Cyclic Codes from the Two-Prime Sequences

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1 Cunsheng Ding Department of Computer Science and Engineering The Hong Kong University of Science and Technology Kowloon, Hong Kong, CHINA May 2012

2 Outline of this Talk A brief introduction to cyclic codes A general construction of cyclic codes with sequences The two-prime sequences over GF(q) Cyclic codes from the two-prime sequences over GF(q) Concluding remarks Reference: C. Ding, Cyclic codes from the two-prime sequences, IEEE Trans. Inform. Theory, June Page 1 May 2012

3 Part I: Introduction to Cyclic Codes Page 2 May 2012

4 Linear Codes and Cyclic Codes Definition: Let q be a power of a prime p. An[n,k,d] linear code over GF(q) is a k-dimensional subspace of GF(q) n with minimum (Hamming) nonzero weight d. Definition: A linear [n,k] code C over the finite field GF(q) is called cyclic if (c 0,c 1,,c n 1 ) C implies (c n 1,c 0,c 1,,c n 2 ) C. Page 3 May 2012

5 The Generator and Parity-Check Polynomials of Cyclic Codes Let gcd(n,q)=1. By identifying any vector (c 0,c 1,,c n 1 ) GF(q) n with c 0 + c 1 x+c 2 x 2 + +c n 1 x n 1 GF(q)[x]/(x n 1), any code C of length n over GF(q) corresponds to a subset of GF(q)[x]/(x n 1). The linear code C is cyclic iff the corresponding subset in GF(q)[x]/(x n 1) is an ideal of the ring GF(q)[x]/(x n 1). Every ideal of GF(q)[x]/(x n 1) is principal. Let C =(g(x)) be a cyclic code. Then g(x) is called the generator polynomial and h(x)=(x n 1)/g(x) is referred to as the parity-check polynomial of C. Page 4 May 2012

6 Bounds on Parameters of Linear Codes Let A q (n,d) denote the maximum cardinality of any q-ary code of length n and minimum Hamming distance d. There are many bounds on codes: Singleton bound: A q (n,d) q n d+1. ( Sphere packing bound: A q (n,d) (d 1)/2 i=0 ( n i ) (q 1) i) q n. Plotkin bound: A q (n,d) qd/(qd (q 1)n), where (q 1)n<qd. Griesmer bound: n k 1 i=0 d/qi for any [n,k,d;q] linear code. Linear programming bounds. Basic Problem: How to construct optimal or good linear and cyclic codes with respect to some bound? Page 5 May 2012

7 Part II: A Generic Construction of Cyclic Codes with periodic Sequences Page 6 May 2012

8 The Linear Span and Minimal Polynomial of Sequence Let s n = s 0 s 1 s n 1 be a sequence over GF(q). The linear span (also called linear complexity) of s n is the smallest positive integer l such that there are constants c 0 = 1,c 1,,c l GF(q) satisfying c 0 s i = c 1 s i 1 + c 2 s i 2 + +c l s i l for all l i<n. Such a polynomial c(x)=c 0 + c 1 x+ +c l x l is called the feedback polynomial of a shortest LFSR that generates s n. Any feedback polynomial of s is called a characteristic polynomial. The characteristic polynomial with the smallest degree is called the minimal polynomial. Remark: The minimal polynomial defined here is the reciprocal of that defined in some other references. Page 7 May 2012

9 Computation of the Minimal Polynomial of Sequences Lemma 1 Let s be a sequence of period n over GF(q). Define S n (x)=s 0 + s 1 x+ +s n 1 x n 1 GF(q)[x]. Then the minimal polynomial m s (x) of s is given by m s (x)= x n 1 gcd(x n 1,S n (x)) (1) and the linear spanl s of s is given by L s = n deg(gcd(x n 1,S n (x))). (2) Page 8 May 2012

10 A Construction of Cyclic Codes with Periodic Sequences Given any sequence s =(s i ) i=0 of period n over GF(q), construct a cyclic code over GF(q) with length n and generator polynomial x n 1 gcd(s(x),x n 1) (3) where S(x)= n 1 i=0 s ix i GF(q)[x]. We call the cyclic code C s with the generator polynomial of (3) the code defined by the sequence s, and the sequence s the defining sequence of the cyclic code C s. How to choose the sequence s in order to construct good codes? Can the cyclic code be an optimal linear code? Page 9 May 2012

11 Part III: The Page 10 May 2012

12 The Two-Prime Sequences Let n 1 and n 2 be two distinct odd primes, define n=n 1 n 2 and N 1 ={n 1,2n 1,,(n 2 1)n 1 }, N 2 ={n 2,2n 2,,(n 1 1)n 2 }. The two-prime sequence, denoted by λ, is defined by 0, if i mod n {0} N 2 λ i = 1, if i mod n N ( )( 1 1 ( i i n 1 n 2 ))/2 otherwise where ( an1) denotes the Legendre symbol. Traditionally, they are defined as binary sequences and a generalization of the binary twin-prime sequences. Here, we treat them as sequences over any finite field GF(q), where gcd(q,n)=1, and will use them to construct cyclic codes of length n over GF(q). (4) Page 11 May 2012

13 A Cyclotomic Description of the Two-Prime Sequences Define N = gcd(n 1 1,n 2 1) and e=(n 1 1)(n 2 1)/N. The CRT guarantees that there are common primitive roots of both n 1 and n 2. Let π be a fixed common primitive root of both n 1 and n 2, and ρ be an integer satisfying Whiteman proved that ρ π (mod n 1 ), ρ 1 (mod n 2 ). Z n ={π s ρ i : s=0,1,,e 1; i=0,1,,n 1}, where Z n denotes the set of all invertible elements of the residue class ringz n. Page 12 May 2012

14 A Cyclotomic Description of the Two-Prime Sequences The generalized cyclotomic classes W i of order N with respect to n 1 and n 2 are Whiteman proved that W (N) i ={π s ρ i : s=0,1,,e 1}, i=0,1,,n 1. Z n = N 1 i=0 W (N) i, W (N) (N) i W j = /0 for i j. This generalized cyclotomy was introduced by Whiteman in The motivation behind the investigation of the generalized cyclotomy is the search for residue difference sets. The famous twin-prime difference sets are among such a class of difference sets. Page 13 May 2012

15 A Cyclotomic Description of the Two-Prime Sequences Define D (2) 0 = (N 2)/2 i=0 W (N) 2i and D (2) 1 = Clearly D (2) 0 is a subgroup of Z n and D (2) 1 = ρd (2) 0. (N 2)/2 i=0 W (N) 2i+1. The cyclotomic classes D (2) 0 and D (2) 1 of order two and are identical to Whiteman s cyclotomic classes of order N when and only when N = 2. Page 14 May 2012

16 A Cyclotomic Description of the Two-Prime Sequences Define D (2) 0 = (N 2)/2 i=0 W (N) 2i and D (2) 1 = (N 2)/2 i=0 W (N) 2i+1 C 0 ={0} N 2 D (2) 0, C 1 = N 1 D (2) 1. Then{C 0,C 1 } is a partition of Z n and 0, if i mod n C 0 λ i = 1, if i mod n C 1. This is the cyclotomic description of the two-prime sequence λ defined in (4). Page 15 May 2012

17 The Parameters of the Cyclic Codes C λ The generator polynomial of C λ = the minimal polynomial m λ (x) of λ. The dimension k of C λ = n deg(m λ (x)). The minimum weight d =? (exact value or lower bound!) Page 16 May 2012

18 The Generator Polynomial and Dimension of C λ The generator polynomial m λ (x)= x n 1 gcd(x n 1,Λ(x)), where Λ(x)=λ 0 + λ 1 x+ +λ n 1 x n 1 GF(q)[x]. The main task is to work out gcd(x n 1,Λ(x)), which involves the computation of the cyclotomic numbers ( ) D (i, j) 2 = D (2) (2) i + 1 j. We will skip the details of the computation of the generator polynomial m λ (x). Page 17 May 2012

19 The Generator Polynomial and Dimension of C λ Recall gcd(q,n)=1. Let l=ord n (q). Then GF(q l ) has a primitive nth root of unity θ. For i {0,1} define Then d i (x)= (x θ i ) GF(q l )[x]. i D (2) i x n 1= n 1 i=0 (x θ i )= (xn 1 1)(x n 2 1) x 1 d 0 (x)d 1 (x). If q D (2) 0, then d i(x) GF(q)[x] for all i. Different choice of θ may result in a swapping of d 0 (x) and d 1 (x). Question: Under what conditions q D (2) 0? Page 18 May 2012

20 The Generator Polynomial and Dimension of C λ Let θ be the primitive nth root of unity θ in GF(q l ) defined before. Define Λ(θ)= n 1 λ i θ i GF(q l ). i=0 Lemma 2 If q D (2) 0, then Λ(θ) {0,1}. Attention: When q D (2) 0, we select θ such that Λ(θ)=0. Question: Under what conditions q D (2) 0? Page 19 May 2012

21 The Generator Polynomial and Dimension of C λ Question: Under what conditions q D (2) 0? Lemma 3 q D (2) 0 iff n 3 (mod 4) and n+1 4 mod p=0; or n 1 (mod 4) and n 1 4 mod p=0. Remark: To describe the generator polynomial of C λ, we need to define 1 = n mod p, 2 = n mod p, = (n 1+ 1)(n 2 1) 2 mod p. Page 20 May 2012

22 The Generator Polynomial and Dimension of C λ The case q D (2) 1 : d i(x) GF(q)[x] Theorem 4 When n 3 (mod 4) and n+1 4 mod p 0 or n 1 (mod 4) and n 1 4 mod p 0, the generator polynomial of C λ is given by x n 1, if 1 0, 2 0, 0 m λ (x)= x n 1 x 1, if 1 0, 2 0, =0 x n 1 x n 2 1, if 1 = 0, 2 0 x n 1 x n 1 1, if 1 0, 2 = 0 (x n 1)(x 1) (x (n 1 1)(x (n 2 1), if 1 = 0, 2 = 0. (5) Hence the dimension k=n deg(m λ (x)). Page 21 May 2012

23 The case q D (2) 0 : d i(x) GF(q)[x] Theorem 5 When n 3 (mod 4) and n+1 4 mod p=0 or n 1 (mod 4) and n 1 4 mod p=0, the generator polynomial of C λ is given by m λ (x)= x n 1 d 0 (x), if 1 0, 2 0, 0 x n 1 (x 1)d 0 (x), if 1 0, 2 0, =0 x n 1 (x n 2 1)d 0 (x), if 1 = 0, 2 0 x n 1 (x n 1 1)d 0 (x), if 1 0, 2 = 0 (x n 1)(x 1) d 0 (x) 2 i=1 (xn i 1), if 1 = 0, 2 = 0. (6) Then the dimension k=n deg(m λ (x)). Page 22 May 2012

24 The Minimum Weight of the Codes C λ Questions: After determining the dimension and generator polynomial of C λ, we ask the following questions: Can we find out the minimum weight of the code C λ? If not, can we develop good lower bounds on the minimum weight d? Page 23 May 2012

25 The Minimum Weight of the Codes C λ Theorem 6 The cyclic code over GF(q) with generator polynomial g(x)=(x n 1)/(x n i 1) has parameters [n,n i,d i ], where where i {1,2}. d i = n i ( 1) i, (7) Example 7 Let q=2 and(n 1,n 2 )=(7,5). Then the cyclic code over GF(2) with the generator polynomial g(x)=(x n 1)/(x n 1 1) has parameters [35,7,5]. Page 24 May 2012

26 The Minimum Weight of the Codes C λ Theorem 8 Let C (n1,n 2,q) denote the cyclic code over GF(q) with generator polynomial g(x)=(x 1)(x n 1)/(x n 1 1)(x n 2 1). Then the code C (n1,n 2,q) has parameters [n,n 1 + n 2 1,d (n1,n 2 )], where d (n1,n 2 ) = min(n 1,n 2 ). (8) Example 9 Let q=2 and(n 1,n 2 )=(7,5). The cyclic code over GF(2) with the generator polynomial g(x)=(x n 1)/(x n 1 1)(x n 2 1) has parameters[35,11,5]. Page 25 May 2012

27 The Minimum Weight of the Codes C λ Theorem 10 Assume that q D (2) 0. Let C (i,q) denote the cyclic code over GF(q) with the generator polynomial d i (x) for i=0 and i=1. Then the code C (i,q) has parameters [n,((n 1 + 1)(n 2 + 1) 2)/2,d i ], where d i min(n1,n 2 ). (9) If 1 D (2) 1, we have di 2 d i + 1 min(n 1,n 2 ). (10) Page 26 May 2012

28 The Minimum Weight of the Codes C λ Theorem 11 Let q D (2) j) 0. Let C(i, (n 1,n 2,q) denote the cyclic code over GF(q) with the generator polynomial g (i, j) (x)=(x n i 1)d j (x)/(x 1), and let d (n 1,n 2,q) denote the minimum distance of this code, where i {1,2} and j {0,1}. Then the code C (i, j) (n 1,n 2,q) where has parameters [ If 1 D (2) 1, we have ( n, (n i 1)(n i ( 1) i+ 1)+2 2 (i, j) d (n 1,n 2,q) ] (i, j),d(n 1,n 2,q), (i, j) (i, j) d (n 1,n 2,q) n i. (11) ) 2 d (i, j) (n 1,n 2,q) + 1 n i. (12) Page 27 May 2012

29 Examples of the Codes C λ Comment: The code C λ may be very good or even optimal. Example 12 Let (p,m,n 1,n 2 )=(2,1,5,3). Then q=2, n=15, and C λ is an optimal [15,5,7] cyclic code over GF(2) with generator polynomial x 10 + x 9 + x 8 + x 6 + x 5 + x Example 13 Let (p,m,n 1,n 2 )=(3,1,5,7). Then q=3, n=35, and C λ is a [35,23,5] cyclic code over GF(3) with generator polynomial x x x 10 + x 9 + 2x 8 + x 7 + x 5 + 2x 4 + x The best ternary linear code known of length 35 and dimension 23 has minimum distance 6. Page 28 May 2012

30 Examples of the Codes C λ Comment: The code C λ may be bad. Example 14 Let (p,m,n 1,n 2 )=(2,1,3,7). Then q=2, n=21, and C λ is a [21,7,3] cyclic code over GF(2) with generator polynomial x 14 + x This is a bad cyclic code due to its poor minimum distance. The code in this case is bad because q D (2) 0. Page 29 May 2012

31 Part IV: Concluding Remarks Page 30 May 2012

32 Concluding Remarks When n 1 and n 2 are twin primes, λ has ideal autocorrelation and its support is the famous twin-prime difference set. In this case, the code C λ may be very good or bad, depending on q and n 1 and n 2. When n 1 and n 2 are primes and n 2 = n 1 + 4, λ has also optimal autocorrelation and its support is an almost difference set. In this case, the code C λ may be very good or bad, depending on q and n 1 and n 2. The p-rank of the twin-prime difference set and the almost difference set are known now. The idea of constructing cyclic codes using special sequences is simple, but very promising. Page 31 May 2012

Binary Sequences with Optimal Autocorrelation

Cunsheng DING, HKUST, Kowloon, HONG KONG, CHINA September 2008 Outline of this talk Difference sets and almost difference sets Cyclotomic classes Introduction of binary sequences with optimal autocorrelation