊂ Fqn on the
vector obtained from z(Q) by replacement all ∗ for zeros gives a
comma-free code of index ρ.
For this reason we have | ∪q−1
i=0 Qi | being the redundancy of a DSS. We
denote by rq (n, ρ) the minimum redundancy of all DSS of index ρ with
parameters n and q.
Such a DSS with redundancy equal to rq (n, ρ) is referred to as optimal.
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Comma-Free Codes
Difference System of Sets
A DSS of index ρ is called perfect if for every number s,
1 ≤ s ≤ n − 1 equation 4 has exactly ρ required solutions.
A DSS is called regular if all subsets Qi are of the same size.
We use the notation DSS-(n, m, q, ρ) for a regular DSS of index ρ
with q subsets of size m of the set Nn ; its redundancy equals r = qm.
Any cyclic difference set (v , k, λ) is a perfect regular
DSS-(v , 1, q, ρ) with q = k and ρ = λ. Thus a DSS can be seen as a
generalization of cyclic difference sets.
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2011 June
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Comma-Free Codes
Difference System of Sets
Theorem (Levenshtein27 )
For any DSS with parameters n, q, and ρ,
s
qρ(n − 1)
rq (n, ρ) ≥
q−1
(5)
with equality if and only the DSS is perfect and regular.
It follows that any perfect regular DSS is optimal.
27
V. I. Levenshtein. One method of constructing quasilinear codes providing
synchronization in the presence of errors. Probl. Inf. Transm., 7:215–222, 1971.
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2011 June
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Comma-Free Codes
Difference System of Sets
DSS from Projective Geometry
Theorem (Fuji-Hara, Munemasa, and Tonchev
28
)
Let
H0 be a hyperplane of PG(2t + 1, q) containing the t-flat S;
W 0 = {F0 , . . . , Fh−1 } be the set of (t + 1)-flats of H0 containing S,
where h = (q t − 1)/(q − 1), and
set Ai = Fi \ S, v = (q 2t+1 − 1)/(q − 1),
ρ = q 2t−1 + q 2t−2 + · · · + q t+1 .
Then the partition {F0 , A1 , . . . , Ah−1 } of the hyperplane H0 is a DSS with
parameters (v , τ0 , . . . , τh , ρ), where τ0 = · · · = τh−1 = q t+1 ,
τh = (q t+1 − 1)/(q − 1).
28
R. Fuji-Hara, A. Munemasa, and V. D. Tonchev. Hyperplane partitions and
difference systems of sets. J. Combin. Theory Ser. A, 113 (8):1689–1698, 2006.
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2011 June
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Comma-Free Codes
Difference System of Sets
Example
Here we give an example in PG (5, 2). We take a primitive element a of
GF (26 ) with minimal polynomial a6 + a + 1 = 0, so that the point set is
represented by Z63 , and
S = {0, 9, 18, 27, 36, 45, 54}.
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2011 June
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Comma-Free Codes
Difference System of Sets
Example (cont.)
The following partition is a DSS with parameters (63, 15, 8, 8, ρ = 8),
F0 = {0, 1, 6, 8, 9, 14, 18, 27, 36, 38, 45, 48, 49, 52, 54},
A1 = {2, 12, 13, 16, 28, 33, 35, 41},
A2 = {3, 4, 7, 19, 24, 26, 32, 56},
while the following partition is a DSS with (63, 8, 8, 8, 7, ρ = 8):
A0 = {1, 6, 8, 14, 38, 48, 49, 52},
A1 = {2, 12, 13, 16, 28, 33, 35, 41},
A2 = {3, 4, 7, 19, 24, 26, 32, 56},
S = {0, 9, 18, 27, 36, 45, 54}.
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2011 June
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Comma-Free Codes
Difference System of Sets
DSS from Balanced Generalized Weighing Matrix
Let B0 , B1 , . . . , Bq−1 be disjoint subsets of Zn = {0, 1, . . . , n − 1}. We
define a circulant n × n array A having all entries in its first row that are
indexed by elements of Bi equal to i (0 ≤ i ≤ q − 1), and entries that are
not indexed by any α ∈ Bi for some i are left empty.
Lemma
The collection B = {Bi }q−1
i=0 is a DSS of index ρ if and only if for every
pair of rows i, j of A, (i 6= j), there exist at least ρ columns that intersect
rows i, j in two nonempty entries that contain different symbols from
{0, 1, . . . , q − 1}. The DSS is perfect if and only if equality holds for all
pairs i, j (i 6= j).
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2011 June
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Comma-Free Codes
Difference System of Sets
Definition
Let G be a multiplicative group. A balanced generalized weighing
matrix BGW(n, k, µ) with parameters n, k, µ over G is an n × n matrix
W = (gij ) with entries from G = G ∪ {0} such that
each row of W contains exactly k nonzero entries, and
for every a, b ∈ {1, . . . , n}, a 6= b, the multiset
{gai gbi−1 : 1 ≤ i ≤ n, gai , gbi 6= 0} contains every element of G exactly
µ/|G | times.
Note that replacing the nonzero entries in a BGW(n, k, µ) with 1’s yields a
(0, 1)-incidence matrix of a symmetric 2-(n, k, µ) design.
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2011 June
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Comma-Free Codes
Difference System of Sets
Theorem (Fuji-Hara, Munemasa, and Tonchev
29
)
The existence of a circulant balanced generalized weighing matrix
W = BGW (n, k, µ) over a group G of order g implies the existence of a
perfect DSS with parameters n, q = g , ρ = (g − 1)µ/g .
Theorem
For every prime power p e and every integer d ≥ 1 such that
gcd(p e − 1, (p e(d+1 − 1)/(p e − 1))) = 1, there exists a perfect DSS with
parameters n = (p e(d+1) − 1)/(p e − 1), q = p e − 1 and
ρ = p e(d−1) (p e − 2).
29
R. Fuji-Hara, A. Munemasa, and V. D. Tonchev. Hyperplane partitions and
difference systems of sets. J. Combin. Theory Ser. A, 113 (8):1689–1698, 2006.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Comma-Free Codes
Difference System of Sets
Theorem (Tonchev30 )
Let D ⊂ {1, 2, ..., n}, |D| = k, be a cyclic (n, k, λ) difference set. Assume
that D is partitioned into q disjoint subsets Q0 , . . . , Qq−1 that are the
base blocks of a cyclic design B with block sizes τi = |Qi |, i = 0, . . . , q − 1
such that every two points are contained in at most λ1 blocks.
Then the sets Q0 , . . . , Qq−1 form a DSS with parameters
(n, τ0 , . . . , τq−1 , ρ = λ − λ1 ).
The DSS {Qi }q−1
i=0 is perfect if and only if B is a pairwise balanced design
with every two points occurring together in exactly λ1 blocks.
30
V. D. Tonchev. Difference systems of sets and code synchronization. Rend. Sem.
Mat. Messina Ser. II, 9(25):217–226 (2004), 2003.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Comma-Free Codes
Difference System of Sets
Theorem
There exists a DSS (n, m, q, ρ) for
1
n = mq + 1 is a prime and ρ = n − m − 1;
2
n = 2mq + 1 ≡ 3 (mod 4) is a prime and ρ = (n − 2m − 1)/4.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Frequency-Hopping Sequences
8
Frequency-Hopping Sequences
Introduction
OC-FHS and PMP
One-Coincidence FH Sequences
Perfect Mendelsohn Packing
Results
Partitioned Cyclic Difference Packing
PCDPs and Frequency-Hopping Sequences
PCDPs and Comma-Free Codes
Results
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Frequency-Hopping Sequences
Gennian Ge (Zhejiang University)
Introduction
Combinatorial Coding Theory
2011 June
188 / 297
Frequency-Hopping Sequences
8
Introduction
Frequency-Hopping Sequences
Introduction
OC-FHS and PMP
One-Coincidence FH Sequences
Perfect Mendelsohn Packing
Results
Partitioned Cyclic Difference Packing
PCDPs and Frequency-Hopping Sequences
PCDPs and Comma-Free Codes
Results
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Frequency-Hopping Sequences
Introduction
Background
In modern radar and multiple-access communication systems,
frequency-hopping (FH) spread-spectrum techniques have become
very popular.
The FH sequences are used to specify which frequency will be used
for transmission in a given time slot.
The set of all available frequencies used for transmission is called the
frequency library of the FH sequences, and usually the cardinality of
the frequency library is required to be large.
Hamming correlations are among the best FH sequence performance
measures.
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Combinatorial Coding Theory
2011 June
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Frequency-Hopping Sequences
Introduction
Hamming Correlation
For any two periodic sequences x and y of period N, x 6= y, their
Hamming cross-correlation is given by the number of coincidences, or hits,
for relative time delay τ , i.e.
X
λx,y (τ ) =
xn ◦ yn+τ
0≤n≤N−1
where
(
0 if xi =
6 yj
xi ◦ yj =
1 if xi = yj
and subscripts are taken modulo N.
Similarly, the Hamming auto-correlation of any periodic sequence x of
period N can be defined.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Frequency-Hopping Sequences
8
OC-FHS and PMP
Frequency-Hopping Sequences
Introduction
OC-FHS and PMP
One-Coincidence FH Sequences
Perfect Mendelsohn Packing
Results
Partitioned Cyclic Difference Packing
PCDPs and Frequency-Hopping Sequences
PCDPs and Comma-Free Codes
Results
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
191 / 297
Frequency-Hopping Sequences
OC-FHS and PMP
Definition
A set of one-coincidence FH (OC-FH) sequences is a set of
non-repeating FH sequences, for which
the peak of the Hamming cross-correlation λx,y (τ ) equals one, for any
pair of sequences x and y belonging to this set, and further
the peak of the Hamming auto-correlation λx (τ ), τ 6≡ 0 (mod N),
equals zero, for any sequence x belonging to this set.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Frequency-Hopping Sequences
OC-FHS and PMP
Lemma
In any set S = {x(i) : i ∈ I } of OC-FH sequences of period N over a
frequency library F,
|S| ≤ |F |(|F| − 1)/N.
Further, equality occurs if and only if any pair of distinct frequencies
fi , fj ∈ F appears in exactly one sequence of S for any non-zero shift τ ,
1 ≤ τ ≤ N − 1.
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Combinatorial Coding Theory
2011 June
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Frequency-Hopping Sequences
OC-FHS and PMP
Lemma
In any set S = {x(i) : i ∈ I } of OC-FH sequences of period N over a
frequency library F,
N|S|
1
−1 ,
λS ≥ × N|S| ×
2
|F|
where λS is the total number of hits between all pairs of sequences
λS =
1
2
X
X
λx,y (τ ).
x,y∈S,x6=y 0≤τ ≤N−1
Further, equality occurs if and only if r1 = · · · = r|F | =
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
N|S|
|F | .
2011 June
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Frequency-Hopping Sequences
OC-FHS and PMP
Let k > t ≥ 1 be positive integers. The k-tuple of distinct elements
(a0 , a1 , . . . , ak−1 ) cyclically contains the ordered pairs (a0 , a1 ), (a1 , a2 ),
. . . , (ak−2 , ak−1 ), (ak−1 , a0 ), and no others. Elements ai , ai+t are said to
be t-apart in the cyclic k-tuple (a0 , a1 , . . . , ak−1 ), where i + t is taken
modulo k.
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Combinatorial Coding Theory
2011 June
195 / 297
Frequency-Hopping Sequences
OC-FHS and PMP
Mendelsohn Packing
Let v , k, and λ be positive integers. A (v , k, λ) Mendelsohn packing, or
briefly (v , k, λ)-MP, is a pair (X , B), where X is a v -set of elements and B
is a collection of cyclically ordered k-tuples of distinct elements from X
called blocks such that every ordered pair of distinct elements of X is
cyclically contained in at most λ of the blocks of B.
If for all t = 1, 2, . . . , k − 1, every ordered pair of elements of X is t-apart
in at most λ of the blocks of B, then the (v , k, λ)-MP is called perfect
and denoted briefly by (v , k, λ)-PMP.
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Combinatorial Coding Theory
2011 June
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Frequency-Hopping Sequences
OC-FHS and PMP
If for all t = 1, 2, . . . , k − 1, every ordered pair of elements of X is t-apart
in exactly λ of the blocks of B, then the (v , k, λ)-PMP is called a perfect
Mendelsohn design and denoted briefly by (v , k, λ)-PMD.
λv (v −1)
It is clear that in a (v ,jk, λ)-PMP
is always satisfied.
k
k (X , B), |B| ≤
λv (v −1)
blocks is said to be optimal.
A (v , k, λ)-PMP with
k
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Frequency-Hopping Sequences
OC-FHS and PMP
Theorem (Cao, Ge, and Miao31 )
(i)
A set S = {an : i ∈ I } of OC-FH sequences of period N over a frequency
library F is equivalent to an (|F| , N, 1)-PMP. Furthermore,
S is optimal with respect to the number of FH sequences if and only
if its corresponding (|F| , N, 1)-PMP is optimal,
S is optimal with respect to the total number of hits between FH
sequences if and only if its corresponding (|F| , N, 1)-PMP is
equi-replicate, and
S is optimum if and only if its corresponding (|F| , N, 1)-PMP is an
(|F| , N, 1)-PMD.
31
Z. Cao, G. Ge, and Y. Miao. Combinatorial characterizations of one-coincidence
frequency-hopping sequences. Des. Codes Cryptogr., 41(2):177–184, 2006.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Frequency-Hopping Sequences
OC-FHS and PMP
Proof.
We need only prove the last assertion.
Let r1 = r2 = · · · = r|F | = r . From the definition of a PMP, for any τ ,
1 ≤ τ ≤ N − 1, any pair of distinct frequencies is τ -apart in at most one
sequence, which implies that r ≤ |F| − 1. If r = |F | − 1, then we can
immediately know that the PMP is a PMD. Otherwise, r ≤ |F| − 2. Since
|S|N = r |F | and |S| = |F|(|F| − 1)/N, we have
b|F|(|F| − 1)/NcN = r |F| ≤ (|F| − 2)|F|. Hence,
b|F|(|F| − 1)/Nc ≤ |F|(|F| − 2)/N. Let |F| = tN + s, where t, s are two
integers such that t ≥ 1 and 0 ≤ s ≤ N − 1. Then we obtain the
inequality t ≤ s(s − 2)/N − bs(s − 1)/Nc, which implies that
t ≤ s(s − 1)/N − bs(s − 1)/Nc − s/N < 1, leading to a contradiction to
|F| ≥ N. The converse is obviously true.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Frequency-Hopping Sequences
OC-FHS and PMP
Theorem (Mendelsohn32 )
Let p be a prime, m a positive integer, and N > 2 an integer such that N
is a divisor of p m − 1. Then there exists a set of optimum OC-FH
sequences of period N over the frequency library GF (p m ).
Proof.
Suppose w is a primitive element of GF (p m ), where p m = eN + 1. Let
(i,r )
{an
} = {w i +r , w i+e +r , . . . , w i+(N−1)e +r }, 0 ≤ i ≤ e −1, r ∈ GF (p m ).
(i,r )
Then the set of there p m (p m − 1)/N sequences {an } forms a set of
optimum OC-FH sequences of period N over the frequency library
GF (p m ).
32
N. S. Mendelsohn. Perfect cyclic designs. Discrete Math., 20:63–68, 1977.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Frequency-Hopping Sequences
OC-FHS and PMP
Theorem
Let R be a ring with identity, and α ∈ R be an element of R such that its
multiplicative order is a prime number p. Then there exists a set of
OC-FH sequences of period p over the frequency library R \ {0}, where
0 ∈ R is the additive zero of R.
Proof.
{anu } = (αu·0 , αu·1 , . . . , αu·(p−1) ), 1 ≤ u ≤ p − 1.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
201 / 297
Frequency-Hopping Sequences
OC-FHS and PMP
Theorem
Let Zn be the residue ring of integers modulo n. Then there exists a set of
full OC-FH sequences of period n over the frequency library Zn .
Proof.
{anu } = (0, u, 2u, . . . , (n − 1)u), u ∈ DU(Zn ), where Let DU(R) denote
the maximum set of units of R such that the difference between any two
distinct elements of DU(R) is still a unit of R.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
202 / 297
Frequency-Hopping Sequences
8
Partitioned Cyclic Difference Packing
Frequency-Hopping Sequences
Introduction
OC-FHS and PMP
One-Coincidence FH Sequences
Perfect Mendelsohn Packing
Results
Partitioned Cyclic Difference Packing
PCDPs and Frequency-Hopping Sequences
PCDPs and Comma-Free Codes
Results
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Combinatorial Coding Theory
2011 June
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Frequency-Hopping Sequences
Partitioned Cyclic Difference Packing
Definition
Given a collection D = {D1 , D2 , . . . , Dm } of subsets (called base blocks)
of Zn , define the difference function ΦD : Z∗n → Z such that
ΦD (g ) =
m
X
|(Di + g ) ∪ Di |.
i=1
For positive integers n, λ, and multiset of positive integers K , a cyclic
difference packing (CDP), or more precisely an (n, K , λ)-CDP, is a
collection of subsets of such that:
K = [|D| : D ∈ D];
λ = maxg ∈Z∗n ΦD (g ).
If, in addition, D partitions Zn , then D is a PCDP, or more precisely an
(n, K , λ)-PCDP.
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Combinatorial Coding Theory
2011 June
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Frequency-Hopping Sequences
Partitioned Cyclic Difference Packing
It is not hard to verify that the following are equivalent definitions of a
PCDP:
D is an (n, K , λ)-PCDP if and only if D partitions Zn , and for any
fixed g ∈ Z∗n , the equation x − y = g has at most λ solutions
(x, y ) ∈ ∪D∈D D × D.
For a set D ⊆ Zn , let ∆D = {a − b : a, b ∈ D, a 6= b}. Then D is an
(n, K , λ)-PCDP if and only if D partitions Zn , and the multiset
∆D = ∪m
i=1 ∆Di contains each elements of Zn at most λ times.
In the particular case where an (n, K , λ)-PCDP D satisfies ΦD (g ) = λ for
all g ∈ Z∗n , it is known as a partitioned cyclic difference family (PCDF), or
more precisely an (n, K , λ)-PCDF.
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Combinatorial Coding Theory
2011 June
205 / 297
Frequency-Hopping Sequences
Partitioned Cyclic Difference Packing
PCDP and FHS
An FH sequence x ∈ F n is called optimal if H(x) ≤ H(y) for all y ∈ F n ,
where H(x) = maxt∈Z∗n λx (t). Here we assume that all transmitters use
the same FH sequence, starting from different time slots. An FH sequence
x ∈ F n with H(x) = λ is called an (n, m, λ)-FH sequence, where m = |F |.
Theorem (Fuji-Hara, Miao, and Mishima33 )
There exists an optimal (n, m, λ)-FH sequence x over the set of
frequencies F = Zm if and only if there exists an optimal (n, K , λ)-PCDP
of m base blocks.
33
R. Fuji-Hara, Y. Miao, and M. Mishima. Optimal frequency hopping sequences: a
combinatorial approach. IEEE Trans. Inform. Theory, 50(10):2408–2420, 2004.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Frequency-Hopping Sequences
Partitioned Cyclic Difference Packing
PCDP and Comma-Free Codes
Theorem (Chee, Ling, and Yin34 )
Let m and n = mµ + ε > m be positive integers and 0 ≤ ε ≤ m − 1. If an
(mµ + ε, [(µ + 1)ε µm−ε ], µ)-PCDP exists, then so does an
(n, [(µ + 1)ε µm−ε ], η)-DSS of minimum redundancy n, where
η = n − µ = (m − 1)µ + ε.
34
Y. M. Chee, A. C. H. Ling, and J. Yin. Optimal partitioned cyclic difference
packings for frequency hopping and code synchronization. IEEE Trans. Inform. Theory,
56(11):5738–5746, 2010.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
207 / 297
Frequency-Hopping Sequences
Partitioned Cyclic Difference Packing
Results
There exist
an optimal (q − 1, [(q − 1/2)2 ], (q − 1)/2)-PCDP for any prime power
q ≡ 1 (mod 4);
an optimal (n, [(n/2)2 ], (n + 2)/2)-PCDP if n = 2p or n = q − 1
where q ≡ 3 (mod 4) is a prime power and p ≡ 5 (mod 8) is a prime.
an optimal (mµ, [µm ], µ)-PCDP for any µ satisfying 2 ≤ µ ≤ p − 1
where m ≥ 3 is an integer whose prime factors are not less than prime
p.
an optimal (p n , [µm−1 (µ + 1)1 ], µ)-PCDP for any odd prime p and
n −1
n ≥ 2 where m = pp−1
and µ = p − 1.
a (3m, [3m ], 3)-PCDP for all m 6≡ 8, 16 (mod 24), m > 2. And a
(6, [32 ], 3)-PCDP cannot exist.
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Combinatorial Coding Theory
2011 June
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Part IV
Deletion-Correcting Codes
Codebook
Cover-Free Family
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Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
9
Deletion-Correcting Codes
Introduction
Directed Designs
Results
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
210 / 297
Deletion-Correcting Codes
Gennian Ge (Zhejiang University)
Introduction
Combinatorial Coding Theory
2011 June
211 / 297
Deletion-Correcting Codes
9
Introduction
Deletion-Correcting Codes
Introduction
Directed Designs
Results
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
211 / 297
Deletion-Correcting Codes
Introduction
Background
When communication takes place over an insertion-deletion channel, data
is lost and gained between the source and the receiver at unknown
locations in the data stream. Insertion-deletion channels are rather
commonplace as follows:
Magnetic and optical recording: Random fluctuations in the motion
of the recording medium, during both the read and write processes,
cause timing uncertainty in the read-back signal. The resulting
received data stream can miss data at certain locations.
Packet-switched communication: Effects of congestion control
protocols and randomness of packet arrivals give rise to random
packet loss.
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Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
Introduction
DNA replication: DNA can undergo deletion and insertion mutations,
where bases are deleted or added, because of strand slippage and
homologous recombination.
Music performance: In music performance, pitch errors can occur.
For instance, notes can be played that are not in the score (insertion
errors) and notes that are specified in the score are omitted (deletion
errors).
It is therefore not surprising that the study of codes for combating
deletions has continued to this day since its systematic treatment by
Levenshtein.
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Combinatorial Coding Theory
2011 June
213 / 297
Deletion-Correcting Codes
Introduction
Preliminary
Let v be a positive integer and Q be a set of v elements. The set Q is
referred to as an alphabet and its elements are referred to as letters. By
a word of length k over Q, we mean a vector (or sequence) of length k
with coordinates taken from Q.
The set of all words of length k over Q will be denoted by Qk∗ .
Furthermore, we use Qk to denote the set of all words of length k over Q,
with different coordinates, that is, if (x1 , . . . , xk ) ∈ Qk then xi 6= xj for
1 ≤ i 6= j ≤ k.
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Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
Introduction
A word x of length n is said to be a sub-sequence of a word y of length
m (n ≤ m) if x can be obtained from y deleting (m − n) letters, or
equivalently y can be obtained from x by inserting (m − n) letters.
In the sequel we will also say that a word x is contained in a word y if x is
a sub-sequence of y .
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Combinatorial Coding Theory
2011 June
215 / 297
Deletion-Correcting Codes
Introduction
Deletion-Correcting Codes
A t-deletion/insertion-correcting code of length k over Q is a subset
∗
C ∗ ⊂ Qk∗ (resp. C ⊂ Qk ) such that every word of Qk−t
(resp. Qk−t )
∗
appears as a sub-sequence of at most one word of C (resp. C ).
As usual, the words of C ∗ (resp. C ) are referred to as codewords.
Furthermore, C ∗ (resp. C ) is said to be a perfect t-deletion-correcting
∗
code if every word of Qk−t
(resp. Qk−t ) appears as a sub-sequence of
exactly one codeword of C ∗ (resp. C ).
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Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
Introduction
The term “t-deletion/insertion-correcting” means that we can correct any
combination of up to t deletions and insertions of letters occurred in
transmission of codewords.
The definition states that the code is capable of correcting t deletions,
because any two distinct codewords in C ∗ (or C ) cannot share a common
sub-sequence of length (k − t) or longer.
However, this implies that it can correct any combination of up to t
deletions and insertions.
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Combinatorial Coding Theory
2011 June
217 / 297
Deletion-Correcting Codes
Introduction
By definition we see that there are two kinds of perfect
t-deletion-correcting codes, those where the coordinates may be equal and
those where all coordinates must be different. We call C ∗ a
T ∗ (k − t, k, v ) code and C a T (k − t, k, v ) code respectively.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
218 / 297
Deletion-Correcting Codes
Introduction
Example
Consider the codes over Q = {1, 2, 3}
C1 = {(1, 2, 3), (3, 2, 1)}
C2 = {(1, 2, 3), (2, 1, 1), (3, 2, 2), (3, 3, 1)},
C3 = {(1, 2, 3), (3, 2, 1), (1, 1, 1), (2, 2, 2), (3, 3, 3)}.
C1 is a T (2, 3, 3) code, C2 and C3 are both T ∗ (2, 3, 3) codes.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
219 / 297
Deletion-Correcting Codes
Introduction
The number of codewords in a T (k − t, k, v ) code is determined by its
parameters. It is easy to prove that each T (k − t, k, v ) code contains
exactly
v −1
v (k − t)! k−t−1
k−1
k k−t−1
different codewords. However, the above example shows that, in contrast
to perfect substitution error correcting codes, T ∗ (k − t, k, v ) codes do not
necessarily have a fixed size.
Let N(k − t, k, v ) denote the maximum number of codewords in any
T ∗ (k − t, k, v ) code. We say that a T ∗ (k − t, k, v ) code is optimal if it
contains N(k − t, k, v ) codewords.
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Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
9
Directed Designs
Deletion-Correcting Codes
Introduction
Directed Designs
Results
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Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
Directed Designs
Directed Packing Designs
A directed packing with parameters v , k and λ, denoted by DP(k, λ; v ),
is a pair (X , A), where X is a v -set (of points) and A is a collection of
transitively ordered k-tuples (called blocks) of X such that every ordered
pair of distinct points of X is contained in at most λ blocks of A.
The directed packing number DDλ (k, v ) is the maximum number of
blocks in any DP(k, λ; v ). When λ = 1, we often write DD(k, v ). A
DP(k, λ; v ) is said to be optimal if it contains DDλ (k, v ) blocks.
Note that if we ignore the order of the blocks, a directed packing
DP(k, λ; v ) becomes a packing P(k, 2λ; v ).
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Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
Directed Designs
DPs and DDCs
The directed packings with λ = 1 are closely related to the perfect
deletion-correcting codes.
Given a T ∗ (2, k, v )-code over X , if we take its alphabet as point set and
the codewords with different coordinates as blocks, then we get a
DP(k, 1; v ) over X because every word of X2 appears as a sub-sequence of
at most one of those codewords.
From this fact we can see that a T ∗ (2, k, v )-code contains at most
DD(k, v ) codewords which contain no repeated letters. In the sequel every
arc in the leave graph of the DP(k, 1; v ) is regarded as a word of X2 ,
which is contained in exactly one codeword with some equal coordinates.
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Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
Directed Designs
Complementary Set
It is not necessary that a T ∗ (2, k, v )-code can be derived from a given
DP(k, 1; v ) by reversing the above process unless there exists a subset
R ⊂ Xk∗ , which satisfies the following properties:
Each word of R has some equal coordinates;
No word of R contains any arc used in the DP(k, 1; v );
Each arc in the leave graph of the DP(k, 1; v ) or each word (x, x) for
all x ∈ X is contained in exactly one word of R.
Such a subset R will be called a complementary set of the DP(k, 1; v ).
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Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
Directed Designs
Thus, the existence of a DP(k, 1; v ) having a complementary set is
equivalent to the existence of a T ∗ (2, k, v )-code.
The number of codewords in the derived code is not fixed, which usually
depends on the structure of the given DP(k, 1; v ) and the construction of
the complementary set.
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Combinatorial Coding Theory
2011 June
225 / 297
Deletion-Correcting Codes
Directed Designs
Perfect DP
The above discussion leads us to define a useful generalization of a
DP(k, 1; v ) for the construction of optimal T ∗ (2, k, v )-codes.
Definition
A DP(k, 1; v ) is called a perfect directed packing (PDP), denoted by
PDP(k, 1; v ), if it is optimal and has a complementary set of size v .
From the definition, we can see that there is exactly one letter occurs at
least twice in each word of the complementary set of a PDP(k, 1; v ).
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2011 June
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Deletion-Correcting Codes
Directed Designs
Theorem (Wang35 )
If there exists a PDP(k, 1; v ), then there exists an optimal T ∗ (2, k, v )-code
with DD(k, v ) + v codewords, that is, N(2, k, v ) = DD(k, v ) + v .
Proof.
By the definition of a PDP, we can obtain a T ∗ (2, k, v )-code by taking as
codewords all DD(k, v ) blocks of the given PDP(k, 1; v ) and all v words of
the complementary set. Since a T ∗ (2, k, v )-code contains at most
DD(k, v ) codewords where each letter occurs at most once per codeword
and v codewords where some letter occurs at least twice per codeword,
i.e., N(2, k, v ) ≤ DD(k, v ) + v , we can claim that the derived code is an
optimal T ∗ (2, k, v )-code.
35
J. Wang. Some Combinatorial constructions for optimal perfect deletion-correcting
codes. Des. Codes Cryptogr. 48:331–347, 2008.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
Directed Designs
Theorem
Suppose that there exists an optimal DP(k, 1; v ) whose leave graph has n
(n ≤ v ) arcs A1 , A2 , . . . , An . If the vertex sets V (A1 ), V (A2 ), . . . , V (An )
possess a system of distinct representatives (SDR), then there exists an
optimal T ∗ (2, k, v )-code with DD(k, v ) + v codewords.
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Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
Directed Designs
Proof.
It is sufficient for us to construct a complementary set of size v for the
given DP(k, 1; v ). Let X be the point set of the optimal DP(k, 1; v ) and
S = {a1 , . . . , an } be the SDR of V (A1 ), V (A2 ), . . . , V (An ). For any i
(1 ≤ i ≤ n), we keep the order of vertices of Ai and replace ai by aik−1 to
produce a word of length k. For every point x ∈ X \ S, we add the word
x k . Then the resulting v words form a complementary set of the optimal
DP(k, 1; v ).
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Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
Directed Designs
Directed Group Divisible Design
A directed group divisible design (DGDD). A DGDD(v , K ; M) is a triple
(X , G, A), where X and G are as defined for GDDs, and
A is a set of transitively ordered k-subsets of X (called blocks),
where k ∈ K ,
no block meets a group in more than one point, and
every ordered pair (x, y ) of points not contained in some group is a
sub-sequence of a unique block.
The group type of a DGDD is the multiset T = {|G | : G ∈ G}, and we
usually use the exponential notation for its description. We wish to remark
that a k-DGDD of type 1v is indeed a T (2, k, v ), or equivalently a
directed BIBD (DBIBD).
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Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
Directed Designs
Lemma
If there exists a DGDD(v , k; M) and for every m ∈ M there is a
T ∗ (2, k, m), then there is a T ∗ (2, k, v ).
Lemma
If there are a GDD(v , K ; M), a T(2, k, t) for each t ∈ K , and a
T ∗ (2, k, m) for each m ∈ M, then there is a T ∗ (2, k, v ).
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Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
9
Results
Deletion-Correcting Codes
Introduction
Directed Designs
Results
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Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
Results
Lemma
Let
v 2(v − 1)
DU(v , k) := b b
cc + v
k k −1
v 2v
DL(v , k) := d d
ee
k k −1
Then for all v ≥ 2 and k ≥ 2
DL(v , k) ≤ N(2, k, v ) ≤ DU(v , k).
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
Results
Theorem (Bours36 )
There is a T ∗ (2, 4, v ) of size DU(v , 4) − ∆(v , 4) for any positive integer
v , where


if v ≡ 0 (mod 3)
bv /4c
∆(v , 4) = 0
if v ≡ 1 (mod 3)


b(v − 2)/6c if v ≡ 2 (mod 3)
36
P. A. H. Bours. On the construction of perfect deletion-correcting codes using
design theory. Des. Codes Cryptogr.. 6:5–20, 1995.
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Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
Results
Theorem (Wang37 )
The following codes exist.
A T ∗ (2, 4, v ) of size DU(v , 4) − ∆(v , 4), where
(
1 if v = 9
∆(v , 4) =
0 otherwise
A T ∗ (2, 5, v ) of size DU(v , 5) − ∆(v , 5), where
(
1 if v ≡ 7, 9 (mod 10)
∆(v , 5) =
0 otherwise
37
J. Wang. Some combinatorial constructions for optimal perfect deletion-correcting
codes. Des. Codes Cryptogr.. 48:331–347, 2008.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
Results
If we define the spectrum of sizes of a perfect T ∗ (k − t, k, v ) as
Spec(v , k, t) = {|C | : C is a T ∗ (k − t, k, v )}.
Theorem (Chee, Ge, and Ling38 )
Let I (v , k) = [DL(v , k), DU(v , k)]. Then
Spec(v , 3, 1) = I (v , 3) for all v .
Spec(v , 4, 2) = I (v , 4) for all positive integers v , except when
v ∈ {4, 6}, and except possibly when v ∈
{9, 10, 11, 13, 15, 16, 18, 19, 21, 22, 24, 25, 31, 34, 37, 40, 43, 46, 52}.
38
Y. M. Chee, G. Ge, and A. C. H. Ling. Spectrum of sizes for perfect
deletion-correction codes. SIAM J. Discrete Math. 24(1):33–55, 2010.
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Deletion-Correcting Codes
Results
Theorem (Levensthein39 )
A T(3, 4, v ) code (or a DD(3, 4, v )) exists if and only if v is even.
Theorem (Wang and Ji40 )
There exists a T ∗ (3, 4, v ) code for all integers v .
39
V. I. Levenshtein. On perfect codes in deletion and insertion metric. Discrete Math.
Appl.. 2:241–258, 1992.
40
J. Wang and L. Ji. Existence of T ∗ (3, 4, v )-codes. J. Combin. Des.. 13:42–53,
2005.
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Combinatorial Coding Theory
2011 June
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Codebook
10
Codebook
Introduction
Difference Sets
Codebooks from Almost Difference Sets
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Combinatorial Coding Theory
2011 June
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Codebook
10
Introduction
Codebook
Introduction
Difference Sets
Codebooks from Almost Difference Sets
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Combinatorial Coding Theory
2011 June
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Codebook
Introduction
Definition
An (N, K ) codebook C is a set {c0 , . . . , cN−1 } of N unit norm 1 × K
complex vectors ci , which are called codewords of the codebook.
The alphabet of the codebook is the set of all different complex
values that the coordinates of all the codewords of C take.
The alphabet size is the number of elements in the alphabet.
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Combinatorial Coding Theory
2011 June
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Codebook
Introduction
The root-mean-square (RMS) cross-correlation and the maximum
cross-correlation amplitudes of such a codebook C are defined as
s
X
1
H 2
Irms (C) :=
ci cj ,
N(N − 1)
0≤i,j≤N−1,i6=j
H
Imax (C) :=
max ci cj ,
0≤i<j≤N−1
where cjH denotes the Hermite transpose.
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Combinatorial Coding Theory
2011 June
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Codebook
Introduction
Theorem (Welch41 )
For any complex codebook C with N > K , we have
s
N −K
Irms (C) ≥
,
(N − 1)K
(6)
P
H
with equality if and only if N
i=0 ci ci = (N/K )IK , where IK denotes the
K × K identity matrix.We have also
s
N −K
,
(7)
Imax (C) ≥
(N − 1)K
q
N−K
with equality if and only if ci cjH = (N−1)K
for all pairs (i, j) with i 6= j.
41
L. Welch. Lower bounds on the maximum cross correlation of signals. IEEE Trans.
Inform. Theory, 20(3):397–399, 1974.
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Combinatorial Coding Theory
2011 June
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Codebook
Introduction
If the equality holds in Eq 6, the codebook CD is referred to as a
Welch-bound-equality (WBE) codebook. A codebook meeting the
bound of Eq 7 is called a maximum-Welch-bound-equality (MWBE)
codebook.
An MWBE codebook must be a WBE codebook, but a WBE codebook
may not be an MWBE codebook. MWBE codebooks form a subset of
WBE codebooks.In the sequel, Welch bound always refers to the one of
Eq 7, unless otherwise stated.
Every linear error correcting code whose dual code with Hamming distance
at least three yields a WBE codebook. So it is very easy to construct
WBE codebooks. There are many constructions of WBE codebooks.
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Combinatorial Coding Theory
2011 June
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Codebook
Introduction
However, MWBE codebooks are extremely hard to construct. The
following are the known classes of MWBE codebooks.
(N, N) orthogonal MWBE codebooks for any N > 1.
(N, N − 1) MWBE codebooks for any N > 1 obtained from the FFT
matrix, and some (N, N − 1) MWBE codebooks from the m-sequence
codes.
(N, K ) MWBE codebooks based on conference matrices, when
N = 2K = 2d+1 and d is a positive integer, and N = 2K = p d + 1
with p a prime number and d a positive integer.
(N, K ) MWBE codebooks based on (N, K , λ) cyclic difference sets in
ZN and those based on (N, K , λ) difference sets in finite fields GF (q).
Gennian Ge (Zhejiang University)
Combinatorial Coding Theory
2011 June
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Codebook
Introduction
In 2005, Xia, Zhou, and Giannakis42 developed a generic construction of
MWBE codebooks using cyclic difference sets in ZN . Shortly after, Ding43
a construction of MWBE codebooks using both cyclic and non-cyclic
difference sets in finite fields GF (q) was proposed. Recently, Ding and
Feng44 generalized these constructions in.
42
P. Xia, S. Zhou, and G. B. Giannakis. Achieving the Welch bound with difference
sets. IEEE Trans. Inform. Theory, 51(5):1900–1907, 2005.
43
C. Ding. Complex codebooks from combinatorial designs. IEEE Trans. Inform.
Theory, 52 (9):4229–4235, 2006.
44
C. Ding and T. Feng. A generic construction of complex codebooks meeting the
Welch bound. IEEE Trans. Inform. Theory, 53(11):4245–4250, 2007.
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Combinatorial Coding Theory
2011 June
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Codebook
10
Difference Sets
Codebook
Introduction
Difference Sets
Codebooks from Almost Difference Sets
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Combinatorial Coding Theory
2011 June
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Codebook
Difference Sets
Difference Set
Let G be an abelian group of order v , and let D be a k-element subset of
G . We define the difference function
ΓD (w ) := |(D + w ) ∩ D|,
where D + w := {d + w : d ∈ D}.
D is called a (v , k, λ)-difference set in G provided that the difference
function D(w ) = λ for all nonzero elements w of G . If D is a
(v , k, λ)-difference set in G , its complement D := G \ D is a
(v , v − k, v − 2k + λ)-difference set.
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Combinatorial Coding Theory
2011 June
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Codebook
Difference Sets
Almost Difference Set
D is called a (v , k, λ, t) almost difference set in G if ΓD (w ) takes on λ
altogether t times and λ + 1 altogether v − 1 − t times when w ranges
over all the nonzero elements of G .
D is a (v , k, λ, t) almost difference set in (G , +) if and only if the
complement D = G \ D is a (v , v − k, v − 2k + λ, t) almost difference set.
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Combinatorial Coding Theory
2011 June
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Codebook
Difference Sets
Codebook From Difference Set
Let (G , +) be an Abelian group of order N and exponent eG , and let
χ0 , χ1 , . . . , χN−1 denote all the characters of G . Given any K -subset
D := {d1 , d2 , . . . , dK } of G , we define a codebook
where for each i
Gennian Ge (Zhejiang University)
CD := {ci : i = 0, . . . , N − 1},
(8)
1
ci := √ (χi (d1 ), . . . , χi (dK )).
K
(9)
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2011 June
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Codebook
Difference Sets
Theorem (Ding and Feng45 )
The set CD is an (N, K ) MWBE codebook with alphabet size eG if and
only if the set D is an (N, K , λ) difference set in (G , +), where K > 1.
45
C. Ding and T. Feng. A generic construction of complex codebooks meeting the
Welch bound. IEEE Trans. Inform. Theory, 53(11):4245–4250, 2007.
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Combinatorial Coding Theory
2011 June
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Codebook
10
Codebooks from Almost Difference Sets
Codebook
Introduction
Difference Sets
Codebooks from Almost Difference Sets
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Combinatorial Coding Theory
2011 June
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Codebook
Codebooks from Almost Difference Sets
Cyclotomy
A class of cyclic almost difference sets is constructed from Whiteman’s
generalized cyclotomy. To introduce the almost difference sets, we have to
deal with Whitemans generalized cyclotomy of order 2.
Let g be a fixed common primitive root of both primes p and q. Assume
that 2 = gcd(p − 1, q − 1), and let f = (p − 1)(q − 1)/2. Then there
exists an integer x such that x ≡ g (mod p), x ≡ 1 (mod q) and
Z∗pq = {g s x i : s = 0, 1, . . . , f − 1; i = 0, 1},
where Z∗pq denotes the set of all multiplicatively invertible elements of Zpq .
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Combinatorial Coding Theory
2011 June
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Codebook
Codebooks from Almost Difference Sets
Whitemans generalized cyclotomic classes Di of order 2 are defined as
Di = {g s x i : s = 0, 1, . . . , f − 1}, i = 0, 1.
The generalized cyclotomic numbers of order 2 are defined by
(i, j)2 = |(Di + 1) ∩ Dj | .
This generalized cyclotomy was used by Whiteman to find the twin-prime
difference sets.
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Combinatorial Coding Theory
2011 June
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Codebook
Codebooks from Almost Difference Sets
Lemma (Ding46 )
Suppose that gcd(p − 1, q − 1) = 2. Let D0 and D1 be the cyclotomic
classes of order 2. Define
D = D1 ∪ {p, 2p, . . . , (q − 1)p}.
(10)
If q − p = 4 and (p − 1)(q − 1)/4 is odd, then D is a
(p(p + 4), (p + 3)(p + 1)/2, (p + 3)(p + 1)/4, (p − 1)(p + 5)/4)
almost difference set of Zp(p+4) .
46
C. Ding. Auto-correlation values of generalized cyclotomic sequences of order two.
IEEE Trans. Inform. Theory, 44(4):1699–1702, 1998.
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Combinatorial Coding Theory
2011 June
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Codebook
Codebooks from Almost Difference Sets
Theorem
Let p and q = p + 4 both be prime, where p ≡ 3 (mod 4). For the
(p(p + 4), (p + 3)(p + 1)/2) codebook CD of Eq 8 defined by the almost
difference set D of Eq 10, we have
Imax (CD ) =
Gennian Ge (Zhejiang University)
1
.
p+1
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2011 June
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Codebook
Codebooks from Almost Difference Sets
Let N1 > 1 and N2 > 1 be two relatively prime integers. Let ΨNi be the
group character of the Abelian group ZNi defined in (7). For any
(n1 , n2 ) ∈ ZN1 × ZN2 , we define
Ψ ((n1 , n2 )) = ΨN1 (n1 )ΨN2 (n2 ).
(11)
Then Ψ is a group character of the Abelian group ZN1 × ZN2 .
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Combinatorial Coding Theory
2011 June
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Codebook
Codebooks from Almost Difference Sets
For any subset D := {d1 , d2 , . . . , dK } of ZN1 × ZN2 , where di = (di,1 , di,2 ),
we define a codebook
UD := {ui : i = 0, . . . , N1 N2 − 1},
(12)
where
1
ui1 ,i2 = √ (Ψ ((i1 d1,1 , i2 d1,2 )) , Ψ ((i1 d2,1 , i2 d2,2 )) , . . . , Ψ ((i1 dK ,1 , i2 dK ,2 )))
K
(13)
for each (i1 , i2 ) ∈ ZN1 × ZN2 .
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Combinatorial Coding Theory
2011 June
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Codebook
Codebooks from Almost Difference Sets
Codebooks from Almost Difference Sets
Below we will describe complex codebooks UD by selecting almost
difference sets D in ZN1 × ZN2 , where N1 and N2 are properly selected
positive integers.
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Combinatorial Coding Theory
2011 June
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Codebook
Codebooks from Almost Difference Sets
Lemma
Let q ≡ 5 (mod 8) be a prime. It is known that q = s 2 + 4t 2 for some s
and t with s ≡ ±1 (mod 4). Set N = 2q. Let i, j, ∈ {0, 1, 2, 3} be three
pairwise distinct integers, and define
h
i h
i
(4,q)
(4,q)
(4,q)
(4,q)
D(i,j,l) = {0} × (Di
∪ Dj
) ∪ {1} × (Dl
∪ Dj
) . (14)
N−6 3N−6
Then D(i,j,l) is an (N, N−2
2 , 4 , 4 ) almost difference set in Z2 × Zq if
t = 1 and (i, j, l) = (0, 1, 3) or (0, 2, 1); or
s = 1 and (i, j, l) = (1, 0, 3) or (0, 1, 2).
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Combinatorial Coding Theory
2011 June
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Codebook
Codebooks from Almost Difference Sets
Theorem
Let t = 1. For the almost difference set D(0,1,3) of Eq 14, UD(0,1,3) is a
(2q, q − 1) codebook with
p
√
2q + 2 q
Imax (UD(0,1,3) ) =
.
2(q − 1)
Remark
Note that the Welch bound for (2p, p − 1) codebook is
q
s
2q + 2 2q−1
q−1
N −K
=
.
(N − 1)K
2q − 1
Hence Imax (UD(0,1,3) ) nearly meets the Welch bound.
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Combinatorial Coding Theory
2011 June
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Codebook
Codebooks from Almost Difference Sets
Theorem
Let s = 1. For the almost difference set D(0,1,2) of Eq 14, UD(0,1,2) is a
(2q, q − 1) codebook with
p
√
2q + 2 q
Imax (UD(0,1,2) ) =
.
2(q − 1)
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Combinatorial Coding Theory
2011 June
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Codebook
Codebooks from Almost Difference Sets
Lemma
Let E be an (l, (l − 1)/2, (l − 3)/4) difference set in Zl . Then
D = ({0} × E ) ∪ {1, 2, 3} × Ē
(15)
is a (4l, 2l + 1, l, l − 1) almost difference set in Z4 × Zl , where Ē is the
complement of E .
Theorem
For the (4l, 2l + 1) codebooks UD of Eq 12 defined by the almost
difference sets D of Eq 15, we have
√
l +1
Imax (UD ) =
.
2l + 1
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Combinatorial Coding Theory
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Codebook
Codebooks from Almost Difference Sets
The following cyclic almost difference sets are stated in terms of optimal
auto-correlation values of a class of binary sequences.
Lemma
(2,q)
Let q be odd. Define D = logα (D1
− 1). Then the set D is
a (q − 1, (q − 1)/2, (q − 3)/4, (3q − 5)/4) almost difference set in
Zq−1 if q ≡ 3 (mod 4), and
a (q − 1, (q − 1)/2, (q − 5)/4, (q − 1)/4) almost difference set in
Zq−1 if q ≡ 1 (mod 4).
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Combinatorial Coding Theory
2011 June
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Codebook
Codebooks from Almost Difference Sets
The proof of these theorems can be found in:
C. Ding and T. Feng. Codebooks from almost difference sets. Des.
Codes Cryptogr., 46(1):113–126, 2008.
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Combinatorial Coding Theory
2011 June
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Cover-Free Family
11
Cover-Free Family
Introduction
Background
Definition
Equivalent Objects
Bounds
Constructions
Construction from Combinatorial Design
Construction from Coding Theory
Construction from Probabilistic Methods
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Combinatorial Coding Theory
2011 June
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Cover-Free Family
11
Introduction
Cover-Free Family
Introduction
Background
Definition
Equivalent Objects
Bounds
Constructions
Construction from Combinatorial Design
Construction from Coding Theory
Construction from Probabilistic Methods
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Combinatorial Coding Theory
2011 June
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Cover-Free Family
Introduction
Background
Cover-free families were considered from different subjects such as
information theory, combinatorics and group testing.
In 1964, cover-free families were first introduced by Kautz and
Singleton to investigate nonrandom superimposed binary codes.
These codes may be used for retrieval files, data communication and
magnetic memories.
In 1985, Erdös, Frankl and Füuredi discussed cover-free families as
combinatorial objects which generalized the Sperner systems.
In 1987, Hwang and S̀os defined cover-free families for non-adaptive
group testing.
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Combinatorial Coding Theory
2011 June
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Cover-Free Family
Introduction
Since then, cover-free families have been discussed in several equivalent
formulations in subjects such as information theory, combinatorics and
group testing by numerous researchers.
Recently, cover-free families were also used for many cryptographic
problems such as frame-proof codes and traceability schemes, broadcast
encryption, key storage, multi-receiver authentication, etc.
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2011 June
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Cover-Free Family
Introduction
Definition
Let X be an n-set and let F be a set of subsets (blocks) of X .
Definition
A set system (X , F ) is called a r -cover-free family (or r -CFF) provided
that, for any r blocks A1 , . . . , Ar ∈ F and any other block B0 ∈ F, we
have
r
[
B0 6⊆
Aj .
j=1
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Combinatorial Coding Theory
2011 June
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Cover-Free Family
Introduction
Generalized Definitions
In 1988, Mitchell and Piper defined the concept of key distribution
patterns, which is in fact a generalized type of cover-free family. Their
definition can be described as follows.
Definition
A set system (X , F ) is called a (w , r )-cover-free family (or (w , r )-CFF)
provided that, for any w blocks B1 , . . . , Bw ∈ F and any other r blocks
A1 , . . . , Ar ∈ F, we have
w
r
\
[
Bi 6⊆
Aj .
i=1
j=1
In the classical definition of cover-free family is the case w = 1.
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Combinatorial Coding Theory
2011 June
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Cover-Free Family
Introduction
In 1989, Dyachkov et. al considered another generalization of cover-free
families when they investigated superimposed distance codes.
Definition
A set system (X , F ) is called a (r , d)-cover-free family (or (r , d)-CFF)
provided that, for any block B0 ∈ F and any other r blocks
A1 , . . . , Ar ∈ F, we have


r
[
|B0 \  Aj  | > d
j=1
where d is a positive integer.
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Combinatorial Coding Theory
2011 June
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Cover-Free Family
Introduction
A set system (X , F) is called a (w , r ; d)-cover-free family (or
(w , r ; d)-CFF) provided that, for any w blocks B1 , . . . , Bw ∈ F and any
other r blocks A1 , . . . , Ar ∈ F, we have

!  r
w
\
[
|
Bi \  Aj  | > d
i=1
j=1
where d is a non-negative integer.
(w , r ; d)-CFF(N, T ) is used to denote a cover-free family to specify that
|X | = N and |F | = T .
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Cover-Free Family
11
Equivalent Objects
Cover-Free Family
Introduction
Background
Definition
Equivalent Objects
Bounds
Constructions
Construction from Combinatorial Design
Construction from Coding Theory
Construction from Probabilistic Methods
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Combinatorial Coding Theory
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Cover-Free Family
Equivalent Objects
There are many equivalent objects of cover-free families, such as
superimposed codes,
key distribution patterns,
non-adaptive group testing algorithms, etc.
In this subsection, we will introduce two combinatorial objects,
coverings of order-interval hypergraph, and
disjunct set systems
which are also equivalent to cover-free families. These two equivalent
objects will be used in later sections.
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Cover-Free Family
Equivalent Objects
Order-Interval Hypergraph
Let [n] = {1, . . . , n} and let Pn;l,u = {Y ⊂ [n] : l ≤ |Y | ≤ u}, where
0 < l < u < n, ordered by inclusion. Define a class of order-interval
hypergraphs Gn;l,u = (P, E ) as follows. Let the set of points P = Pn;l,u
and let the set of edges E be the maximal intervals, i.e.,
E = {I = {C ⊂ [n] : Y1 ⊂ C ⊂ Y2 } : |Y1 | = l, |Y2 | = u, Y1 , Y2 ⊂ [n]}.
Definition
A (point) covering of a hypergraph is a subset of points S such that each
edge of the hypergraph contains at least one point of S.
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Cover-Free Family
Equivalent Objects
Disjunct System
Definition
A set system (X , B) is an (i, j)-disjunct system provided that, for any
P, Q ⊆ X such that |P| ≤ i, |Q| ≤ j and P ∩ Q = ∅, there exists a B ∈ B
such that P ⊆ B and Q ∩ B = ∅. An (i, j)-disjunct system, (X , B), will be
denoted as an (i, j)-DS(v , b) if |X | = v and |B| = b.
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Cover-Free Family
Equivalent Objects
Theorem
There exists a covering of Gn,l,u of size b if and only if there exits a
(l, n − u)-DS(n, b).
Proof.
S is a covering of Gn,l,u if and only if for any Y1 , Y2 ⊂ [n], Y1 ⊂ Y2 ,
|Y1 | = l, |Y2 | = u, there is a C ∈ S such that Y1 ⊂ C ⊂ Y2 . This is
equivalent to that for any Y1 , Y3 ⊂ [n], |Y1 | = l, |Y3 | = n − u,
Y1 ∩ Y3 = ∅, there is some C ∈ S such that Y1 ⊂ C and Y3 ∩ C = ∅.
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Cover-Free Family
Equivalent Objects
Disjunct systems and cover-free families are dual incidence structures. If A
is an incidence matrix of a disjunct system, the AT , the transpose of A, is
an incidence matrix of a cover-free family. Thus we have the following
theorem.
Theorem
There exists an (i, j)-CFF(N, T ) if and only if there exists an
(i, j)-DS(T , N).
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Combinatorial Coding Theory
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Cover-Free Family
11
Bounds
Cover-Free Family
Introduction
Background
Definition
Equivalent Objects
Bounds
Constructions
Construction from Combinatorial Design
Construction from Coding Theory
Construction from Probabilistic Methods
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Combinatorial Coding Theory
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Cover-Free Family
Bounds
Trade-off on Bounds
It is easy to see that there is a trade-off between the values of N and T in
a cover-free family.
We want to maximize the value of T or minimize the value of N in case
that other parameters are given.
The bounds of N or T in cover-free families were discussed by many
researchers.
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Cover-Free Family
Bounds
NP-Completeness
The first question about the bound of CFF is whether we can find an
efficient algorithm to determine the bound. Unfortunately, this is not the
case.
Let Gn,l,u be an order-interval hypergraph. Define
τ (Gn,l,u ) = min{|S| : S is a covering of Gn,l,u }.
Hence τ is an upper bound of N for certain (i, j)-CFF.
Theorem (Bouchemakh and Engel47 )
The problem of deciding τ (Gn,l,u ) ≤ k (with input Gn,l,u and k) is
NP-complete.
47
I. Bouchemakh and K. Engel. The order-interval hypergraph of a finite poset and
the König property. Discrete Math., 170:51–61, 1997.
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Combinatorial Coding Theory
2011 June
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Cover-Free Family
Bounds
Bounds for r -CFF
Theorem (Erdös, Frankl, and Füredi48 )
In a k-uniform r -CFF,
T ≤
N
k −1
/
.
dk/r e
dk/r e − 1
48
P. Erdös, P. Frankl and Z. Füredi. Families of finite sets in which no set is covered
by the union of r others. Israel J. Math., 51:75–89, 1985.
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Combinatorial Coding Theory
2011 June
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Cover-Free Family
Bounds
Bounds for (r ; d)-CFF
Theorem (Füredi49 )
In a (r ; d)-CFF,
N
T <r+
m
where m =
49
l
2(N−r −d(r +1))
r (r +1
m
.
Z. Föredi. On r -cover-free families. J. Combin. Theory Ser. A, 73:172–173, 1996.
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Cover-Free Family
Bounds
Bounds for (w , r )-CFF
Let N((w , r ), T ) denote the minimum number of points in any (w , r )-CFF
having T blocks.
Theorem (Stinson, Wei, and Zhu50 )
For w , r ≥ 1 and T ≥ w + r > 2, we have
N((w , r ), T ) ≥ 2c
w +r
w
log(w + r )
log T ,
where c is a constant.
50
D. R. Stinson, R. Wei and L. Zhu. Some New Bounds for Cover-Free Families, J.
Combin. Theory Ser. A, 90:224–234, 2000.
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Cover-Free Family
Bounds
Theorem (Stinson, Wei, and Zhu51 )
For w , r ≥ 1 and T ≥ w + r > 2, we have
N((w , r ), T ) ≥ 2c
w +r
r
log w + r
log T
for some constant c.
51
D. R. Stinson, R. Wei and L. Zhu, Some New Bounds for Cover-Free Families, J.
Combin. Theory A, 90:224–234, 2000.
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Combinatorial Coding Theory
2011 June
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Cover-Free Family
11
Constructions
Cover-Free Family
Introduction
Background
Definition
Equivalent Objects
Bounds
Constructions
Construction from Combinatorial Design
Construction from Coding Theory
Construction from Probabilistic Methods
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Combinatorial Coding Theory
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Cover-Free Family
Constructions
For the purpose of applications, researchers tried to find efficient
constructions of cover-free families. Three basic methods were used to
construct CFF:
combinatorial methods,
methods from coding theory, and
probabilistic method.
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Combinatorial Coding Theory
2011 June
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Cover-Free Family
Constructions
Packing Design
Lemma
If there exists a t-(v , k, 1) packing design having b blocks, then there
exists a (r ; d)-CFF(v , b), where r = b(k − d − 1)/(t − 1)c.
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Cover-Free Family
Constructions
Orthogonal Array52
Suppose {s1 , s2 , . . . , sk } is a column in an OA(t, k, s). Define a block as
{(s1 , 1), (s2 , 2), . . . , (sk , k)} accordingly. In this way, we can obtain a
t-(ks, k, 1) packing design from an OA(t, k, s). It is known that if q is a
prime power and t < q, then there exists an OA(t, q + 1, q), and hence a
t-(q 2 + q, q + 1, 1) packing design with q t blocks exists.
Lemma (Stinson and Wei)
For any prime power q and any integer t < q, there exists a
2
t
(b q−d
t−1 c; d)-CFF(q + q, q ).
52
D. R. Stinson and R. Wei. Combinatorial properties and constructions of
traceability schemes and frame-proof codes. SIAM J. Discrete Math., 11:41–53, 1998.
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Combinatorial Coding Theory
2011 June
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Cover-Free Family
Constructions
Separating Hash Family53
Definition
An (n, m, {w1 , w2 })-separating hash family is a set of functions F, such
that |Y | = n, |X | = m, f : Y → X for each f ∈ F, and for any
C1 , C2 ⊆ {1, 2, . . . , n} such that |C1 | = w1 , |C2 | = w2 and C1 ∩ C2 = ∅,
there exists at least one f ∈ F such that
{f (y ) : y ∈ C1 } ∩ {f (y ) : y ∈ C2 } = ∅.
The notation SHF(N; n, m, {w1 , w2 }) will be used to denote an
(n, m, {w1 , w2 })-separating hash family with |F| = N.
53
D. R. Stinson, Tran van Trung and R. Wei. Secure frame-proof codes, key
distribution patterns, group testing algorithms and related structures. J. Statist. Plann.
Inference, 86:595–617, 2000.
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Combinatorial Coding Theory
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Cover-Free Family
Constructions
An SHF(N; n, m, {w1 , w2 }) can be depicted as an N × n matrix with
entries from {1, 2, . . . , m}, such that in any two disjoint sets C1 and C2 of
w1 and w2 columns (respectively), there exists at least one row such that
the entries in the columns of C1 are distinct from the entries in the
columns of C2 .
Now suppose that A is an n × N matrix which is depicted from an
SHF(N; n, m, {w , r }) (The transpose of the matrix). The elements in A
are 1, 2, . . . , m. Suppose B is the incidence matrix of a
(w , r ; d)-CFF(v , m). Denote b1 , b2 , . . . , bm the rows of B. We construct a
Nv × n matrix A0 by substituting the element i in A by bi .
It can be verified that A0 is an incidence matrix of a (w , r ; d)-CFF(vN, n).
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Combinatorial Coding Theory
2011 June
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Cover-Free Family
Constructions
Theorem
If there exists an (w , r ; d)-CFF(v , m) and an SHF(N; n, m, {w , r }), then
there exists a (w , r ; d)-CFF(vN, n).
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Cover-Free Family
Constructions
Construction from Coding Theory
Consider a code C of length N on an alphabet Q with |Q| = q. Then
C ⊆ Q N and we will call it an (N, n, q)-code if |C| = n. The elements of C
are called codewords; each codeword is x = (x1 , . . . , xN ), where xi ∈ Q,
1 ≤ i ≤ N.
Suppose C is an (N, n, q) code on an alphabet Q. Define
X = {1, . . . , N} × Q, and for each codeword c = (c1 , . . . , cN ) ∈ C, define
an N-subset of X as follows:
Bc = {(i, ci ) : 1 ≤ i ≤ N}.
Finally, define B = {Bc : c ∈ C}. Then we obtain an N-uniform set system
(X , B).
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Combinatorial Coding Theory
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Cover-Free Family
Constructions
Using this correspondence between a code and a set system, we are able to
construct uniform CFF from codes which satisfy certain properties.
Lemma
Suppose that C is an (N, n, q)-code having minimum distance D. Then
there is a (r ; d)-CFF(Nq, n), where r = b N−d−1
N−D c.
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Cover-Free Family
Constructions
A Reed-Solomon code is an (N, q t , q)-code with minimum distance
D = N − t + 1.
Theorem
Suppose N, q, r and d are given, with q a prime power and N ≤ q + 1.
Then there exists an (r ; d)-CFF(qN, T ) where T = q d(N+r −d−1)/r e .
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Combinatorial Coding Theory
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Cover-Free Family
Constructions
Construction from Probabilistic Methods
Theorem (Stinson, van Trung, and Wei54 )
Suppose that T , w and r are positive integers. Define
p =1−
wwrr
.
(w + r )w +r
Then there exists a (w , r )-CFF(N, T ) if
N >
N >
(w + r ) log T
, or
− log p
(w + r − 1) log(2T )
.
− log p
54
D. R. Stinson, Tran van Trung and R. Wei. Secure frame-proof codes, key
distribution patterns, group testing algorithms and related structures. J. Statist. Plann.
Inference, 86:595–617, 2000.
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Combinatorial Coding Theory
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The End
THAN K
Y OU
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