On Binary Sequences of Period n
with Optimal Autocorrelation *
= pm -
1
Tor Helleseth 1 and Kyeongcheol Yang2
1
2
Dept. of Informatics, University of Bergen, Hlilyteknologisenteret,
N-5020 Bergen, Norway
Dept. of Electronic and Electrical Engineering, Pohang University of Science
and Technology, Pohang, Kyungbuk 790-784, Korea
Summary. Binary sequences of period n = pm - 1 for an odd prime p are introduced in [4] by taking the characteristic sequence of the image set of the polynomial
(z + 1)d + azd + b over the finite field F1Jm of pm elements. It was shown in [4] that
they are (almost) balanced and have optimal autocorrelation in the case where the
polynomial can be transformed into the form Z2 - c. In this paper, we show that
the sequences are (almost) balanced and have optimal autocorrelation in the case
of d
(pm + 1)/2, a
(_1)d-l and b
±l. Furthermore, we show that they
are equivalent to the Lempel-Cohn-Eastman sequence in [2] in the balanced case.
We also give a direct proof of the autocorrelation property of the Lempel-CohnEastman sequence and discuss its linear complexity.
=
1
=
=
Introduction
Given a binary sequence {s(t)} of period n, let D be the difference between
the number of l's and that ofO's in its period. {s(t)} is said to be balanced if
D is zero when n is even, and if D is one when n is odd. It is said to be almost
balanced if D is 2. The number D will be referred to as the discrepancy of
the sequence {s(t)}.
The periodic autocorrelation function R( r) of {s (t)} is defined by
R(r)
n-l
= L(_l)s(t)+B(t+
T
)
t=o
where the sum t + r is computed mod n. Clearly, R(O) = n. It is also easily
checked that R( r) = n (mod 4) and
n-l
LR(r)
= D2•
T=O
* This work was supported in part by the Norwegian Research Council and by
the BK21 Program of the Ministry of Education of Korea and the Com2Mac of
Postech.
T. Helleseth et al. (eds.), Sequences and their Applications
© Springer-Verlag London Limited 2002
210
T. Helleseth and K. Yang
The sequence {s(t)} of even period n is said to have optimal autocon-elation
if
R(r) _ {O or -4 if n = 0 (mod 4),
2 or -2 if n = 2 (mod 4)
for any ri'O (mod n).
Let {Sl(t)} and {S2(t)} be two binary sequences of period n. IT there
exists a constant r such that S2(t) = Sl(t + r) for any t, then {S2(t)} is
called a cyclic shift of {Sl (t)}. IT there is an r with (r, n) = 1 such that
S2(t) = Sl(rt) for all t, then {S2(t)} is called the r-decimation of {Sl(t)}.
If S2(t) = Sl (t) + 1 (mod 2) for all t, then {S2(t)} is called the complement
of {Sl(t)}. If {S2(t)} is a cyclic shift of any of decimations or complement
of {Sl (t)} or their combination, then they are said to be equivalent. Two
equivalent sequences have the same distribution of autocorrelation and the
same discrepancy.
For an odd prime p, let a be a primitive element of the finite field F of
pm elements and F· = F\{O}. For a nonempty S in F·, let {s(t)} be the
sequence of length n = pm - 1, defined by
s(t)
= {I
if at
E.S,
o otherwise.
The sequence {s(t)} has period n = pm - 1 and is called the characteristic
sequence of S in F·.
Binary sequences of period n = pm - 1 for an odd prime p are introduced
in [4] by taking the characteristic sequence of the image set of the polynomial
(z + l)d + az d + b over F. It was shown in [4] that they are (almost) balanced
and have optimal autocorrelation in the case where the polynomial can be
transformed into the form Z2 - c.
In this paper, we show that the sequences are (almost) balanced and have
optimal autocorrelation in the case of d = (pm + 1)/2, a = (_I)d-1 and
b = ±l. Furthermore, we show that they are equivalent to the Lempel-CohnEastman sequence in [2]. We also give a direct proof of the autocorrelation
property of the Lempel-Cohn-Eastman sequence and discuss its linear complexity.
2
Cyclotomic Numbers and Quadratic Character
From now on, we let F be the finite field of pm elements and F·
and let a be a primitive element of F. For i = 0 and 1, let
= F\ {O}
In other words, Co (C1 , resp.) is the set of squares (non-squares, resp.) in F·.
For fixed i and i, the cyclotomic number (i,i) is defined to be the number
On Binary Sequences with Optimal Autocorrelation
211
of solutions to the equation
1 +Zi
= Zj
where Zi E Gi , Zj E Gj . From the theory of cyclotomic numbers (cr. [6]), it is
easy to determine (i,i) as in the following lemma.
Lemma 1. (Lemma 6, [6]) The cyclotomic numbers are given as follows:
(a) Ifn == 0 (mod 4), then (0,0)
(pm -1)/4.
(b) If n == 2 (mod 4), then (0,0)
(pm + 1)/4.
= (pm
- 5)/4; (0,1)
= (1,0) = (1,1) =
= (1,0) = (1,1) = (pm -
Let I be the indicator function such that 1(0) = 1 and I(x)
Let X be the quadratic character of F, defined by
X(z) =
3)/4; (0,1)
=
= 0 for x =f. o.
+1 if Z EGo,
-1 ~f Z E Gl ,
{
o If Z = O.
The sum r(a) = LZEF x(az 2 + 1) can be easily determined using the theory
of cyclotomic numbers.
Lemma 2. For a E F,
r(a) -
{p-x(a)
m
ila =0
'I,
otherwise.
Proof. Clearly, r(O) = pm. Assume that a
numbers n+, no, and n_ by
=f. O. First, we define the three
= I{ Z E F IX( az 2 + 1) = +1} I,
= I{z E F Ix(az 2 + 1) = O}I,
= I{z E F I x(az 2 + 1) = -I}I.
Note that n+ + no + n_ = pm.
Case a E Go: Note that Z = 0 is a solution to x(az 2 +1) = +1. When Z runs
n+
no
n_
through F*, az 2 takes each element in Go twice. Therefore, n+ = 1 + 2(0,0).
Note that x(az 2 + 1) = 0 if and only if az 2 + 1 = 0 and that az 2 + 1 = 0 has
exactly two solutions in F if and only if -1 E Co. Hence,
n
o
= {2
if n
0 if n
=0
=2
(mod 4),
(mod 4)
= 1 + (_I)n/2.
Since n_ = pm - (n+ + no), we have
r(a) = n+ . (+1) + no ·0 + n_ . (-1)
= 4(0,0) + 3 + (_I)n/2 _ pm
= -1
T. Helleseth and K. Yang
212
and so r(a) = -x(a).
Case a E C1 : Similarly, it is easily shown that n+ = 2(1,0)
no = 1- (_I)n/2. Therefore, we have r(a) = +1 = -x(a).
+1
and
0
More generally (d. [5]),
Lemma 3. For a, band c E F,
~ ( 2 b
) {I(b)X(C)pm
~ X az + z + C = x(a) (pm I(b2
zEF
3
_
if a = 0,
4ac) - 1) otherwise.
Lempel-Cohn-Eastman Sequences
11 i = 0, I, ... , (pm - 1) /2 - 1}. The characteristic sequence
{SL(t)} of period n = pm - 1 is defined by
Let S
= {a2i+1 t)
SL ( =
{10 otherwise.
if at ES,
This sequence is referred to as the Lempel-Cohn-Eastman Sequence. In [2],
it was shown that {SL(t)} is balanced and has optimal autocorrelation.
Using the indicator function I and the quadratic character X, {sL{t)} can
be expressed in a closed form. First, we note that
sL(t)
= 1 {:} at = a 2i+l -
1 for some i
{:} X(at + 1) = -1 and at + 1 t6 o.
It is ah!o easily checked that
1 + x(a t + 1) = 2(1 - SL(t)) - I(a t + 1).
Therefore, we have
SL(t)
= 21 (1- I(a t + 1) -
x(at + 1)).
(1)
It is possible to give a direct proof on the autocorrelation properties of
{SL(t)} in the following, which is different from that in [2].
Theorem 4. The autocorrelation of {sL{t)} is given as follows: If n
(mod 4), then
RL{r)
= {-4
if x(aT~ = 1 and X(1- aT)
o otherunse.
= -I,
If n = 2 (mod 4), then
RL(r)
={
2 if x(aT~ = 1 and X(1- aT)
- 2 otherunse.
= 1,
=0
On Binary Sequences with Optimal Autocorrelation
213
Proof. Since the ± 1 version of {s d t)} is given by
(_l)8L(t)
= 1- 2sL(t) = I((i + 1) + x(at + 1),
we have
n-I
RL(r)
=L
(I(a HT
+ 1) + x(a HT + 1)) (I(a t + 1) + x(at + 1))
t=o
= x(l- aT) + X(l- a- T) + Lx((at + l)(at+T + 1))
t
by the properties of I and X. For simple notation, let z
Then the last term leads to
Lx((at + l)(a HT
= at
and a
= aT.
+ 1)) = L x(az 2 + (a + l)z + 1)
t
zEF*
=L
x(az2
+ (a + l)z + 1) - X(l)
zEF
= -x(a) -
X(l)
by Lemma 3. Therefore we have
RL(r)
= X(l = X(l -
+ X(l - a-I) - x(a) - x(l)
a) (1 + (-It/ 2 x(a)) - x(a) - X(l)
a)
which leads to the theorem.
4
o
Sequences from the Image sets of Some Polynomials
In [4], it was shown that polynomials introduced in [3] and [1] can be generalized to generate binary sequences of period pm - 1 with optimal autocorrelation for any prime p and an integer m. For a, bE F and a positive integer
d, consider the subset of F* given by
{x I x
= (z + l)d + az d + b,z E F}\{O}
(2)
and its characteristic sequence {s(t)}. For d = 2,3,4 or d = pm - pm-I - 1,
there are some combinations of p, a, and b such that the polynomial (z +
l)d + az d + b can be transformed ino the form Z2 - c. In these cases, the corresponding characteristic sequences are (almost) balanced binary sequences
with optimal autocorrelation. In particular, they are all equivalent to the
Lempel-Cohn-Eastman sequence in the balanced case.
For d = (pm + 1)/2, a = (_l)d-l and b = ±1, it was shown in [4] that
the corresponding characteristic sequence is balanced when n = 0 (mod 4)
and is almost balanced when n = 2 (mod 4). Here we will give an expression
T. Helleseth and K. Yang
214
of {s(t)} and its derived sequences in a closed form and show that they have
optimal autocorrelation.
For any z E F*, we have zd-1 = x(z) since d = (pm - 1)/2 + 1. Therefore,
the polynomial j(z) = (z+l)d+( _1)d-1 zd+b over F can be simply expressed
as
j(z)
= (x(z + 1) + x(-z))z + x(z + 1) + b.
(3)
Lemma 5. Let 8+ and 8_ be the image sets in F* defined by j(z) with
b = +1 and b = -1 in (9), respectively. Then
8+
8_
= {2} U {z Iz/2 E Co, 1- z/2 E Co} U {z Iz/2 E C1, 1- z/2 E Ct},
= {-2}U {zlz/2 E Co, 1 +z/2 E Co} U {zlz/2 E C1,1 +z/2 E Ct}.
Furthermore,
°
18 I = 18 I = { n/2
if n = (mod 4),
+
n/2+ 1 ifn = 2 (mod 4).
°
Proof. First, we consider 8+. Since -1 E Co for n = (mod 4), we have
8+
= {2} U {2z + 21 z E Co, 1 + z E Co} U {-2z I z E C1, 1 + z E Ct}
= {2} U {z I z/2 E Co, 1- z/2 E Co} U {z Iz/2 E C1, 1 - z/2 E Ct}
and 18+1 = 1 + (0,0)
-1 E C1 • Therefore,
+ (1,1) =
(pm - 1)/2. For n = 2 (mod 4), we have
= {2} U {2z + 21 z E C1, 1 + z E Co} U {-2z I z E Co, 1 + z E Ct}
= {2} U {z I z/2 E Co, 1 - z/2 E Co} U {z Iz/2 E C1, 1 - z/2 E Ct}
and 18+1 = 1 + (1,0) + (0,1) = (pm + 1)/2. Similar works hold for 8_.
0
8+
Theorem 6. Let {s+(t)} and {s_(t)} be the characteristic sequences of 8+
and 8_, respectively. Then
s+(t) = ~ (1 + 1(1 + o:-(t-'Y)) + (-1)n/2 X(1 + o:-(t-'Y))) ,
s_(t) = ~ (1 + 1(1 + o:-(t-A)) + (-1)n/2 X(1 + o:-(t-A)))
where
0:1'
= -2 and o:A
= 2.
Proof. First, consider s+(t). Note that s+(t)
Then it can be eaily checked that
(1 + X(o:t /2}) (1 + X(l- o:t /2})
= 1 if and only if o:t E 8+.
+ (1- X(o:t /2}) (1- X(l- o:t /2})
=
4 if o:t E 8(1)\{2},
2 if o:t = 2,
{
otherwise
°
On Binary Sequences with Optimal Autocorrelation
215
and this leads to
2 + 2X((1 - at /2)a t /2)
Note that l(at - 2)
2a- t ). Therefore,
s+(t)
= 4s+(t) -
= 1(1 -
= ~ (1 + 1(1 -
21(at - 2).
2at -t) and X((l - at /2)a t /2)
= (-1)n/2 X(1-
2a- t ) + (-1)n/2 X(1 - 2a- t ))
Similar works hold for the case {s_(t)}.
0
= 0 (mod 4), the sequences {s+(t)} and {s_(t)} are
equivalent to the Lempel-Cohn-Eastman sequence {s£(t)}. More precisely,
Corollary 1. For n
s+(t)
L(t)
where a"Y
= 1- SL(r = 1- s£(,\ -
t),
t)
= -2 and a A = 2.
o
Proof. It follows from and (1) and Theorem 6.
= 2 (mod 4), the sequences {s+(t)} and {L(t)} are
almost balanced and have optimal autocorrelation given by
Corollary 8. For n
R (r)=R (r)={ 2 ifx(aT~=-landX(l-aT)=-l,
'+
,-2 otherwISe.
Proof. The sequences {s+ (t)} and {s _(t)} are almost balanced in the case
of n = 2 (mod 4) by Lemma 5. By Theorem 9 and Lemma 3, it can be easily
checked that
R,+ (r)
= R,_ (r) = x(a)x(1- a) -
X(l - a) - x(a) - 1
o
where a = aT.
= S+\{2} andS~ = S_\{-2}. Let {s+(t)}
be their characteristic sequences, respectively. Then
Theorem 9. LetS+
s+ (t)
= ~ (1 -
and{s~(t)}
1(1 + a-(t-"Y)) + (-1)n/2 X(1 + a-(t-"Y))) ,
s~(t) = ~ (1- 1(1 + a-(t-A)) + (-1)n/2 X(1 + a-(t-A)))
where a"Y = -2 and a A = 2.
Proof. In the case of S+, it can be easily checked that
(1 + x(at /2)) (1 + X(l- at /2)) + (1- x(a t /2)) (1- X(l= 4s+(t) + 2I(at - 2).
The case S+ is also similar.
ci /2))
o
216
T. Helleseth and K. Yang
= 0 (mod 4), the sequences {s+(t)} and
almost balanced and have optimal autocomlation given by
Corollary 10. For n
_ R () _
R,. ()
r - ,.- r +
{-40
{s~(t)}
are
if x(aT) = 1 and X(I- aT) = 1,
.
othenmse.
Proof. The sequences {s+(t)} and {s~(t)} are almost balanced in the case
of n = 0 (mod 4) by Lemma 5. By Theorem 9 and Lemma 3, it can be easily
checked that
R,+ (r)
= R,_ (r) = -(x(a) + I)(X(I- a) + 1)
where a = aT.
o
= 2 (mod 4), the sequences {s+(t)} and {s~(t)} are
equivalent to the Lempel-Cohn-Eastman sequence {SL(t)}. More precisely,
Corollary 11. For n
= sd'}' s~(t) = SL(A s+(t)
where
a"Y
t),
t)
= -2 and a..\ = 2.
Proof. It follows from (1) and Theorem 9.
5
o
Linear Complexity of Lempel-Cohn-Eastman
Sequences
Any binary sequence {s(t)} of period n satisfies a linear recursion, say,
s(t + m) = am-ls(t + m - 1) + ... + als(t + 1) + aos(t)
(4)
for all t, where ai E {a, I} and ao 1:- O. Here, m is called the degree of the
linear recusion. There are an infinite number of linear recursions of {s(t)},
including s(t + n) = s(t) for all t. The degree of the minimum-degree linear
recursion is called the linear complexity or linear span of {s(t)}.
For convenience, assume that (4) is the minimum-degree linear recursion
of {s(t)}. Then the polynomial a(x) = xm + am_1Xm - 1 + ... + ao is called
the characteristic polynomial of {s(t)}. If we write G(x) =E:o s(t)xt , it is
easily checked that
G(x)
(t) t
= ",n-l
L..t=o 8 X
n
x +l
and G(x) can be reduced to b(x)fa(x), where b(x) is a polynomial of degree
< m and relatively prime to a(x).
Theorem 12. Ifn = 0 (mod 4), the linear complexity of {SL(t)} is at most
n -1.
On Binary Sequences with Optimal Autocorrelation
217
Proof. Since {sL(t)} is balanced, it has n/2 l's in one period. This implies
that
n-l
I: sL(t)xt
t=o
is divisible by x + 1. On the other hand, x n + I has x + I as its factor.
Therefore, the degree of the charcteristic polynomial is at most n - 1.
0
Theorem 13. If n = 2p for an odd prime p and 2 is a primitive element
mod p, then the linear complexity of {sL(t)} is n.
Proof. Clearly, zn + 1 = (zn/2 + 1)2. If we let C(x) = I:~:~ SL(t)x t , it is
sufficient to show that C(f3) =f- 0 for any f3 such that f3n/2 = 1.
C(f3)
=
n/2-1
I: (sL(t) + sL(t + n/2))f3t
t=o
= l-X (2)
2
n/2-1
+
(
~ 2-X I
LJ
t=o
(
t)
+ a t) -xl-a
f3t
2
Note that the minimal poynomial of f3 over F2 is x p Therefore, C(f3) = 0 implies that X(2) = -1 and
X(l
+ at) + X(1 -
at)
1
+ x p - 2 + ... + x + 1.
=0
for any t, 1 :$ t :$ n/2 - 1, which is impossible.
o
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