Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Linear Codes
over Finite
Chain Rings
Linear Codes over Finite Chain
Rings—Algebraic Theory
Thomas Honold
Institute of Information and Communication Engineering
Zhejiang University
CIMPA Summer School
August 2008
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Outline
Thomas
Honold
Linear Codes
over Finite
Chain Rings
1 Linear Codes over Finite Chain Rings
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Linear Codes
over Finite
Chain Rings
Today’s Lecture: #4
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Linear Codes
over Finite
Chain Rings
Structure of Linear Codes
For linear codes over R (i.e. submodules C ≤ R R n ) the structure
theorem for R-modules says:
There exists a uniquely determined partition λ into at most n
parts, all of which are ≤ m, such that R C has a basis c1 , . . . , ck
(k = λ′1 = rk C) with corresponding periods θλ1 , . . . , θλk .
Since R R n is free, ci has height m − λi in R R n , i.e.
ci = (θm−λi ci1 , θm−λi ci2 , . . . , θm−λi cin )
with at least one cij not divisible by θ.
The number of codewords of C is
|C| = q |λ| = q λ1 +···+λk .
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Linear Codes
over Finite
Chain Rings
Theorem
Every linear code C ≤ R R n is permutation equivalent to a linear
code generated by a matrix of the following form:


Ik0 A01 A02 . . .
A0,m−1
A0,m
 0 θIk1 θA12 . . . θA1,m−1

θA1,m


2
2
2
0
0
θ Ik2 . . . θ A2,m−1
θ A2,m 
G := 

 ..

..
..
..
..
..
.

.
.
.
.
.
0
0
...
0
θm−1 Ikm−1
θm−1 Am−1,m
Here ki ≥ 0 are integers satisfying k0 + k1 + · · · + km−1 ≤ n, Iki
denote ki × ki identity matrices over R, and with
P
ki ×kj
km := n − m−1
for all i < j.
i=0 ki we have Aij ∈ R
Proof.
Suppose R C has shape λ. Let ki := λ′m−i − λ′m−i+1 denote the
number of parts of λ equal to m − i. Then
k = rk(R C) = k0 + k1 + · · · + km−1 .
Arrange the basis c1 , . . . , ck of R C as rows of a matrix G ∈ R k ×n
(in that order). Then the first k0 rows of G have height 0, the next
k1 rows have height 1, etc.
Performing Gaussian elimination and permuting columns, if
necessary, we can transform G into the required form.
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Linear Codes
over Finite
Chain Rings
Remarks
• In terms of k0 , k1 , . . . , km−1 , the number of codewords of C is
(q m )k0 (q m−1 )k1 · · · q km−1 = q
Pm−1
i=0
(m−i)ki
.
• The submodules Ci := C[θi ] ∩ θi−1 C of the Ulm-Kaplansky
series of C are visible in G as follows: If we define


Ik0 A01 A02 . . . A0,m−1
A0,m
 0 Ik1 A12 . . . A1,m−1
A1,m 



′
m−1  0
0
I
.
.
.
A
A
k2
2,m−1
2,m  , (⋆)
G := θ

 ..
..
.
.
.
.
..
..
..
.. 

.
.
0
0
... 0
Ikm−1
Am−1,m
then C1 = C[θ] is generated by G′ (which has k0 + · · · + km−1
rows), C2 = C[θ] ∩ θC by the first k0 + · · · + km−2 rows of G′ ,
. . . , and finally Cm = C[θ] ∩ θm−1 C by the first k0 rows of G′ .
• Using the isomorphism θm−1 R = N m−1 ∼
= R/N, we can
consider the Ci ’s as classical linear codes over Fq . (Simply
omit the factor θm−1 in (⋆) and read the rest modulo N.) This
viewpoint was adopted in [2, 3].
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Duality
Thomas
Honold
Linear Codes
over Finite
Chain Rings
For x, y ∈ R n write x · y := x1 y1 + · · · + xn yn .
For S ⊆ R n define the left (resp. right) dual code of S as
⊥
S := {x ∈ R n ; x · y = 0 for all y ∈ S},
S ⊥ := {x ∈ R n ; y · x = 0 for all y ∈ S}.
Remark
We have ⊥ S ≤ R R n regardless of whether S is (one-sided) linear
or not, and similarly for S ⊥ .
Hence we can expect a suitable duality theory only if we define
the double dual of C ≤ R R n as (⊥ C)⊥ .
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Theorem
Let C ≤ R R n be a left linear code over R of shape λ.
1
Thomas
Honold
Linear Codes
over Finite
Chain Rings
The right linear code C ⊥ has complementary shape
(m − λn , m − λn−1 , . . . , m −
λ1).
n
In particular we have |C| · C ⊥ = |R| , and C is free as an
⊥
R-module iff C is free iff rk(C) + rk(C ⊥ ) = n.
2 ⊥ (C ⊥ )
3
=C
C 7→ C ⊥ defines an antiisomorphism between the lattices of
left resp. right linear codes of length n over R, and hence
⊥
⊥
(C ∩ C ′ )⊥ = C ⊥ + C ′ , (C + C ′ )⊥ = C ⊥ ∩ C ′ .
Remark
Parts (2) and (3) are more generally true for quasi-Frobenius
rings.
Proof.
Since C ⊆ ⊥ (C ⊥ ), Part (2) follows from (1), and clearly implies
(3).
(1) It is possible to derive this result working with standard
generator matrices, but the following is more conceptual and will
be needed later.
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Linear Codes
over Finite
Chain Rings
Proof cont’d.
Each y ∈ R n induces a linear map R C → R R, x 7→ x · y.
In this way we obtain a homomorphism from RRn to
C ♯ = Hom(R C, R R)R with kernel C ⊥ .
Any linear map f : R C → R R can be extended to R R n (using, for
example, stacked bases for R C and R R n ) and hence has the form
x 7→ x · y.
This proves RRn /C ⊥ ∼
= C ♯ and that the shapes of C ⊥ , C ♯ are
complementary. The proof of (1) is finished by the following
Lemma.
Lemma
A module R M and its dual M ♯ have the same shape (and in
particular the same number of elements).
Proof.
Lr
Write R M ∼
= i=1 R/N λi , and let x1 , . . . , xr be a basis of R M.
A linear map f : R M → R R is determined once f (x1 ), . . . , f (xr )
have been specified. For f (xi ) we can choose any ai ∈ R of
period dividing θλi , i.e. any ai ∈ N m−λi .
Lr
Lr
(N m−λi )R ∼
(R/N λi )R .
This leads to M ♯ ∼
=
=
i=1
i=1
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Linear Codes
over Finite
Chain Rings
Remark
A self-dual linear code over R (i.e. C = C ⊥ ) necessarily has
self-complementary shape. This gives certain restrictions on the
parameters of C.
Examples
The linear codes C8 and O over Z4 generated by
0
1
B0
B
B0
B
B0
B
B0
B
@0
0
1
2
0
0
0
0
0
1
2
2
0
0
0
0
1
0
2
2
0
0
0
1
0
0
2
2
0
0
1
0
0
0
2
2
0
1
0
0
0
0
2
2
1
1
0C
C
0C
C
0C
C,
0C
C
0A
2
0
1
B0
B
@0
0
are both selfdual with shapes
resp.
The code O is the famous Octacode.
0
1
0
0
0
0
1
0
0
0
0
1
2
1
1
1
1
2
1
3
1
3
2
1
1
1
1C
C
3A
2
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Linear Codes
over Finite
Chain Rings
Weight Spectra of Linear Codes
For linear codes over a proper chain ring R the Hamming
distance dHam is not a good performance parameter. This is due to
the following
Fact
For C ≤ R R n we have dHam (C) = dHam C[θ] .
So C cannot be better than the (usually much smaller) code C[θ]
(which is a classical linear code over R/N ∼
= Fq ).
Proof.
Let c ∈ C with wHam (c) = dHam (C). If c has period θi , then
θi−1 c = (θi−1 c1 , . . . , θi−1 cn ) ∈ C[θ] \ {0}.
ObviouslywHam (θi−1 c) ≤ wHam (c), so we must have equality and
dHam C[θ] = wHam (c) = dHam (C).
Consequence
To produce good linear codes over chain rings, we should assign
larger weights to those elements of R which generate small
ideals. Hence we must be able to distinguish between elements
of R ∗ , N \ N 2 , . . . , N m−1 \ {0}, and {0}.
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Linear Codes
over Finite
Chain Rings
For x = (x1 , . . . , xn ) ∈ R n and 0 ≤ i ≤ m define
ai (x) := {i; 1 ≤ i ≤ n and xi ∈ N i \ N i+1 } .
i.e. ai (x) counts the entries of x which are “exactly” divisible by θi .
Definition
(i) The weight composition of x ∈ Rn is the (m + 1)-tuple of
integers a0 (x), a1 (x), . . . , am (x) .
(ii) The weight enumerator of S ⊆ R n is the polynomial
X a (x)
a (x)
AS (X0 , . . . , Xm ) :=
X0 0 · · · Xmm ∈ C[X0 , . . . , Xm ].
x∈S
Example
The Z4 -linear code C ≤ Z44 of shape


1 1 1 1
0 2 2 0
0 0 2 2
has weight enumerator
AC (X0 , X1 , X2 ) = 8X04 + X14 + 6X12 X22 + X24 .
generated by
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Linear Codes
over Finite
Chain Rings
Isomorphism of Linear Codes
Definition
(i) Two linear codes C1 , C2 ≤ R R n are said to be linearly
isomorphic if there exists a monomial matrix A ∈ R n×n (a
matrix having exactly one nonzero entry aij ∈ R × in each row
and column) such that C2 = {xA; x ∈ C1 }.
(ii) C1 and C2 are said to be semilinearly isomorphic if
C2 = {σ(x)A; x ∈ C1 } with A as in (i) and σ ∈ Aut R.
Remark
The monomial transformations R n → R n , x 7→ xA, are exactly
those automorphisms of the module R R n which preserve the
weight composition. A similar remark applies to the
“semimonomial” transformations of (ii).
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Linear Codes
over Finite
Chain Rings
Multisets in PG(MR )
Deviating a little from our previous notation, we denote the lattice
of submodules of an arbitrary (not necessarily free) module MR by
PG(MR ) and the set of nonzero cyclic submodules of PG(MR ) by
P. The elements of P are called points. A point xR is degenerate
if xR is not free, i.e. |xR| < |R|.
Definition
A multiset in PG(MR ) is a mapping K : P → N0 .
The mapping K is extended to the power set 2P by defining
X
K(Q) =
K(P) for Q ⊆ P.
P∈Q
For P = xR ∈ P the integer K(P) ≥ 0 is called the multiplicity of P
in K.
P
The integer K(P) = P∈P K(P) is called the cardinality of K.
The support of K is defined as supp K := P ∈ P; K(P) > 0 , the
P
hull of K as the module hKi := xR∈supp K Rx ≤ MR , and the
shape of K as the shape of hKi.
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Linear Codes
over Finite
Chain Rings
Definition
Two multisets K, K′ in PG(MR ) resp. PG(MR′ ) are said to be
equivalent if there exists asemilinear bijection φ : hKi → hK′ i
satisfying K(P) = K′ φ(P) for every point P = xR ≤ hKi.
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Linear Codes
over Finite
Chain Rings
Codes Associated to Multisets
Suppose C ≤ R R n is any left linear code of length n over R.
Take any generator matrix G ∈ R k ×n of C (i.e. C = {xG; x ∈ R k },
but the rows of G need not be independent).
Write G = (g1 | . . . |gn ) and define a multiset K in PG(RRk ) by
setting
K(P) := |{j; 1 ≤ j ≤ n ∧ P = gj R}|
for P = gR ∈ P.
Definition
We say that the multiset K and the code C are associated.
Remarks
• If C has full length (i.e. no universal zero coordinate), then
K(P) = n.
• One can restrict the definition to multisets in PHG(RRk ) (i.e.
allow only nondegenerate points) and so-called fat linear
codes.
• In general the relation C 7→ K is many-to-many.
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Linear Codes
over Finite
Chain Rings
The Correspondence Theorem
Theorem
For every multiset K of cardinality n in PG(RRk ) there exists an
associated linear code C ≤ R R n (which is necessarily of full
length). Two multisets K1 in PG(RRk1 ) and K2 in PG(RRk2 )
associated with full-length linear codes C1 and C2 over R,
respectively, are equivalent if and only if the codes C1 and C2 are
semilinearly isomorphic.
Proof.
The first part is easy: Choose a sequence g1 , . . . , gn over R k
such that the multiplicities in K of all points match those in the
sequence g1 R, . . . , gn R, and define C ≤ R R n as the code
generated by (g1 | . . . |gn ).
The proof of the second part is quite technical and can be found
in [?].
Remark
The theorem holds for a more general class of rings (including
finite Frobenius rings).
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Linear Codes
over Finite
Chain Rings
How to Compute the Parameters of C
from Properties of K?
Parameters of C: length, shape (cardinality), weight enumerator
The length of C equals K(P)).
For the shape use the following
Proposition
A multiset K in PG(RRk ) and an associated code C have the same
shape. In particular, |hki| = |C|.
Proof.
Let G ∈ R k ×n be such that
C = {xG; x ∈ R k }
hki = D : = {Gy; y ∈ R n }
(Left row space of G)
(Right column space of G)
Define f : R n → C, x 7→ xG.
C = Im f ∼
= D♯
= R n / Ker f = R n /⊥ D ∼
Hence C, D ♯ and D all have the same shape.
(as left R-modules)
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Linear Codes
over Finite
Chain Rings
Remark
The following example shows that the left row space and the left
column space of a matrix G ∈ R k ×n usually have different shape.
Example
Let a, b ∈ R such that ab 6= ba and
1 a
G=
.
b ab
a
Left column space of G: R( b1 ) + R( ab
) = R( b1 )
Left row space of G: R(1, a) + R(b, ab) ) R(1, a).
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Linear Codes
over Finite
Chain Rings
G. J. Janusz.
Separable algebras over commutative rings.
Transactions of the American Mathematical Society,
122:461–479, 1966.
G. H. Norton and A. Sălăgean.
On the Hamming distance of linear codes over a finite
chain ring.
IEEE Transactions on Information Theory,
46(3):1060–1067, May 2000.
G. H. Norton and A. Sălăgean.
On the structure of linear and cyclic codes over a finite
chain ring.
AAECC, 10:489–506, 2000.
R. Raghavendran.
Finite associative rings.
Compositio Mathematica, 21:195–229, 1969.
J. A. Wood.
Linear Codes
over Finite
Chain Rings—
Algebraic
Theory
Thomas
Honold
Linear Codes
over Finite
Chain Rings
Duality for modules over finite rings and applications to
coding theory.
American Journal of Mathematics, 121(3):555–575,
1999.