AUTOCORRELATION VALUES AND LINEAR COMPLEXITY OF ...

AUTOCORRELATION VALUES AND LINEAR COMPLEXITY OF GENERALIZED CYCLOTOMIC SEQUENCE OF ORDER FOUR, AND CONSTRUCTION OF CYCLIC CODES

arXiv:1510.05467v1 [cs.IT] 19 Oct 2015

PRITI KUMARI AND PRAMOD KUMAR KEWAT

Abstract. Let n1 and n2 be two distinct primes with gcd(n1 − 1, n2 − 1) = 4. In this paper, we compute the autocorrelation values of generalized cyclotomic sequence of order 4. Our results show that this sequence can have very good autocorrelation property. We determine the linear complexity and minimal polynomial of the generalized cyclotomic sequence over GF(q) where q = pm and p is an odd prime. Our results show that this sequence possesses large linear complexity. So, the sequence can be used in many domains such as cryptography and coding theory. We employ this sequence of order 4 to construct several classes of cyclic codes over GF(q) with length n1 n2 . We also obtain the lower bounds on the minimum distance of these cyclic codes.

1. Introduction There are different kinds of cyclotomic sequences and they have quite good randomness properties. Example of such sequence is the generalized cyclotomic sequence. The generalized cyclotomic sequences of order two has several good randomness properties like large linear span and good autocorrelation property. Whiteman [15] studied the generalized cyclotomy of order two and four only for the purpose of searching the residue difference sets. Whiteman did not mention anything about the application of this generalized cyclotomy in the sequences. Ding [5] introduced the generalized cyclotomy of order two in sequences and its coding properties were studied in [7] and [13]. In this correspondence we calculate the exact autocorrelation values of certain generalized cyclotomic sequences of order four. Then we discuss how to choose the parameters in order to ensure good autocorrelation property for these sequences. We also show that the characteristic sets of these sequences are almost difference sets under certain conditions. Let n1 , n2 be two distinct odd primes with gcd(n1 −1, n2 −1) = 4. We determine the linear complexity and minimal polynomial of the generalized cyclotomic sequences of order four over GF(q), where q be a power of a prime. The linear span of this generalized cyclotomic sequence takes the value n1 n2 − 1, (n1 − 1)n2 , (n2 − 1)n1 and (n1 − 1)(n2 − 1) depending on the different values of n1 and n2 such that gcd(n1 − 1, n2 − 1) = 4. Our results show that for every value of n1 and n2 , these sequences possesses large linear complexity. An [n, k, d] linear code C over a finite field GF(q) is a k−dimensional subspace of the vector space GF(q)n with the minimum distance d. A linear code C is a cyclic code if the cyclic shift of a codeword in C is again a codeword in C, i.e., if (c0 , · · · , cn−1 ) ∈ C then (cn−1 , c0 · · · , cn−2 ) ∈ C. Let gcd(n, q) = 1. We consider the univariate polynomial ring GF(q)[x] and the ideal I = hxn − 1i of GF(q)[x]. We denote by R the ring GF(q)[x]/I. We can consider a cyclic code of length n over GF(q) as an ideal in R via the following correspondence GF(q)n → R,

(c0 , c1 , · · · , cn−1 ) 7→ c0 + c1 x + · · · + cn−1 xn−1 .

Then, a linear code C over GF(q) is a cyclic code if and only if C is an ideal of R. Since R is a principal ideal ring, if C is not trivial, there exists a unique monic polynomial g(x) dividing xn − 1 in GF(q)[x] Key words and phrases. Cyclic codes, finite fields, cyclotomic sequences. 1

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Priti Kumari and P.K. Kewat

and C = hg(x)i. The polynomials g(x) and h(x) = (xn − 1)/g(x) are called the generator polynomial and the parity-check polynomial of C respectively. If the dimension of the code C is k, the generator polynomial has degree n − k. An [n, k, d] cyclic code C is capable of encoding q−ary messages of length k and requires n − k redundancy symbols. The total number of cyclic codes over GF(q) and their construction are closely related to the cyclotomic cosets modulo n. One way to construct cyclic codes over GF(q) with length n is to use the generator polynomial

where S(x) =

n−1 P i=0

xn − 1 , gcd(xn − 1, S(x))

(1)

si xi ∈ GF(q)[x] and s∞ = (si )∞ i=0 is a sequence of period n over GF(q). The cyclic

code Cs generated by the polynomial in Eq.(1) is called the cyclic code defined by the sequence s∞ , and the sequence s∞ is called the defining sequence of the cyclic code Cs . Cyclic codes have been studied in a series of papers and a lot of progress have been accomplished (see, for example [2], [8], [10], [12] and [14]). Ding [7] and Sun et al.[13] constructed number of classes of cyclic codes over GF(q) with length n = n1 n2 from the two-prime Whiteman’s generalized cyclotomic sequence of order 2 and 4 respectively and gave the lower bounds on the minimum weight of these cyclic codes under certain conditions. We employ the first class two-prime Whiteman’s generalized cyclotomic sequence of order 4 to construct several classes of cyclic codes over GF(q). We also obtain the lower bounds on the minimum distance of these cyclic codes. 2. Preliminaries In this section, we present basic notations and results of residue difference set, Whiteman’s cyclotomy and sequences based on Whiteman’s generalized cyclotomic classes. ∞

2.1. Linear complexity and minimal polynomial. If (si )i=0 is a sequence over a finite field GF(q) and f (x) is a polynomial with coefficients in GF(q) given by f (x) = c0 + c1 x + · · · + cL−1 xL−1 , then we define f (E)sj = c0 sj + c1 sj−1 + · · · + cL−1 sj−L+1 , where E is a left shift operator defined by Esi = si−1 for i ≥ 1. Let sn be a sequence s0 s1 · · · sn−1 of length n over a finite field GF(q). For a finite sequence, the n is finite; for a semi-infinite sequence, the n is ∞. A polynomial f (x) ∈ GF(q)[x] of degree 6 l with c0 6= 0 is called a characteristic polynomial of the sequence sn if f (E)sj = 0 for all j with j ≥ l. For every characteristic polynomial there is a least l ≥ deg(f ) such that the above equation hold. The smallest l is called the associated recurrence length of f (x) with respect to the sequence sn . The characteristic polynomial with smallest length is known as minimal polynomial of the sequence sn and the associated recurrence length is called the linear span or linear complexity of the sequence sn . If a semi-infinite sequence s∞ is periodic, then its minimal polynomial is unique if c0 = 1. The linear complexity of a periodic sequence is equal to the degree of its minimal polynomial. For the periodic sequences, there are few ways to determine their linear spans and minimal polynomials. One of them is given in the following lemma. Lemma 1. [11] Let s∞ be a sequence of a period n over GF (q). Define S n (x) = s0 + s1 x + · · · + sn−1 xn−1 ∈ GF(q)[x].

Then the minimal polynomial ms of s∞ is given by xn − 1 . gcd(xn − 1, S n (x))

(2)

Consequently, the linear span Ls of s∞ is given by

Ls = n − deg(gcd(xn − 1, S n (x))).

(3)

Autocorrelation values and Linear complexity of generalized cyclotomic sequences and construction of cyclic codes

3

2.2. Almost difference sets. Let C be a k-subset of an Abelian group (A, +) of order n. The set C is called a difference set of A if dC (a) = λ for every nonzero element a of A, where dC (a) is the difference function defined by dC (a) = |C ∩ (C + a)|, C + a = {c + a : c ∈ C}. The integers n, k and λ are called the parameters of the set and satisfy the relation k(k − 1) = λ(n − 1). Let C be a k-subset of an Abelian group (A, +) of order n. The set C is an (n, k, λ, t) almost difference set of A if dC (a) takes on λ altogether t times and λ + 1 altogether n − 1 − t times when a ranges over all nonzero elements of A. We refer [1, 9] for the detailed informations on almost difference sets. The parameters n, k, λ and t of almost difference sets of A satisfy the relation k(k − 1) = tλ + (n − 1 − t)(λ + 1). If n is an odd integer and C be a k subset of Zn , following Ding [9], we say that C is an (n, k, λ) almost difference set if t = (n − 1)/2. In this case we have the following relation k(k − 1) = (2λ + 1)(n − 1)/2. It is obvious that every odd integer n must be of the forms 4f + 1 or 4f − 1 for some f ∈ Z. If n = 4f − 1, then (n − 1)/2 = 2f − 1 is an odd integer, this implies that (2λ + 1)(n − 1)/2 must be odd. This contradicts the above relation. Therefore, if Zn has an (n, k, λ) almost difference set, then n must be of the form 4f + 1. In this paper, we present one class of almost difference set. 2.3. The Whiteman’s generalized cyclotomic sequences and its construction. An integer a is called a primitive root modulo n if the multiplicative order of a modulo n, denoted by ordn (a), is equal to φ(n), where φ(n) is the Euler phi function and gcd(a, n) = 1. Let n1 and n2 be two distinct odd prime numbers, define n = n1 n2 , d = gcd(n1 − 1, n2 − 1) = 4 and e = (n1 − 1)(n2 − 1)/4. From the Chinese Remainder Theorem, there are common primitive roots of both n1 and n2 . Let g be a fixed common primitive root of both n1 and n2 . Let u be an integer satisfying u ≡ g (mod n1 ), u ≡ 1 (mod n2 ).

(4)

Whiteman [15] proved that Z∗n = {g s ui : s = 0, 1, · · · , e − 1; i = 0, 1, 2, 3}.

where Z∗n denotes the set of all invertible elements of the residue class ring Zn and e is the order of g modulo n. The Whiteman’s generalized cyclotomic classes Wi of order 4 are defined by Wi = {g s ui (mod n) : s = 0, 1, · · · , e − 1}, i = 0, 1, 2, 3. The classes Wi , 0 ≤ i ≤ 3 give a partition of Z∗n , i.e., Z∗n = ∪3i=0 Wi , Wi ∩Wj = ∅ for i 6= j; i, j = 0, 1, 2, 3. Let P = {n1 , 2n1 , 3n1 , · · · , (n2 − 1)n1 }, Q = {n2 , 2n2 , 3n2 , · · · , (n1 − 1)n2 }, C0∗ = {0} ∪ Q ∪ W0 ∪ W1 , C1∗ = P ∪ W2 ∪ W3 ,

C0 = {0} ∪ Q ∪ W0 ∪ W2 and C1 = P ∪ W1 ∪ W3 .

It is easy to see that if d > 2, then C0 6= C0∗ and C1 6= C1∗ . Now, we introduce two kinds of Whiteman’s generalized cyclotomic sequences of order 4 (see [3]). n−1 of order 4 and Definition. The two-prime Whiteman’s generalized cyclotomic sequence λ∞ = (λi )i=0 period n, which is called two-prime WGCS-I, is defined by ( 0, if i ∈ C0 λi = (5) 1, if i ∈ C1 .

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Priti Kumari and P.K. Kewat

The set C1 ⊆ Zn is called the characteristic set of the sequence λ∞ . n−1 The two-prime Whiteman’s generalized cyclotomic sequence s∞ = (si )i=0 of order 4 and period n, which is called two-prime WGCS-II, is defined by ( 0, if i ∈ C0∗ si = 1, if i ∈ C1∗ . The set C1∗ ⊆ Zn is called the characteristic set of the sequence s∞ .

The cyclotomic numbers corresponding to these cyclotomic classes are defined as (i, j)d = |(Wi + 1) ∩ Wj |, where 0 ≤ i, j ≤ 3.

Additionally, for any t ∈ Zn , we define

d(i, j; t) = |(Wi + t) ∩ Wj |,

where Wi + t = {w + t|w ∈ Wi }. By a well-known theorem of Gauss, we have exactly two representations of n in the form n = a2 + 4b2 , n = a′2 + 4b′2 (a ≡ a′ ≡ 1 (mod 4))

(6)

where the sign of b indeterminate.

2.4. Properties of Whiteman’s cyclotomy of order 4. In this subsection, we summarize number of properties of Whiteman’s generalized cyclotomy of order d = gcd(n1 − 1, n2 − 1) = 4.

Lemma 2. [15] The sixteen cyclotomic numbers (i, j); i, j = 0, 1, 2, 3 depend solely upon one of two decompositions in Eq.(6): Table 1. {(n1 − 1)(n2 − 1)}/16 even 0

1

2

3

0 A

B

C

D

1

E

E

D

2 A

E

3

D

E

Table 2. {(n1 − 1)(n2 − 1)}/16 odd 0

1

2

3

0

A

B

C D

B

1

B

D

E

E

A

E

2

C

E

C

E

B

E

3 D

E

E

B

2 −1) is even or odd, the relation between the 16 cyclotomic numbers is given by Table When (n1 −1)(n 16 1 or Table 2 respectively. Thus, there are five possible different cyclotomic numbers in these cases, 2 −1) is even, then in Table 1, 8A = −a + 2M + 3, 8B = −a − 4b + 2M − 1, 8C = i.e., if (n1 −1)(n 16 2 −1) 3a + 2M − 1, 8D = −a + 4b + 2M − 1, 8E = a + 2M + 1, and if (n1 −1)(n is odd, then in Table 2, 16 8A = 3a + 2M + 5, 8B = −a + 4b + 2M + 1, 8C = −a + 2M + 1, 8D = −a − 4b + 2M + 1, 8E = a + 2M − 1 where M = (n1 −2)(n4 2 −2)−1 .

A change in the choice of g may lead to the replacement of a and b in Lemma 2 by a′ and b′ respectively. The proof of the following lemma follows from the Theorem 4.4.6 of [4]. Lemma 3. Let the notations be same as before and t 6= 0. We have  (n −1)(n −1) 1 2  , i 6= j, t ∈ P ∪ Q  16   (n −1)(n  1 2 −1−d) , i = j, t ∈ P, t ∈ /Q 16 d(i, j; t) = (n1 −1−d)(n2 −1)  , i = j, t ∈ Q, t ∈ /P  16    ′ ′ (i , j )d for some (i′ , j′ ), otherwise.

Autocorrelation values and Linear complexity of generalized cyclotomic sequences and construction of cyclic codes

5

Lemma 4. Let the symbols be defined as before. The following four statements are equivalent: (1) −1 ∈ W2 . 2 −1) (2) (n1 −1)(n is even. 16 (3) One of the following ( ( sets of equations is satisfied: n1 ≡ 1 (mod 8) n1 ≡ 5 (mod 8) n2 ≡ 5 (mod 8), n2 ≡ 1 (mod 8). (4) n1 n2 ≡ 5 (mod 8).

Proof. (1) ⇔ (2) The result follows from (2.3) in [15]. (2) ⇒ (3) Let n1 = 4f +1, n2 = 4f ′ +1 and e = 4f f ′ , where f and f ′ are integer. Since gcd(f, f ′ ) = 1, f 2 −1) is even. So, f or f ′ is even. Let f is even and f ′ and f ′ can not both be even. Here f f ′ = (n1 −1)(n 16 is odd. If f is even, then n1 − 1 = 8k1 , where k1 is an integer. Therefore, n1 ≡ 1 (mod 8). If f ′ is odd, then n1 − 1 = 4(2k2 + 1), where k2 is an integer. Therefore, n2 ≡ 5 (mod 8). Similarly, when f is odd and f ′ is even. We get n1 ≡ 5 (mod 8) and n2 ≡ 1 (mod 8). (3) ⇒ (2) and (3) ⇒ (4) are obvious. (4) ⇒ (3) Since gcd(n1 − 1, n2 − 1) = 4, let n1 − 1 = 4f and n2 − 1 = 4f ′ . We have n1 n2 ≡ 5 (mod 8), this gives (f + f ′ ) ≡ 1 (mod 8). So, n1 = 4(8k + 1 − f ′ ) + 1, this gives n1 ≡ 4(1 − f ′ ) + 1 (mod 8). If f ′ is odd, then n1 ≡ 1 (mod 8) and n2 ≡ 5 (mod 8). If f ′ is even, then n1 ≡ 5 (mod 8) and n2 ≡ 1(mod 8).  Lemma 5. Let the symbols be defined as before. The following four statements are equivalent: (1) −1 ∈ W0 . 2 −1) is odd. (2) (n1 −1)(n 16 (3) The following set of equation is satisfied: ( n1 ≡ 5 (mod 8)

n2 ≡ 5 (mod 8), (4) n1 n2 ≡ 1 (mod 8).

Proof. Similar to the proof of the above Lemma.



3. Autocorrelation values In this section, we discuss about the autocorrelation value of the two-prime WGCS-I of order 4. Given a binary sequence λ∞ of period n, the periodic autocorrelation value of the binary sequence λ∞ is defined by Cλ (w) =

n−1 X

(−1)λi+w +λi ,

i=0

where 0 6 w 6 n − 1. We recall the following lemma from [9].

Lemma 6. Let C be an (n, k, λ) almost difference sequence s∞ , i.e., si = 1 if and only if i mod n ∈ C.     n, Cs (a) = n − 4(k − λ),    n − 4(k − λ − 1),

set of Zn and the characteristic set of a binary Then the autocorrelation value if a = 0 for half of these a ∈ Z∗n

for the other half.

Thus, each (n, k, λ) almost difference set of Zn gives a binary sequence with three-level autocorrelation. Let the symbols be same as above. We define dλ (i, j; w) = |Ci ∩ (Cj + w)|, 0 < w 6 n − 1, i, j = 0, 1.

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Priti Kumari and P.K. Kewat

Lemma 7. [6] For each w ∈ Zn . We have

Cλ (w) = n − 4dλ (1, 0; w),

where 0 < w 6 n − 1.

Let the set C1 ⊆ Zn be the characteristic set of the sequence λ∞ . The following result will be needed in the sequel: dλ (1, 0; w) = |C1 ∩ (C0 + w)|

= |(P ∪ W1 ∪ W3 ) ∩ ((R ∪ Q ∪ W0 ∪ W2 ) + w)|

= |P ∩ ((R ∪ Q) + w)| + |(W1 ∪ W3 ) ∩ ((W0 ∪ W2 ) + w)|

+ |(W1 ∪ W3 ) ∩ ((R ∪ Q) + w)| + |P ∩ ((W0 ∪ W2 ) + w)|.

(7)

To calculate the value of dλ (1, 0; w), we need the following lemmas. Lemma 8. [6] If w ∈ Zn , then we have     1, if w ∈ P |P ∩ ((Q ∪ R) + w)| = 0, if w ∈ Q    1, if w ∈ Z∗ . n

Lemma 9. Let the notations be same as before. For each w ∈ Z∗n . We have |(W1 ∪ W3 ) ∩ ((W0 ∪ W2 ) + w)| = M. Proof. We know that |(W1 ∪ W3 ) ∩ ((W0 ∪ W2 ) + w)| = |W1 ∩ (W0 + w)| + |W3 ∩ (W0 + w)| + |W1 ∩ (W2 + w)| + |W3 ∩ (W2 + w)|. From Lemma 8 in [16], we have |Wi ∩ (Wj + w)| = (j − k, i − k)4 for each w ∈ Wk , k = 0, 1, 2, 3. We have      (0, 3)4 , if w ∈ W0  (0, 1)4 , if w ∈ W0      (3, 2) , if w ∈ W  (3, 0) , if w ∈ W 4 1 4 1 (b) |W3 ∩ (W0 + w)| = (a) |W1 ∩ (W0 + w)| =   (2, 1)4 , if w ∈ W2 (2, 3)4 , if w ∈ W2         (1, 0)4 , if w ∈ W3 (1, 2)4 , if w ∈ W3     (2, 1)4 , if w ∈ W0 (2, 3)4 , if w ∈ W0        (1, 0) , if w ∈ W  (1, 2) , if w ∈ W 4 1 4 1 (c) |W1 ∩ (W2 + w)| = (d) |W3 ∩ (W2 + w)| =   (0, 3)4 , if w ∈ W2 (0, 1)4 , if w ∈ W2         (3, 2)4 , if w ∈ W3 (3, 0)4 , if w ∈ W3 From the above discussion and using Lemma 2, we get |(W1 ∪ W3 ) ∩ ((W0 ∪ W2 ) + w)| = M .  Lemma 10. If w ∈ Zn , then we have |(W1 ∪ W3 ) ∩ ((R ∪ Q) + w)| =

(

0, n1 −1 2 ,

if w ∈ Q ∪ R

otherwise.

Proof. By Lemma 4 in [6], we have |Wi ∩ ((R ∪ Q) + w)| = This implies the lemma.

(

0, n1 −1 4 ,

if w ∈ Q ∪ R

otherwise.



Autocorrelation values and Linear complexity of generalized cyclotomic sequences and construction of cyclic codes

7

Lemma 11. Let symbols and notations be same as before. (i) If (n1 − 1)(n2 − 1)/36 is even, then   if w ∈ P   0, |P ∩ ((W0 ∪ W2 ) + w)| = (n2 − 1)/2, if w ∈ Q ∪ W0 ∪ W2    (n − 1)/2 − 1, if w ∈ W ∪ W . 2 1 3 (ii) If (n1 − 1)(n2 − 1)/36 is odd, then

|P ∩ ((W0 ∪ W2 ) + w)| =

    0,

if w ∈ P

(n2 − 1)/2 − 1, if w ∈ W0 ∪ W2    (n − 1)/2, if w ∈ Q ∪ W1 ∪ W3 . 2

Proof. We have |P ∩ ((W0 ∪ W2 ) + w)| = |P ∩ (W0 + w)| + |P ∩ (W2 + w)| and |(P ∪ R) ∩ (Wi + w)| = |P ∩ (Wi + w)| + |R ∩ (Wi + w)|. If (n1 − 1)(n2 − 1)/36 is even, then −1 ∈ W2 and if (n1 − 1)(n2 − 1)/36 is odd, then −1 ∈ W0 . By Lemma 3 and Corollary 1 of [16], we have |(P ∪ R) ∩ (Wi + w)| =

(

0, n2 −1 4 ,

if w ∈ P ∪ R

otherwise.

and     1, |R ∩ (Wi + w)| = 1,    0,

if w ∈ Wi

and (n1 − 1)(n2 − 1)/36 is odd

if w ∈ Wi+2 and (n1 − 1)(n2 − 1)/36 is even

otherwise.

Clearly, the lemma follows from above discussion.



We are now ready to compute the value of dλ (1, 0; w) in the following 2 −1) is even, by Lemmas 8-11 and Eq.(7), we obtain Case I: If (n1 −1)(n 16  2 −1)  1 + (n1 −1)(n + n12−1 + 0, if  4    0 + (n1 −1)(n2 −1) + 0 + n2 −1 , if 4 2 dλ (1, 0; w) = (n1 −2)(n2 −2)−1 n1 −1 n2 −1  + 2 + 2 , if  4  1+   (n1 −2)(n2 −2)−1 n1 −1 n2 −1 + 2 + 2 − 1, if 1+ 4 =

Case II: If

(n1 −1)(n2 −1) 16

          

(n1 n2 +n1 −n2 +3) , 4 (n1 n2 +n2 −n1 −1) , 4 (n1 n2 +3) , 4 (n1 n2 −1) , 4

two cases. w∈P

w∈Q

w ∈ W0 ∪ W2

w ∈ W1 ∪ W3 .

if w ∈ P

if w ∈ Q

if w ∈ W0 ∪ W2

if w ∈ W1 ∪ W3 .

is odd, we get  (n n +n −n +3) 1 2 1 2  ,  4    (n1 n2 +n2 −n1 −1) , 4 dλ (1, 0; w) = (n1 n2 −1)  ,  4    (n1 n2 +3) , 4

if w ∈ P

if w ∈ Q

if w ∈ W0 ∪ W2

if w ∈ W1 ∪ W3 .

By substituting the value of dλ (1, 0; w) in Lemma 7, we get the following result:

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Priti Kumari and P.K. Kewat

Theorem 1. (i) Let (n1 − 1)(n2 − 1)/16 be even. Then   n2 − n1 − 3,     n − n + 1, 1 2 Cλ (w) =  −3,     1,

if w ∈ P

if w ∈ Q

if w ∈ W0 ∪ W2

if w ∈ W1 ∪ W3 .

(ii) Let (n1 − 1)(n2 − 1)/16 be odd. Then

  n2 − n1 − 3, if w ∈ P     n − n + 1, if w ∈ Q 1 2 Cλ (w) =  1, if w ∈ W0 ∪ W2     −3, if w ∈ W1 ∪ W3 .

By Theorem 1, the autocorrelation values of the two prime WGCS-I of order 4 are quite good when n2 − n1 is very small. When n2 − n1 = 4, the Cλ (w) is three-valued and the characteristic set C1 is the almost difference set. So, this sequence λ∞ has optimal autocorrelation under certain condition. 4. A class of cyclic codes over GF(q) defined by two-prime WGCS-I of order 4 Let gcd(n, q) = 1. Let m be the order of q modulo n. Then the field GF(q m ) has a primitive nth root of unity β. We define ! X X X X i Λ(x) = x = + xi ∈ GF(q)[x], (8) + i∈C1

i∈P

i∈W1

i∈W3

Our main aim in this section is to find the generator polynomial gλ (x) =

xn − 1 gcd(xn − 1, Λ(x))

of the cyclic code Cλ defined by the sequence λ∞ . To compute the parameters of the cyclic code Cλ defined by the sequence λ∞ , we need to compute gcd(xn − 1, Λ(x)). Since β is a primitive nth root of unity, we need only to find such t’s that Λ(β t ) = 0, where 0 ≤ t ≤ n − 1. To this end, we need number of auxiliary results. We have 0 = β n − 1 = (β n1 )n2 − 1 = (β n1 − 1)(1 + β n1 + β 2n1 + · · · + β (n2 −1)n1 ). It follows that β n1 + β 2n1 + · · · + β (n2 −1)n1 = −1, i.e.,

X

β i = −1.

(9)

X

β i = −1.

(10)

i∈P

By symmetry we get β n2 + β 2n2 + · · · + β (n1 −1)n2 = −1, i.e.,

i∈Q

Lemma 12. Let the symbols be same as before. For 0 ≤ j ≤ 3, we have ( X − n14−1 (mod p), if t ∈ P it β = − n24−1 (mod p), if t ∈ Q. i∈Wj

Autocorrelation values and Linear complexity of generalized cyclotomic sequences and construction of cyclic codes

9

Proof. Suppose that t ∈ Q. Since g is a common primitive roots of n1 and n2 and the order of g modulo n is e, by the definition of u, we have Wj mod n1 = {g s uj mod n1 : s = 0, 1, 2, · · · , e − 1} = {g s+j mod n1 : s = 0, 1, 2, · · · , e − 1} n2 − 1 ∗ {1, 2, · · · , n1 − 1}, = 4

where n24−1 denotes the multiplicity of each element in the set {1, 2, · · · , n1 − 1}. We can write g s xj in the form 1 + k11 n1 , 1 + k12 n1 , · · · , 1 + k1(n2 −1)/4 n1 ,

2 + k21 n1 , 2 + k22 n1 , · · · , 2 + k2(n2 −1)/4 n1 ,

.. .

n1 − 1 + k(n1 −1)1 n1 , n1 − 1 + k(n1 −1)2 n1 , · · · , n1 − 1 + k(n1 −1)(n2 −1)/4 n1 .

(11)

where kli is an positive integer, 1 ≤ l ≤ n1 − 1 and 1 ≤ i ≤ (n2 − 1)/4. Since s ranges over {0, 1, · · · , e−1}, we divides the Wj set into (n2 −1)/4 subsets each of which contains n1 − 1 consecutive integers, i.e., g s+j mod n1 takes on each element of {1, 2, · · · , n1 − 1} exactly n24−1 times. From Eq.(11), it follows that if t ∈ Q, we have β (m+kli n1 )t = β mt , where 1 ≤ m ≤ n1 − 1. It follows from Eq.(10) that   X n2 − 1 n2 − 1 X j β =− (mod p). β it = 4 4 j∈Q

i∈Wj

For t ∈ P , we can get the result by similar argument. Lemma 13. For any r ∈ Wi , we have rWj = W(i+j)(mod

 d) ,

where rWj = {rt | t ∈ Wj }.

Proof. We have Wi = {g s ui : s = 0, 1, 2, · · · , e − 1}, i = 0, 1, · · · , d − 1 and let r = g s1 ui ∈ Wi . Then rWj = g s1 ui {uj , g 1 uj , · · · , g e−1 uj } = {g s1 ui+j , g s1 +1 ui+j , · · · , g s1 +e−1 ui+j }. Since u ∈ Z∗n , there must exist an integer υ with 0 6 υ 6 e − 1 such that ud = g υ , therefore, we must have rWj = W(i+j)(mod d) .  Lemma 14. Let D0 = W0 ∪ W2 and D1 = W1 ∪ W3 and rest of the symbols be same as before. For all t ∈ Zn we have   − n12+1 (mod p), if t ∈ P     n2 −1 (mod p), if t ∈ Q 2 Λ(β t ) =  Λ(β), if t ∈ D0     −(Λ(β) + 1), if t ∈ D1 .

Proof. Since gcd(n1 , n2 ) = 1, then tP = P if t ∈ P . By Eq.(8), Eq.(9) and Lemma 12, we get ! X X X X t ti Λ(β ) = β = + β ti + i∈C1

i∈P

i∈W1

= (−1 mod p) − =−

i∈W3



n1 + 1 mod p. 2

   n1 − 1 n1 − 1 mod p − mod p 4 4

10

Priti Kumari and P.K. Kewat

If t ∈ Q, then tP = 0. By Eq.(8), Eq.(9) and Lemma 12, we get t

S(β ) =

X

X

ti

β =

i∈C1

+

i∈P

X

+

i∈W1

X

i∈W3

= (n2 − 1 mod p) − =

n2 − 1 mod p. 2

!



β ti

   n2 − 1 n2 − 1 mod p − mod p 4 4

If t ∈ D0 , we have two cases: Case I: Let t ∈ W0 , then by Lemma 13, we have tWi = Wi . Since gcd(t, n2 ) = 1, then tP = P if t ∈ W0 . Hence

Λ(β t ) =

X

β ti =

X i∈P

i∈W1

i∈W3

=

X

X

X

i∈C1

+

+

i∈P

X

+

+

i∈W1

= Λ(β).

X

i∈W3

Case II: Let t ∈ W2 , then by Lemma 13, we have tWi = W(i+2)(mod then tP = P if t ∈ W2 . Hence t

Λ(β ) =

X

X

ti

β =

i∈C1

=

+

X

+

4)

X

i∈P

i∈W1

i∈W3

X

X

X

i∈P

= Λ(β).

+

i∈W3

+

i∈W1

!

β ti

!

βi

for 0 6 i 6 5. Since gcd(t, n2 ) = 1,

!

β ti

!

βi

Similarly, if t ∈ D1 , we have two cases: Case I: Let t ∈ W1 then by Lemma 13, we have tWi = W(i+1) . Since gcd(t, n2 ) = 1, then tP = P n−1 n−1 P i P i β = 0. Therefore, β ) = 0 and β − 1 6= 0, this give if t ∈ W1 . We have β n − 1 = (β − 1)( i=0

i=0

n−1 P i=0

i

β = 1+

P

i∈P

i

β +

P

i∈Q

i

β +

i∈

P

3 S

i

β = 0. From Eq.(9) and Eq.(10), we get

Wj

j=0

i∈

X 3 S

j=0

Wj

β i = 1.

(12)

Autocorrelation values and Linear complexity of generalized cyclotomic sequences and construction of cyclic codes

11

Hence t

Λ(β ) =

X

X

ti

β =

i∈C1

=

=

From Eq.(9), we know that

P

i∈P

+

+

X

i∈P

i∈W1

i∈W3

X

X

X

+

+

i∈P

i∈W2

i∈W0

X

X

X

i∈P

−

=

X

−

X i∈P

i∈W1

−

−

X

i∈W1

i∈W3

−

!

β ti

!

βi

!

βi + 1

X

i∈W3

!

βi + 2

X

βi + 1

i∈P

β i = −1 and by the definition of Λ(β), we have Λ(β t ) = −(Λ(β)+1). 

Lemma 15. If q ∈ D0 , we have Λ(β) ∈ GF(q) and Λ(β)q = Λ(β). If q ∈ D1 , we have Λ(β)q = −(Λ(β) + 1). Proof. We have gcd(n, q) = 1, i.e., q ∈ Z∗n , then q ∈

3 S

i=1

Wi . If q ∈ D0 , by Lemma 14, we have

(Λ(β))q = Λ(β q ) = Λ(β). So, Λ(β) ∈ GF(q). If q ∈ D1 , similarly, the result follows from Lemma 14. 

Lemma 16. If n1 n2 ≡ 5 (mod 8) or n1 n2 ≡ 1 (mod 8), we have Λ(β)(Λ(β) + 1) =

n−1 . 4

Proof. We have Λ(β) = −1 +

X

βi +

i∈W1

X

βi.

i∈W3

Then, we get Λ(β)(Λ(β) + 1) = −

X

i

β +

i∈W1

X

β

i∈W3

i

!

+

X X

β i+j +

i∈W1 j∈W1

X X

β i+j + 2

i∈W3 j∈W3

X X

β i+j . (13)

i∈W1 j∈W3

First suppose that n1 n2 ≡ 5 (mod 8). By Lemma 4, −1 ∈ W2 and that −Wj = {−t : t ∈ Wj } = W(j+2) mod 4 . X X X X β i+j = β i−j i∈W1 j∈W1

=

i∈W1 j∈W3

X

d(3, 1; r)β r + (3, 1)4

r∈P ∪Q

X X =

X

r∈P ∪Q

β i + (2, 0)4

X

β i + (0, 2)4

i∈W0

β i+j =

i∈W3 j∈W3

X

X X

X

β i + (1, 3)4

X

β i + (3, 1)4

i∈W1

X

β i + (0, 2)4

X

β i + (2, 0)4

i∈W2

X

βi,

(14)

X

βi,

(15)

i∈W3

β i−j

i∈W3 j∈W1

d(1, 3; r)β r + (1, 3)4

i∈W0

i∈W1

i∈W2

i∈W3

12

2

Priti Kumari and P.K. Kewat

X X

β i+j = 2

i∈W1 j∈W3



= 2 |W1 | +

X X

β i−j

i∈W1 j∈W1

X

d(1, 1; r)β r + (1, 1)4

r∈P ∪Q

X

β i + (0, 0)4

i∈W0

X

β i + (3, 3)4

i∈W1

X

β i + (2, 2)4

i∈W2

X

i∈W3



βi .

(16)

Substituting the values frpm Eq.(14) - Eq.(16) into Eq.(13) and combining Lemma 3, Lemma 2 and Eq.(12), we get ! X X X X X X Λ(β)(Λ(β) + 1) = − βi + βi + M β i + (M + 1) βi + M β i + (M + 1) βi i∈W1

i∈W3

i∈W0

i∈W1

i∈W2

i∈W3

(n1 − 1)(n2 − 1) (n1 − 5)(n2 − 1) (n1 − 1)(n2 − 5) (n1 − 1)(n2 − 1) +2 −2 −2 −4 16 4 16 16 n−1 . = 4

Similarly, we prove that if n ≡ 1 (mod 8), then Λ(β)(Λ(β) + 1) =

n−1 4 .



Note that Λ(1) = Lemma 17. If n ≡ 5 (mod 8) and q (mod n) ∈ D0 .

n−1 4

(n1 + 1)(n2 − 1) mod p. 2

≡ 0 (mod p) or n ≡ 1 (mod 8) and

(17) n−1 4

≡ 0 (mod p), then

Proof. Clearly, D0 is a subgroup of Z∗n . Hence, D0 is a multiplicative group. Since q is a power of p, it is sufficient to prove that p ∈ D0 . First, we assume that n ≡ 5 (mod 8) and n−1 4 ≡ 0 (mod p). For p = 2, we do not have any possibility. Therefore, we suppose that n ≡ 1 (mod 8) and n−1 4 ≡ 0 (mod p). For p = 2, by Lemma 5, we have only one possibility: ( n1 ≡ 5 (mod 8) (18) n2 ≡ 5 (mod 8), .

Suppose on the contrary that 2 ∈ D1 . By definition of Whiteman’s generalized cyclotomic classes 2 = us g i , 0 6 i 6 e − 1 and s is odd. From Eq.(4), we have 2 ≡ g s+i (mod n1 )

and

2 ≡ g i (mod n2 ).

Therefore, 2 must be a quadratic residue (non residue, respectively) modulo n1 if it is a quadratic non residue (residue, respectively) modulo n2 . It follows that the possibility in Eq.(18) is not possible. This gives a contradiction, therefore 2 ∈ D0 . We now prove this lemma for the case that p is an odd prime. First, we assume that n ≡ 5 (mod 8) and n−1 4 ≡ 0 (mod p). By Lemma 4, we have n1 ≡ 1 (mod 8) and n2 ≡ 5 (mod 8) or n1 ≡ 5 (mod 8) and n2 ≡ 1 (mod 8). Thus, (n1 + n2 )/2 is odd. If n−1 4 ≡ 0 (mod p), then n = n1 n2 ≡ 1 (mod p). By the Law of Quadratic Reciprocity, if p and ni are odd prime, then we have     ni −1 p−1 ni p for i = 1, 2, = (−1)( 2 )( 2 ) ni p

Autocorrelation values and Linear complexity of generalized cyclotomic sequences and construction of cyclic codes

where (−) is the Legendre symbol. Now,       n1 +n2 −2 p−1 n2 p p n1 ( ) 2 )( 2 = (−1) n1 n2 p p   n +n −2 p−1 1 2 n1 n2 = (−1)( 2 )( 2 ) p   n +n −2 p−1 1 2 1 = (−1)( 2 )( 2 ) p = 1.

13

(19)

Suppose on the contrary that p ∈ D1 . By the definition, p = us g i , 0 6 i 6 e − 1 and s is odd. We have p ≡ g s+i (mod n1 )

p ≡ g i (mod n2 ).

and

Since s is odd, then we must have 

p n1



p n2



= −1,

This contradicts Eq.(19). Thus, p ∈ D0 . Similarly, we prove that if n ≡ 1 (mod 8) and then q (mod n) ∈ D0 .

n−1 4

≡ 0 (mod p), 

we need to discuss the factorization of xn − 1 over GF(q). Let β and other symbols be the same as before. Define for each i; 0 ≤ i ≤ 3, Y ωi (x) = (x − β j ), j∈Wi

where Wi denote the Whiteman’s cyclotomic classes of order 4. Among the nth roots of unity β i , where 0 ≤ i ≤ n − 1, the n2 elements β i , i ∈ P ∪ {0}, are n2 th roots of unity, the n1 elements β i , i ∈ Q ∪ {0}, are n1 th roots of unity. Hence, xn2 − 1 =

Y

(x − β i )

Y

(x − β i ).

i∈P ∪{0}

and xn1 − 1 = n−1 Q

i∈Q∪{0}

3 Q ωi (x). Also, it can be where ω(x) = i=0 i=0 n1 n2 Q Q −1) (x − β i ). It (x − β i ) and d1 (x) = d0 (x)d1 (x), where d0 (x) = written as xn − 1 = (x −1)(x x−1

Then, we have xn − 1 =

(x − β i ) =

(xn1 −1)(xn2 −1) ω(x), x−1

i∈D0

i∈D1

is straightforward to prove that if q ∈ D0 then di (x) ∈ GF(q) for all i.

In the sequel, let Lλ and mλ (x) be the linear complexity and minimal polynomial of binary Whiteman’s generalized cyclotomic sequence of order 4 with respect to the two primes n1 and n2 . Let 2 −1) (mod p). We have the following △1 = n12+1 (mod p), △2 = n22−1 (mod p) and △ = (n1 +1)(n 2 theorem.

14

Priti Kumari and P.K. Kewat

Theorem 2. (I) When n ≡ 5 (mod 8) and n−1 4 6≡ 0 (mod p) or n ≡ 1 (mod 8) and then  n  if △1 6= 0, △2 6= 0, △ = 6 0  x − 1,    xn −1 , if △1 = 0, △2 6= 0 xn2 −1 mλ (x) = xn −1  , if △1 6= 0, △2 = 0  xn1 −1    (xn −1)(x−1) , if △1 = △2 = 0. (xn1 −1 )(xn2 −1)

n−1 4

6≡ 0 (mod p),

and

  n,     n−n , 2 Lλ (x) =  n − n1 ,     n − (n1 + n2 − 1),

if △1 6= 0, △2 6= 0, △ = 6 0 if △1 = 0, △2 6= 0

if △1 6= 0, △2 = 0

if △1 = △2 = 0.

In this case, the cyclic code Cλ over GF(q) defined by the sequence λ∞ has parameters [n, k, d] and generator polynomial mλ (x) is given above, where the dimension k = n − deg(mλ (x)). n−1 (II) When n ≡ 5 (mod 8) and n−1 4 ≡ 0 (mod p) or n ≡ 1 (mod 8) and 4 ≡ 0 (mod p), then

mλ (x) =

and

                                

  n−     n− Lλ (x) =  n−     n−

xn −1 d0 (x) , xn −1 d1 (x) , xn −1 , (xn2 −1 )d0 (x) n x −1 , (xn2 −1 )d1 (x) n x −1 , (xn1 −1 )d0 (x) xn −1 , (xn1 −1 )d1 (x) (xn −1)(x−1) , (xn1 −1 )(xn2 −1)d0 (x) (xn −1)(x−1) , (xn1 −1 )(xn2 −1)d1 (x)

(n1 −1)(n2 −1) , 2 (n1 +1)(n2 −1)+2 , 2 (n1 −1)(n2 +1)+2 , 2 (n1 +1)(n2 +1)−2 , 2

if △1 6= 0, △2 6= 0, △ = 6 0, Λ(β) = 0

if △1 6= 0, △2 6= 0, △ = 6 0, Λ(β) = −1 if △1 = 0, △2 6= 0, Λ(β) = 0

if △1 = 0, △2 6= 0, Λ(β) = −1 if △1 6= 0, △2 = 0, Λ(β) = 0

if △1 6= 0, △2 = 0, Λ(β) = −1

if △1 = △2 = 0, Λ(β) = 0

if △1 = △2 = 0, Λ(β) = −1.

if △1 6= 0, △2 6= 0, △ = 6 0, Λ(β) = 0 or Λ(β) = −1

if △1 = 0, △2 6= 0, Λ(β) = 0 or Λ(β) = −1 if △1 6= 0, △2 = 0, Λ(β) = 0 or Λ(β) = −1 if △1 = △2 = 0, Λ(β) = 0 or Λ(β) = −1.

In this case, the cyclic code Cλ over GF(q) defined by the sequence λ∞ has parameters [n, k, d] and generator polynomial mλ (x) as given above, where the dimension k = n − deg(mλ (x)). n−1 Proof. (I) When n ≡ 5 (mod 8) and n−1 4 6≡ 0 (mod p) or n ≡ 1 (mod 8) and 4 6≡ 0 (mod p), then t by Lemma 16, we have Λ(β) 6= 0, −1. Therefore, from Lemma 14, Λ(β ) = 0 only when t is in P or Q or both. So, the conclusion on the minimal polynomial mλ (x) of the sequence λ∞ follows from above discussion. The linear complexity of the sequence λ∞ is equal to deg(mλ (x)). (II) When n ≡ 5 (mod 8) and n−1 ≡ 0 (mod p) or n ≡ 1 (mod 8) and n−1 ≡ 0 (mod p), then by 4 4 Lemma 16, we have Λ(β) ∈ {0, −1}, by Lemma 17, q ∈ D0 and di (x) ∈ GF(q)[x] for each i if q ∈ D0 . So, the conclusion on the minimal polynomial mλ (x) of the sequence λ∞ follows from (17), Lemma 15 and 14 . The linear complexity of the sequence λ∞ is equal to deg(mλ (x)). 

Autocorrelation values and Linear complexity of generalized cyclotomic sequences and construction of cyclic codes

15

Corollary 1. Let q = 2. We have the following conclusions: (1) If n1 ≡ 1 (mod 8) and n2 ≡ −3 (mod 8) or if n1 ≡ −3 (mod 8) and n2 ≡ 1 (mod 8), we have gλ (x) =

xn − 1 and xn1 − 1

L λ = n − n1 .

In this case, the cyclic code Cλ over GF(q) defined by the sequence λ∞ has parameters [n, n1 , n2 ] (From Theorem 3, d = n2 ) and generator polynomial gλ (x) as above. (2) If n1 ≡ −3 (mod 8) and n2 ≡ −3 (mod 8), we have ( (xn −1) if Λ(β) = 0 (n1 − 1)(n2 + 1) + 2 (xn1 −1)d0 (x) , . gλ (x) = and Lλ = n − (xn −1) 2 , if Λ(β) = 1 (xn1 −1)d1 (x) 2 +1)+2 In this case, the cyclic code Cλ over GF(q) defined by the sequence λ∞ has parameters [n, (n1 −1)(n , d] 2 and generator polynomial gλ (x) as above.

Corollary 2. Let q = 3. We have the following conclusions: (1) If n1 ≡ 1 (mod 24) and n2 ≡ 5 (mod 24) or if n1 ≡ 13 (mod 24) and n2 ≡ 5 (mod 24) or if n1 ≡ 13 (mod 24) and n2 ≡ 17 (mod 24), we have gλ (x) = xn − 1 and

Lλ = n.

In this case, the cyclic code Cλ over GF(q) defined by the sequence λ∞ has parameters [n, 0, 0] and generator polynomial gλ (x) as above. (2) If n1 ≡ 5 (mod 24) and n2 ≡ 1 (mod 24) or if n1 ≡ 5 (mod 24) and n2 ≡ 13 (mod 24) or if n1 ≡ 17 (mod 24) and n2 ≡ 13 (mod 24), we have gλ (x) =

(xn − 1)(x − 1) (xn1 − 1(xn2 − 1)

and Lλ = n − (n1 + n2 − 1).

In this case, the cyclic code Cλ over GF(q) defined by the sequence λ∞ has parameters [n, n1 + n2 − 1, d] (From Theorem 4, d = min(n1 , n2 )) and generator polynomial gλ (x) as above. (3) If n1 ≡ 5 (mod 24) and n2 ≡ 5 (mod 24) or n1 ≡ 5 (mod 24) and n2 ≡ 17 (mod 24) or n1 ≡ 17 (mod 24) and n2 ≡ 5 (mod 24) , we have ( (xn −1) if Λ(β) = 0 (n1 + 1)(n2 − 1) + 2 (xn2 −1)d0 (x) , gλ (x) = . and Lλ = n − (xn −1) 2 , if Λ(β) = 1 n2 (x

−1)d1 (x)

In this case, the cyclic code Cλ over GF(q) defined by the sequence λ∞ has parameters [n, (n1 +1)(n2 2 −1)+2 , d] and generator polynomial gλ (x) as above.

5. The minimum distance of the cyclic codes In this section, we determine the lower bounds on the minimum distance of some of the cyclic codes of this paper. Theorem 3. [7] Let Ci denote the cyclic code over GF(q) with the generator polynomial gi (x) = The cyclic code Ci has parameters [n, ni , di ], where di = ni−(−1)i and i = 1, 2.

xn −1 xni −1 .

Theorem 4. [7] Let C(n1 ,n2 ,q) denote the cyclic code over GF(q) with the generator polynomial g(x) = (xn −1)(x−1) (xn1 −1)(xn2 −1) . The cyclic code C(n1 ,n2 ,q) has parameters [n, n1 + n2 − 1, d(n1 ,n2 ,q) ], where d(n1 ,n2 ,q) = min(n1 , n2 ).

16

Priti Kumari and P.K. Kewat

Theorem 5. Assume that q ∈ D0 . Let C (i,j) denote the cyclic code over GF(q) with the generator n −1 and let d(i,j) denote the minimum distance of this code, where i ∈ polynomial g (i,j) (x) = (xnix−1)d j (x)

{1, 2} and j ∈ {0, 1}. The cyclic code C (i,j) has parameters [n, ni + √ ⌈ ni−(−1)i ⌉. If −1 ∈ D1 , we have (d(i,j) )2 − d(i,j) + 1 ≥ ni−(−1)i .

(n1 −1)(n2 −1) , d(i,j) ], 2

where d(i,j) ≥

Proof. Let c(x) ∈ GF(q)[x]/(xn − 1) be a codeword of Hamming weight ω in C (i,j) . Take any r ∈ D1 . The cyclic code c(xr ) is a codeword of Hamming weight ω in C (i,(j+1) mod 2) . It then follows that d(i,j) = d(i,(j+1) mod 2) . Let c(x) ∈ GF(q)[x]/(xn − 1) be a codeword of minimum weight in C (i,j) . Then c(xr ) is a codeword of same weight in C (i,(j+1) mod 2) . Hence, c(x)c(xr ) is a codeword of Ci , where Ci n denote the cyclic code over GF(q) with the generator polynomial gi (x) = xxni−1 −1 and minimum distance (i,j) 2 di = ni−(−1)i . Hence, from Theorem 3, we have (d ) ≥ di = ni−(−1)i , and (d(i,j) )2 − d(i,j) + 1 ≥ ni−(−1)i if −1 ∈ D1 .  (j)

Theorem 6. Assume that q ∈ D0 . Let C(n1 ,n2 ) denote the cyclic code over GF(q) with the gener(j)

ator polynomial g(n1 ,n2 ) (x) =

n

(x −1)(x−1) (xn1 −1)(xn2 −1)dj (x)

(j)

and let dn1 ,n2 denote the minimum distance of this (j)

code, where i ∈ {1, 2} and j ∈ {0, 1}. The cyclic code C(n1 ,n2 ) has parameters [n, n1 + n2 − 1 + p (j) (j) (n1 −1)(n2 −1) , d(n1 ,n2 ) ], where d(n1 ,n2 ) ≥ ⌈ min(n1 , n2 )⌉. 2 (j)

(j)

If −1 ∈ D1 , we have (d(n1 ,n2 ) )2 − d(n1 ,n2 ) + 1 ≥ min(n1 , n2 ).

(j)

Proof. Let c(x) ∈ GF(q)[x]/(xn − 1) be a codeword of Hamming weight ω in C(n1 ,n2 ) . Take any ((j+1) mod 2)

r ∈ D1 . The cyclic code c(xr ) is a codeword of Hamming weight ω in C(n1 ,n2 )

lows that

(j) d(n1 ,n2 )

(j) C(n1 ,n2 ) .

=

((j+1) mod 2) . d(n1 ,n2 ) r

. It then fol-

n

Let c(x) ∈ GF(q)[x]/(x − 1) be a codeword of minimum weight ((j+1) mod 2)

. Hence, c(x)c(xr ) is a codeThen c(x ) is a codeword of same weight in C(n1 ,n2 ) in word of C(n1 ,n2 ,q) , where C(n1 ,n2 ,q) denote the cyclic code over GF(q) with the generator polynomial n −1)(x−1) g(x) = (x(x n1 −1)(xn2 −1) and minimum distance d(n1 ,n2 ,q) = min(n1 , n2 ). Hence, from Theorem 4, we have (j)

(j)

(j)

(d(n1 ,n2 ) )2 ≥ d(n1 ,n2 )j = min(n1 , n2 ), and (d(n1 ,n2 ) )2 − d(n1 ,n2 ) + 1 ≥ min(n1 , n2 ) if −1 ∈ D1 .



Example 1. Let (p, m, n1 , n2 ) = (2, 1, 5, 13). Then q = 2, n = 65 and Cλ is a [65, 29, 12] cyclic code n −1 over GF(q) with generator polynomial g(x) = (xn1x−1)d = x36 + x33 + x31 + x28 + x27 + x25 + x19 + 1 (x) x18 + x17 + x11 + x9 + x8 + x5 + x3 + 1. Example 2. Let (p, m, n1 , n2 ) = (3, 1, 17, 5). Then q = 3, n = 85 and Cλ is a [85, 37, 17] cyclic code n −1 = x48 + 2x46 + 2x45 + 2x44 + x43 + x42 + x41 + over GF(q) with generator polynomial g(x) = (xn2x−1)d 1 (x) 2x40 + x39 + x37 + 2x36 + x34 + 2x33 + 2x32 + 2x31 + x30 + x28 + 2x27 + 2x26 + x25 + 2x24 + x23 + 2x22 + 2x21 + x20 + x18 + 2x17 + 2x16 + 2x15 + x14 + 2x12 + x11 + x9 + 2x8 + x7 + x6 + x5 + 2x4 + 2x3 + 2x2 + 1. Example 3. Let (p, m, n1 , n2 ) = (2, 1, 5, 17). Then q = 2, n = 85 and Cλ is a [85, 5, 17] cyclic code 85 −1) 80 75 70 65 60 55 50 45 over GF(q) with generator polynomial g(x) = (x (x5 −1) = x + x + x + x + x + x + x + x + x40 + x35 + x30 + x25 + x20 + x25 + x20 + x15 + x10 + x5 + 1. This is a bad cyclic code due to its poor minimum distance. The code in this case is bad because q ∈ / D0 . Example 4. Let (p, m, n1 , n2 ) = (2, 1, 5, 29). Then q = 2, n = 145 and Cλ is a [145, 61, 22] cyclic code over GF(q) with generator polynomial g(x) = x84 + x83 + x82 + x81 + x77 + x74 + x73 + x71 + x70 + x69 + x68 + x67 + x65 + x60 + x56 + x53 + x51 + x50 + x49 + x48 + x47 + x45 + x44 + x42 + x40 + x39 + x37 + x36 + x35 + x34 + x33 + x31 + x28 + x24 + x19 + x17 + x16 + x15 + x14 + x13 + x11 + x10 + x7 + x3 + x2 + x + 1.

Autocorrelation values and Linear complexity of generalized cyclotomic sequences and construction of cyclic codes

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Example 5. Let (p, m, n1 , n2 ) = (5, 1, 13, 17). Then q = 5, n = 221 and Cλ is a [221, 96] cyclic code n −1 = x125 + 3x123 + 2x122 + 2x121 + 4x119 + x118 + over GF(q) with generator polynomial g(x) = xd0 (x) 115 114 112 111 110 109 108 4x + 2x + 4x + 3x + 3x + 4x + x + 2x106 + 4x105 + 3x104 + 2x103 + 2x102 + x99 + x98 + x97 + 2x96 + 3x95 + 4x94 + 4x93 + 2x92 + 4x90 + 4x89 + 2x88 + 3x86 + 3x85 + x84 + x83 + 4x82 + 3x81 + 3x80 + 4x79 + 4x78 + 3x77 + 4x76 + 2x74 + 3x73 + 4x72 + 4x71 + 3x70 + 2x69 + 4x67 + 3x64 + 2x61 + x58 + 3x56 + 2x55 + x54 + x53 + 2x52 + 3x51 + x49 + 2x48 + x47 + x46 + 2x45 + 2x44 + x43 + 4x42 + 4x41 + 2x40 + 2x39 + 3x37 + x36 + x35 + 3x33 + x32 + x31 + 2x30 + 3x29 + 4x28 + 4x27 + 4x26 + 3x23 + 3x22 + 2x21 + x20 + 3x19 + 4x17 + x16 + 2x15 + 2x14 + x13 + 3x11 + x10 + 4x7 + x6 + 3x4 + 3x3 + 2x2 + 4. We did some computation and our computation shows that upper bound on the minimum distance for this code is 81. Remark. It was proved in Theorem 1 that the characteristic set C1 = P ∪ W1 ∪ W3 is an almost difference set when n2 = n1 + 4. Example 5 give an almost difference set of Whiteman’s generalized cyclotomic sequence of order 4. Conclusion. We can see by Theorem 1 that the autocorrelation values of the WGCS-I of order 4 has good autocorrelation property under certain condition. The best case is when n2 − n1 = 4, in this case, the periodic autocorrelation value is three-valued, i.e., its characteristic set C1 is an almost difference set. In the case Λ(β) ∈ / {0, 1}, the least value of linear complexity is n − (n1 + n2 − 1) and in the case Λ(β) ∈ {0, 1}, the least value of linear complexity is n − (n1 +1)(n2 2 +1)−2 > n2 . Our results show that these binary sequence of order 4 over GF(q) presented in this paper are very good in terms of linear complexity. We have constructed several classes of cyclic codes over GF(q) using the WGCS-I of order 4 and also obtained the lower bounds on the minimum distance of these cyclic codes References [1] K. Arasu, C. Ding, T. Helleseth, P. V. Kumar, and H. M. Martinsen. Almost difference sets and their sequences with optimal autocorrelation. Information Theory, IEEE Transactions on, 47(7):2934–2943, 2001. [2] E. Betti and M. Sala. A new bound for the minimum distance of a cyclic code from its defining set. IEEE Transactions on Information Theory, 52(8):3700–3706, 2006. [3] Z. Chen and S. Li. Some notes on generalized cyclotomic sequences of length pq. Journal of Computer Science and Technology, 23:843–850, 2008. [4] T. Cusik, C. Ding, and A. Renvall. Stream Ciphers and Number Theory. North-Holland Mathematical Lib., NorthHolland, 2003. [5] C. Ding. Linear complexity of generalized cyclotomic binary sequences of order 2. Finite Fields Appl., 3(2):159–174, 1997. [6] C. Ding. Autocorrelation values of generalized cyclotomic sequences of order two. IEEE Transactions on Information Theory, 44(4):1699–1702, 1998. [7] C. Ding. Cyclic codes from the two-prime sequences. IEEE Transactions on Information Theory, 58(6):3881–3891, 2012. [8] C. Ding, X. Du, and Z. Zhou. The bose and minimum distance of a class of BCH codes. IEEE Transactions on Information Theory, 61(5):2351–2356, 2015. [9] C. Ding, T. Helleseth, and K.-Y. Lam. Several classes of binary sequences with three-level autocorrelation. IEEE Transactions on Information Theory, 45(7):2606–2612, 1999. [10] M. Eupen and J. van Lint. On the minimum distance of ternary cyclic codes. IEEE Transactions on Information Theory, 39(2):409–416, 1993. [11] R. Lidl and H. Niederreiter. Finite fields. Cambridge Univ. Press, 1997. [12] F. Macwilliams and N. Sloane. The theory of error correcting codes. North-Holland Mathematical Lib., NorthHolland, 1977. [13] Y. Sun, T. Yan, and H. Li. Cyclic code from the first class whiteman’s generalized cyclotomic sequence with order 4. CoRR, abs/1303.6378, 2013. [14] J. van Lint and R. Wilson. On the minimum distance of cyclic codes. IEEE Transactions on Information Theory, 32(1):23–40, 1986. [15] A. L. Whiteman. A family of difference sets. Illinois J. Math., 6:107–121, 1962. [16] C. Zhao, W. Ma, T. Yan, and Y. Sun. Autocorrelation values of generalized cyclotomic sequences of order six. IEICE Transactions, 96-A(10):2045–2048, 2013.

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Priti Kumari and P.K. Kewat

Department of Applied Mathematics, Indian School of Mines, Dhanbad 826 004, India E-mail address: [email protected],[email protected]