On cyclic codes of composite length and the minimal distance

Transcription

1 1 On cyclic codes of composite length and the minimal distance Maosheng Xiong arxiv: v1 [cs.it] 31 Mar 2017 Abstract In an interesting paper Professor Cunsheng Ding provided three constructions of cyclic codes of length being a product of two primes. Numerical data shows that many codes from these constructions are best cyclic codes of the same length and dimension over the same finite field. However, not much is known about these codes. In this paper we explain some of the mysteries of the numerical data by developing a general method on cyclic codes of composite length and on estimating the minimal distance. Inspired by the new method, we also provide a general construction of cyclic codes of composite length. Numerical data shows that it produces many best cyclic codes as well. Finally, we point out how these cyclic codes can be used to construct convolutional codes with large free distance. Index terms Quadratic residue code, cyclic code of composite length, minimum distance, convolutional code. I. INTRODUCTION The theory of error-correction codes is an important research area, and it has many applications in modern life. For example, error-correction codes are widely used in cell phones to correct errors arising from fading noise during high frequency radio transmission. One of the major challenges in coding theory remains to construct new error-correction codes with good parameters and to study various problems such as finding the minimum distance and designing efficient decoding and encoding algorithms. Let F q be a finite field of order q. Let, be two distinct odd primes such that,q = 1 and q is a quadratic residue for both and. In an interesting paper [1], Ding provided a general construction of cyclic codes of length and dimension + 1/2 over F q by using generalised cyclotomies M. Xiong is with the Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong mamsxiong@ust.hk.

2 2 of order two in Z, the multiplicative subgroup of the ring Z n1 := Z/ Z. This construction is reminiscent of that of quadratic residue codes of prime length, which can be defined by using cyclotomy of order two in Z n when n is a prime number. There are three different generalised cyclotomies of order two in Z n when n is a product of two distinct odd primes, corresponding to Ding s first, second and third construction, each of which yields 8 cyclic codes of length n and dimension n+ 1/2. More interestingly, by using Magma and extensive numerical computation, Ding found some striking information on these 24 cyclic codes, which we list below in the table see [1]:,,q 7,17,2 11,13,3 5,7,4 [ n1, ] +1 2 [119, 60] [143, 72] [35, 18] Constructio 4 codes d = 12* 4 codes d = 12* 4 codes d = 8* 4 codes d = 11 4 codes d = 11 4 codes d = 7 Constructio 4 codes d = 12* 4 codes d = 12* 4 codes d = 7 4 codes d = 6 4 codes d = 6 4 codes d = 4 Construction 3 4 codes d = 8 4 codes d = 12* 4 codes d = 8* 4 codes d = 4 4 codes d = 6 4 codes d < 8 *: best cyclic codes of the same length & dim over F q. Here d is the min distance. The above data clearly demonstrates that Ding s three constructions of cyclic codes are promising and may produce many best cyclic codes. Hence it is worth studying these cyclic codes. In this paper we provide a theory on Ding s three constructions that partially explains some mysteries of the above data: first, we prove that under permutation equivalence, there are indeed two codes in each construction; second, we prove an almost square-root bound on the minimum distance which is satisfied by all these codes; third, for Ding s second and third construction, we illustrate why half of the cyclic codes have relatively small minimal distance see Theorems 4, 5 and 6 in Section IV. In order to study these codes, we develop a general method to study cyclic codes of composite length and to estimate the minimum distance see Theorem 1 in Section II. This method seems new and may be useful for other problems in coding theory. Inspired by this method, we provide a general construction of cyclic codes of composite length which are also related to quadratic residue codes of prime length and study their properties see Theorems 2 and 3 in Section III. Numerical data shows that many of these codes are the best cyclic codes of the same length and dimension over F q and hence are interesting. Finally, in light of [8, Theorem 3], cyclic codes of composite length and the estimate of the minimal distance are very useful

3 3 in the construction of convolutional codes with a prescribed free distance. As applications of results in the paper, we construct two families of convolutional codes with large free distance by cyclic codes from our construction and from Ding s constructions Theorem 8 in Section V. In Section VI we conclude the paper and propose two open problems. II. CYCLIC CODES OF COMPOSITE LENGTH In this section we study cyclic codes of composite length. We first introduce some standard notation which will be used throughout the paper. Let F q be the finite field of order q, where q is a prime power. A linear [n,k,d;q] code C is a k- dimensional subspace of F n q with minimum Hamming distance d = dc. A linear [n,k] code C over F q is called a cyclic code of length n if any c 0,c 1,...,c n 1 C implies c n 1,c 0,c 1,...,c n 2 C. By identifying any vector c 0,c 1,...,c n 1 F n q with cx = c 0 +c 1 x+ +c n 1 x n 1 R n := F q [x]/x n 1, it is known that C is a cyclic code of length n over F q if and only if the corresponding subset of R n, still written as C, is an idea of the ring R n. Since every ideal of R n is principal, there is a monic polynomial gx F q [x] of least degree such that C = gx R n. This gx is unique, satisfying gx x n 1 and is called the generator polynomial of C, and hx := x n 1/gx is called the parity-check polynomial of C. Two codes C 1 and C 2 are called permutation equivalent, written as C 1 C 2 if there is a permutation of coordinates that sends C 1 to C 2. The permutation of coordinates is called a permutation equivalence. For any cx F q [x], define Suppcx to be the set of integers i such that the term x i appears in cx. Define the weight wtcx to be the cardinality of the set Suppcx. The main theorem is the following. Theorem 1. Let n,r 2 be positive integers such that gcdnr,q = gcdn,r = 1. Assume that r q 1. Let θ be a primitive nr-th root of unity in some extension of F q. Define λ := θ n and let n be a positive integer such that n n 1 mod r. For any 0 t r 1, let θ t := λ nt. We define the map φ : r F q [x] x nr 1 Fq [x] x n 1

4 4 by φ : cx 1 r r 1 r 1 c t xλ tk k=0, here for any cx F q [x]/x nr 1, the polynomial c t x F q [x]/x n 1 is given by c t x cxθ t mod x n 1 0 t r 1. Then: 1 the map φ is a permutation equivalence, and cx 0 if and only if c t x r 1 0; 2 if cx 0, then Moreover, wtcx min t {wtc t x : c t x 0}. 2.1 if c 0 x = c 1 x = = c r 1 x, then wtcx = wtc 0 x; 2.2 if c t x = 0 for some t, then wtcx 2min t {wtc t x : c t x 0}. Let C = gx F q [x]/x nr 1 be a cyclic code with the generator polynomial gx. Then 3 C is permutation equivalent to φc, which is given by r 1 r 1 φc = c t xλ tk : c t x C t 0 t r 1, k=0 where C t = g t x F q [x]/x n 1 is a cyclic code with the generator polynomial g t x given by g t x = gcdgxθ t,x n 1 0 t r 1. 4 If C 0, then Moreover, 4.1 if C 0 = C 1 = = C r 1, then dc = dc 0 ; 4.2 if C t = 0 for some t, then dc min t {dc t : C t 0}. dc 2min t {dc t : C t 0}.

5 5 Proof. 1. Writting n 1 c t x = c t,z x z, cx = z=0 then from c t x cxθ t mod x n 1, we find r 1 n 1 c t,z = k=0 z=0 nr 1 i=0 r 1 n 1 c i x i = c kn+z x kn+z, k=0 z=0 c kn+z θt kn+z, t,z. Since θ t = λ nt and λ is a primitive n-th root of unity, we can obtain c kn+z = 1 r r 1 c t,z θt kn z, t,z. Thus we have cx = 1 r 1 n 1 r 1 c t,z x z x nk λ tk+ nz. r z=0 k=0 For any given z, let k k+ nz mod r. Noting that x nk x nk nz mod x nr 1 and as k runs over a complete residue system modulo r, so does k k nz mod r, and it is clear that ψ : x nk x nk induces a permutation of coordinates in F q [x]/x nr 1, thus cx is permutation equivalent to ψcx = 1 r 1 n 1 r 1 c t,z x z x nk λ tk = 1 r 1 r r z=0 k=0 k=0 c t xλ tk. 1 x nk r 1 Hence ψcx is permutation equivalent to φcx, and thus φ is a permutation equivalence. Moreover, noting c t x = 0 x n 1 c t x x n λ t cx, and r 1 x nr 1 = x n λ t, it is obvious that cx 0 if and only if c t x r 1 2. Since c t x cxθ t mod x n 1, we have 0. This proves 1. wtcx = wtcxθ t wtc t x 0 t r 1.

6 6 Taking c 0 x = = c r 1 x i, and using r 1 r : if r k λ tk = 0 : if r k, we find easily that ψcx = c 0 x. This proves 2 and 2.1. As for 2.2, let A = {0 t r 1 : c t x 0}, I = t ASuppc t x. From 1 we find wtcx = z I r 1 wt c t,z λ tk. k=0 t A For each k and z, write h k,z := t A c t,z λ tk, and for eachz, writeh z := h k,z r 1 k=0 andc z := c t,z t A as column vectors. Assume thata = {t 1,t 2,...,t u }. Then we have and λ t 1 λ t 2 λ tu c.... z = h z, 2 λ r 1t 1 λ r 1t 2 λ r 1tu wtcx = z I wth z. Noting that for any z I, we have c z 0, and the matrix on the left side of the equatio is a Vandermond matrix of size r u where 1 u < r, we find wth z 2. Thus wtcx z I 2 2min t {wtc t x : c t x 0}. This proves For any 0 t r 1, define the maps Φ t,f t by F q [x] x nr 1 Φ t F q [x] x n λ t f t F q [x] x n 1 Φ t : cx cx mod x n λ t, f t : x xθ t.

7 7 Let Ψ t := f t Φ t and Ψ = Ψ t r 1. The isomorphism from the Chinese Remainder Theorem induces an isomorphism ΨC = r 1 Ψ : r 1 F q [x] x nr 1 F q [x] x n 1 Ψ t C. Since clearly C t = Ψ t C, hence 3 is proved. Finally, 4, 4.1 and 4.2 are direct consequences of This completes the proof of Theorem 1. Remark. In Theorem 1, the requirement r q 1 is mild and can be removed as follows: suppose C = gx F q [x]/x nr 1 is a cyclic code and r q 1. There is a k such that r q k 1. Let q = q k and C = gx F q [x]/x nr 1 be the extension of C over F q. It is known that wtc = wt C [6, Theorem 3.8.8]. Hence to study the minimum distance of C over F q, it is equivalent to study C on which the condition r q 1 is satisfied. III. CYCLIC CODES OF COMPOSITE LENGTH RELATED TO QUADRATIC RESIDUE CODES Inspired by Theorem 1, we give a construction of cyclic codes of composite length which are closely related to quadratic residue codes of prime length. Throughout this section, we make the following assumptions: 3.1 F q is the finite field of order q, 3.1 n 3 is a prime number such that q n = 1 where n is the Legendre symbol for the prime n, 3.3 r 2 is a positive integer with gcdnr,q = gcdn,r = 1. Let θ be a primitivenr-th root of unity in some extension of F q and let λ := θ n. For each 0 t r 1, we may factor x n λ t as n x n λ t = x θ t+rj. j=1 For each t, there is a unique integer j t where 1 j t n such that t + rj t 0 mod n. Define θ t := θ t+rjt. It is easy to see that θ t = λ nt where n is a positive integer such that nn 1 mod r. We can write x n λ t = x θ t g t,1 xg t, 1 x,

8 8 where for {±1}, g t, x = 1 j n t+rj n = x θ t+rj. For any = 0, 1,..., r 1 {±1} r, define r 1 g x = g t,t x. 3 Let Λ := { = 0, 1,, r 1 {±1} r : j = qj 0 j r 1}. 4 Here the subscript qj in qj is always understood to be qj mod r. For any Λ, since q n = 1, it is easy to see that g x F q [x]. We remark here that a defining set similar to Λ has been used in [3] to construct binary duadic codes of prime length. Definitio. Assume We define C to be the cyclic code of length nr over F q with the generator polynomial g x given in 3, that is, C = g x F q [x]/x nr 1. 5 Remark. The code C in 5 is a cyclic code over F q of length nr and dimension n+1r/2. The total number of these codes is 2 τq,r, where τq,r := #Λ is the number of q-cyclotomic cosets modulo r. In particular there are always at least 4 such codes since τq,r 2 for any r 2, and if q 1 mod r, then τq,r = r, hence in this case the number of such codes is 2 r. In general there may be many such cyclic codes C. We first study the codes C under permutation equivalence. For any = 0, 1,, r 1 Λ, define := 0, 1,, r 1. For any u with gcdu,r = 1, define u := u t r 1 as u t := tu 0 t r 1. We have the following: Theorem 2. Assume For any Λ and any u such that gcdu,r = 1, the codes C,C and

9 9 C u given in 5 are all permutation equivalent. In particular we have dc = dc = dc u. Proof. For C, by Theorem 1, we find that g t x = gcdg xθ t,x n 1 = g 0,t x, 0 t r 1. Hence the code C is permutation equivalent to r 1 r 1 φc = c t xλ tk : c t x C 0,t 0 t r 1, where k=0 C 0,t := g 0,t x F q [x]/x n 1 6 is a quadratic residue code of length n and dimension n+1/2 over F q. Let s be a positive integer such that s n = 1. It is known that the map x x s induces a permutation equivalence from F q [x]/x n 1 to itself and c t x C 0,t if and only if c t x s C 0, t. Thus by taking x x s we find that φc is permutation equivalent to φc. For any cx C, φcx = r 1 r 1 c t xλ tk = k=0 r 1 r 1 c ut xλ utk, where c ut x C o,ut, and the subscript ut is understood to be ut mod r. It is easy to see that the map k uk induces a permutation equivalence between φcx and r 1 r 1 c ut xλ tk. Hence C and C u are permutation equivalent. This proves Theorem 2. k=0 k=0 Now we can estimate the minimum distance of these codes C. Let d n,q be the minimum distance of the quadratic residue code C 0, given in 6 = ±1. It is known that d n,q > n [6].

10 Theorem 3. Assume For any Λ, the code C given in 5 is an following properties: 1 dc d n,q ; 2 if = 0,0,,0 or 1,1,,1, then dc = d n,q ; 3 if 0,0,,0 or 1,1,,1, then { dc min d n,q +1, 8n+1 1 Proof. 1 and 2 are direct consequences of Theorem 1, noting that for any t, 2 }. [ nr, n+1r 2 ] 10 code with the g t x = gcdg xθ t,x n 1 = g 0,t x, and C t = g 0,t x F q [x]/x n 1, dc t = d n,q. As for 3, suppose dc = d n,q. Let cx C with wtcx = d n,q. Then by the proof of Theorem 1, the corresponding c t x r 1 satisfies i c t x 0 for any t, ii let I = r 1 Suppc t x, then #I = d n,q. So for each t, we have 0 c t x = z I c t,zx z C 0,t with wtc t x = d n,q being the minimum weight in a quadratic residue code of length n and dimension n + 1/2. It is known that c t 1 0 for all t. There are t 1,t 2 such that c t1 x C 0,1, c t2 x C 0, 1. So we have x n 1 x 1 c t 1 xc t2 x = wtc t1 xc t2 x = n. On the other hand, c t1 xc t2 x = z 1,z 2 I c t1,zc t2,xx z 1+z 2 mod x n 1. We have n = wtc t1 xc t2 x #I+I d n,q + dn,q 2 = d n,qd n,q +1. 2

11 11 This implies that This completes the proof of Theorem 3. d n,q 8n Example 1. In examples below, we use Magma to compute the parameters of some codes for q = 2,3,4 and a few small values n,r. We only consider codes which are not permutation equivalent by Theorem 2. The -symbol indicates that the code is the best among all cyclic codes, -symbol indicates that the code is the best among all linear codes, and -symbol indicates that the code is optimal among all codes. For q = 2,3, such information is obtained by consulting [2, Appendix A: Tables of best binary and ternary cyclic codes]; for q = 4 we consult the online reference to check the best linear codes. There are also some sporadic examples of best cyclic codes that we exclude from the table. It demonstrates that the construction produces many best cyclic codes, and when the parameters are small, it may even produce best linear codes. q = 2 n = 7 n = 17 n = 23 n = 31 r = 3 [21,12] code [51,27] code [69,36] code [93,48] code 1 code d = 5 1 code d = 9 1 code d = 11 1 code d = 14 1 code d = 3 1 code d = 5 1 code d = 7 1 code d = 7 r = 5 [35,20] code [85,45] code [115,60] code [155,80] code 1 code d = 6 1 code d = 10 1 code d = 14 1 code d = 14 1 code d = 3 1 code d = 5 1 code d = 7 1 code d = 7 r = 7 NA [119, 63] code [161, 84] code [217, 112] code 1 code d = 11 2 codes d = 14 NA 1 code d = 10 1 code d = 7 1 code d = 5

12 12 q = 3 n = 11 n = 13 n = 23 n = 37 r = 2 [22,12] code [26,14] code [46,24] code [74,38] code 1 code d = 7 1 code d = 7 1 code d = 13 1 code d = 14 1 code d = 5 1 code d = 5 1 code d = 8 1 code d = 10 r = 4 [44,24] code [52,28] code [92,48] code [148,76] code 2 codes d = 8 2 codes d = 9 2 codes d = 14 NA 1 code d = 7 1 code d = 7 1 code d = 13 1 code d = 5 1 code d = 5 1 code d = 8 q = 4 n = 5 n = 7 n = 11 n = 13 n = 17 r = 3 [15,9] code [21,12] code [33,18] code [39,21] code [51,27] code 2 codes d = 5 2 codes d = 5 2 codes d = 8 2 codes d = 9 2 codes d = 9 1 code d = 3 1 code d = 3 1 code d = 5 1 code d = 5 1 code d = 5 Throughout this section we assume that 4.1 F q is the finite field of order q, IV. CYCLIC CODES FROM DING S CONSTRUCTIONS 4.2, are two distinct odd primes such that q = q = 1 and gcdn1,q = 1, 4.3 θ is a primitive -th root of unity in some extension of F q, and λ := θ. Ding s three constructions of cyclic codes can be described explicitly by using the three different generalized cyclotomies of order two in Z see [1]. However, it may look more natural to describe these constructions as follows. A. Ding s 1 st construction We write x 1 = where for {±1}, 1 i=0 x θ i = x 1F 1 1,1 xf1 1, 1 xf 2,1xF 2, 1 xf 3,1 xf 3, 1 x, F 1 1, x = 1 i 1 i = x θ i, 7

13 13 here = is the Jacobi symbol for n1, and F 2, x = F 3, x = 1 i 1 i = 1 i 1 i = Since q = q = 1, we have F 1 1, x,f 2,x,F 3, x F q [x] for all. x θ i, 8 x θ i. 9 Denote by d n1,q the minimum distance of the code F 3, x F q [x]/x 1. This is a quadratic residue code of length and dimension +1/2 over F q. Denote by d n1,q the minimum distance of the code x 1F 3, x F q [x]/x 1. It is known that d n1,q = d n1,q +1 and d n1,q >. We also define d n2,q and d n2,q similarly. Definitio Ding s 1 st constructions. Assume For any := 1, 2, 3 {±1} 3, let g 1 x := F 1 1, 1 xf 2,2 xf 3,3 x. Then define C 1 to be the cyclic code of length over F q with the generator polynomial g 1 x, that is, C 1 = g 1 x F q [x]/x Next we state the main results on these codes C 1. Theorem 4. i For any {±1} 3, the cyclic code C 1 over F q given i0 has length and dimension +1/2, satisfying d C 1 > max{dn1,q,d n2,q} > max{, }. ii Write {±1} 3 = A 1 A 2 where A 1 = {1,1,1, 1, 1,1, 1,1, 1,1, 1, 1}, A 2 = { 1, 1, 1,1,1, 1,1, 1,1, 1,1,1}.

14 14 Then for i = 1 or 2, the codes C 1 with A i are all permutation equivalent. Proof. i. Obviously the code C 1 is of length and dimension +1/2. To study the minimum distance, we may assume that q 1. Applying Theorem 1 for n = and r =, we find that C 1 φ C 1 = n2 1 is permutation equivalent to n2 1 c t xλ tk : c t x C t 0 t 1, where for each t, C t = g t x F q [x]/x 1 is a cyclic code with g t x = gcd g 1 xθ t,x 1 = k=0 g 0,3 x : t = 0 x 1g 0,1 2 x : g 0, 1 2 x : t = 2 t = 2. Here θ t = λ t where is a positive integer such that 1 mod, and g 0, x = F 3, x for any. It implies from Theorem 1 that d C 1 d n1,q. A closer look from Theorem 1 easily shows that the equality = can not be achieved, hence we have d C 1 > d n1,q. On the other hand, we may also apply Theorem 1 for n = and r = to obtain d C 1 > d n2,q > n2. Thus we obtain d C 1 > max{dn1,q,d n2,q} > max{, }. ii. For any positive integer u such that gcdu, = 1, the map x x u induces a permutation equivalence φ u : F q [x]/x 1 F q [x]/x 1. Choose two positive integers a,b such that a = 1, and for any = 1, 2, 3 {±1}, let a = 1, b = 1, b = 1, a := 1, 2, 3, b := 1, 2, 3. It is easy to see that φ a C 1 1 = C, φ a b C 1 1 = C. b So the codes C 1,C 1 a and C 1 b are all permutation equivalent. Considering these two actions a and b, it is easy to check that the codes C 1 are

15 15 permutation equivalent for all A i where i = 1 or 2. This completes the proof of Theorem 4. B. Ding s 2 nd construction We write x 1 = where for any {±1}, 1 i=0 x θ i = x 1F 2 1,1 xf2 1, 1 xf 2,1xF 2, 1 xf 3,1 xf 3, 1 x, F 2 1, x = 1 i 1 i = gcdi, =1 and the functions F 2, x and F 3, x are defined in 8 and 9 respectively. x θ i, 11 Definition 3 Ding s 2 nd constructions. Assume For any := 1, 2, 3 {±1} 3, let g 2 x := F 2 1, 1 xf 2,2 xf 3,3 x. Then define C 2 to be the cyclic code of length over F q with the generator polynomial g 2 x, that is, Next we state the main results on these codes C 2. C 2 = g 2 x F q [x]/x Theorem 5. i For any {±1} 3, the cyclic code C 2 over F q given i2 has length and dimension +1/2, satisfying d C 2 > dn1,q >. ii Write {±1} 3 = A 1 A 2 where A 1 = {1,1,1,1, 1,1, 1,1, 1, 1, 1, 1}, A 2 = {1,1, 1,1, 1, 1, 1,1,1, 1, 1,1}. Then for i = 1 or 2, the codes C 2 for A i are all permutation equivalent.

16 16 iii Taking 1 = 1,1,1 A 1, then d C 2 1 = d n1,q +1. iv If 1 mod 4, and q is a prime power such that q = 2 m or q 1 mod 4, let γ F q such that 1+γ 2 = Then for any {±1} 3, the code C 2 := { C 2 given by c 0,...,c n1 1,c : c = γc1, cx = has parameters [ ] +1, +1 2 and is self-dual. 1 i=0 c i x i C 2 } Proof. i. Obviously the code C 2 is of length and dimension +1/2. To study the minimum distance, we may assume that q 1. Applying Theorem 1 for n = and r =, we find that C 2 φ C 2 = n2 1 is permutation equivalent to n2 1 c t xλ tk : c t x C t 0 t 1, where for each t, C t = g t x F q [x]/x 1 is a cyclic code with g t x = gcd g 1 xθ t,x 1 = It implies from Theorem 1 that d C 1 k=0 > d n1,q >. g 0,3 x : t = 0 x 1g 0,1 x : g 0,1 x : t = 2 t = 2. ii. The permutation equivalence of C 2 can be proved in exactly the same way as that of C 1, hence we omit the details. We are contented by stating that for any = 1, 2, 3 {±1} 3, define a := 1, 2, 3, b := 1, 2, 3, then the codes C 2, C 2 a and C 2 b are all permutation equivalent. Considering these two actions a and b, ii is easily verified. iii Let = 1,1,1. Applying Theorem 1 for n = and r =, we find that C 2 is permutation

17 17 equivalent to φ n2 1 n2 1 C 2 = c t xλ tk : c t x C t 0 t 1, k=0 where for each t, C t = g t x F q [x]/x 1 is a cyclic code with g t x = gcd g 2 xθ t,x 1 = g 0,1 x : t = 0 x 1g 0,1 x : g 0,1 x : t = 1 t = 1. We may choose c x x 1g 0,1 x F q [x]/x 1 with wtc x = d n1,q = d n,q + 1, and let c t x = c x for all t. The corresponding codeword cx C 2 proved. iv It was noted in [1] that C 2 satisfies wtcx = d n1,q. Now iii is is a duadic code. When 1 mod 4, then µ 1 defines a splitting of C 2, under the condition the equatio3 is always solvable for γ F q, hence see 6.11 of page 226 in [6] the extended code C 2 is self-dual. This completes the proof of Theorem 5. C. Ding s 3 rd construction We write x 1 = where for any {±1}, 1 i=0 x θ i = x 1F 3 1,1xF 3 1, 1xF 2,1 xf 2, 1 xf 3,1 xf 3, 1 x, F 3 1,x = 1 i 1 i = gcdi, =1 x θ i, 14 and the functions F 2, x and F 3, x are defined in 8 and 9 respectively. Definition 4 Ding s 3 rd constructions. Assume For any := 1, 2, 3 {±1} 3, let g 3 x := F 3 1, 1 xf 2,2 xf 3,3 x. Then define C 3 to be the cyclic code of length over F q with the generator polynomial g 3 x, that

18 18 is, C 3 = g 3 x F q [x]/x By switching the roles, it is easy to see that Ding s 3 rd construction is the same as Ding s 2 nd construction, hence the properties of the codes C 3 follow directly from Theorem 4. For the sake of completeness, we state the results explicitly below. Theorem 6. i For any {±1} 3, the cyclic code C 3 over F q given i5 has length and dimension +1/2, satisfying d C 3 > dn2,q >. ii Write {±1} 3 = A 1 A 2 where A 1 = {1,1,1,1,1, 1, 1, 1,1, 1, 1, 1}, A 2 = {1, 1,1,1, 1, 1, 1,1,1, 1,1, 1}. Then for i = 1 or 2, the codes C 3 iii Taking 1 = 1,1,1 A 1, then for A i are all permutation equivalent. d C 3 1 = d n2,q +1. iv If 1 mod 4, and q is a prime power such that q = 2 m or q 1 mod 4, let γ F q such that 1+γ 2 = 0. Then for any {±1} 3, the code { C 3 := C 3 given by c 0,...,c n1 1,c : c = γc1, cx = has parameters [ ] +1, +1 2 and is self-dual. 1 i=0 c i x i C 3 } V. APPLICATION TO CONVOLUTIONAL CODES The class of convolutional codes was invented by Elias [4] i955 and has been widely in use for wireless, space, and broadcast communications since the 1970s. Compared with linear block codes,

19 19 however, convolutional codes are not so well understood. In particular, there are only a few algebraic constructions of convolutional codes with good designed parameters. Convolutional codes can be defined as subspaces over the rational functional field F q D see [7], [9] or over the field of Laurent series F q D [10], or as submodules over the polynomial ring F q [D] [5], [11], and the theories are all equivalent. Here we follow the presentation of [5]. Interested readers may consult the textbooks [7], [9], [10] for more information. Definition 5. Let F q [D] be the polynomial ring and F q D the field of rational functions. For any 1 k n, let GD F q [D] k n be a k n polynomial matrix with rank FqDGD = k. The rate k/n convolutional code C generated by GD is defined as the set C = { udgd F q D n : ud F q D k}. The matrix GD is called a generator matrix or an encoder of C. Let GD = g i,j D F q [D] k n be an encoder of C. The i-th row degree v i of GD is v i = max j degg i,j D. If the encoders GD and G D generate the same code C, then we say GD and G D are equivalent encoders. It is known that each convolutional code C can be generated by a minimal basic encoder GD, that is, GD has a polynomial right inverse, and the sum k i=1 v i of the row degrees of GD attains the minimal value among all encoders of C. Let GD F q [D] k n be a minimal basic encoder of C, then GD is automatically noncatastrophic, that is, finite-weight codewords can only be produced from finite-weight messages, and the set {v i : 1 i k} of the row degrees of GD is invariant among all minimal basic encoders of C. Hence we can define the degree of C as δ := k i=1 v i by using the minimal basic encoder GD of C. We call this C an n,k,δ q convolutional code. Each vd F q D n can be expanded uniquely as a Laurent series vd = j= v jd j where v j F n q for each j. Define the weight wtvd as wtvd := j= wtv j. Here wtv j denotes the Hamming weight of the vector v j F n q. The free distance of the convolutional code C FD n is defined as d free := min{wtvd : vd C,vD 0}.

20 20 It is known that see [11] d free := min{wtvd : vd C F q [D] n,vd 0}. For a given convolutional code C, the four parameters n,k,δ,d free are of fundamental importance because k/n is the rate of the code, δ and d free determine respectively the decoding complexity and the error correcting capability of C with respect to some decoding algorithms such as the celebrated Viterbi algorithm [12]. For these reasons, for given rate k/n and q, generally speaking, it is desirable to construct convolutional codes with relatively small degree δ and relatively large free distance d free. There are only a few algebraic constructions of convolutional codes. One particularly useful construction of convolutional codes is given in [8, Theorem 3], which we describe below. For any positive integer n and any cx F q [x], write n 1 cx = c i x n x i, where c i x F q [x] for each i. This induces a map φ n given by i=0 φ n : F q [x] F q [x] n, cx c i x n 1 i=0. For simplicity, we write φ n cx F q [x] n as an n 1 column vector. It is easy to see that for any positive integers m,n, this map φ n defines a permutation equivalence from F q [x]/x mn 1 to F q [x]/x m 1 n. For any a,b in some finite extension of F q, we say that a and b are n-equivalent or belong to the same n-equivalence class if a n = b n. Theorem 7. [8, Theorem 3] Let m,n be positive integers such that gcdnm,q = 1. Let C = gx F q [x]/x nm 1 be a cyclic code of length nm with the generator polynomial gx F q [x]. If gx has at most n k roots in each n-equivalence class, then the matrix GD F q [D] k n given by GD T := [ φ n gd,φ n DgD,...,φ n D k 1 gd ] is a minimal basic encoder, and the code generated by GD is a rate k/n convolutional code with the free distance d free satisfying d free dc. Here GD T is the transpose of the matrix GD, and dc is the minimal distance of the code C. We remark that Theorem 7 has been used in the construction of MDS-convolutional codes for any rate

21 21 k/n and any degree δ over a large field F q [11, Theorem 3.3]. In light of Theorem 7, it is readily seen that cyclic codes of composite length mn and the estimate of the minimal distance are very useful in the construction of convolutional codes with prescribed rate and free distance. We also remark that generally speaking, it is not an easy task to construct a generator polynomial gx such that gx has at most n k roots in each n-equivalence class and the corresponding cyclic code has large minimal distance at the same time see [8, Theorem 6]. Using Theorem 7, we note that the cyclic codes from 5 and from Ding s 1 st construction i0 are useful to construct convolutional codes with large free distance. Theorem 8. Let F q be a finite field of order q. i Let n be a prime number such that q n = 1, and r 2 be a positive integer such that gcdnr,q = gcdn,r = 1. Let Λ \ {0,...,0,1,...,1} {±1} r where Λ is given in 4, and let C = g x F q [x]/x nr 1 be the cyclic code of lengthnr given in 5. Then for any1 k n+1/2, the code generated by the encoder GD given by GD T := [ φ n g D,φ n Dg D,...,φ n D k 1 g D ] is a rate k/n convolutional code with the free distance satisfying d free > n. ii Let, be two distinct odd primes such that q = q = 1 and gcdn1,q = 1. For any {±1} 3, let C 1 = g 1 x F q [x]/x 1 be the cyclic code of length given i0. Then for any 1 k 1/2, the code generated by the encoder GD given by GD T := [ φ n1 g D,φ n1 Dg D,...,φ n1 D k 1 g D ] is a rate k/, degree 1k 2 convolutional code over F q with the free distance satisfying d free > max { n1, }. Proof. The only thing we need to check is that the generator polynomial gx in i and ii has at most n k roots in each n-equivalence class, which is almost trivial, so we omit the details. Example 2. For q = 2,n = 7,r = 3, using the cyclic code in 5, for = 1, 1, 1, we find g x = x 9 +x 8 +x 7 +x 5 +x 4 +x+1.

22 22 The code C = g x F 2 [x]/x 21 1 has parameters [21,12,5]. We may write g x = 6 g i x 7 x i, i=0 where g 0 x = 1+x, g 1 x = 1+x, g 2 x = x, g 3 x = 0, g 4 x = 1, g 5 x = 1, g 6 x = 0. Taking k = 4, from 1 of Theorem 8, the matrix GD F 2 [D] 3 7 given by 1+D 1+D D D 1+D D GD = D 0 1+D 1+D D 0 1 D D 0 1+D 1+D D 0 is a minimal basic encoder which generates a rate 4/7, degree 4 convolutional code C over F 2 with free distance d free 5. Actually the codeword c = [1,1,1,0] GD = [1,0,D,1,1+D,0,0] has weight 5, so d free = 5. We would like to mention that the Heller bound [7, Corollary 3.18] implies that the largest free distance d free of any binary, rate 4/7 and unit-memory convolutional code satisfies d free 7. Example 3. For q = 2, = 7, = 17, using Ding s 1 st construction see 10, for = 1,1,1, we find g 1 x = x 59 +x 58 +x 57 +x 56 +x 51 +x 50 +x 49 +x 48 +x 47 +x 46 +x 45 +x 44 +x 43 +x 39 +x 38 +x 37 +x 36 +x 33 +x 32 +x 30 +x 24 +x 22 +x 20 +x 19 +x 18 +x 15 +x 14 +x 9 +x 8 +x 7 +x 6 +x 5 +x 4 +x+1. The code C 1 = g 1 x F 2 [x]/x has parameters [119,60,12]. We may write g 1 x = 6 g i x 7 x i, i=0

23 23 where g 0 x = 1+x+x 2 +x 7 +x 8, g 1 x = 1+x+x 2 +x 3 +x 5 +x 6 +x 7 +x 8, g 2 x = x+x 4 +x 5 +x 6 +x 7 +x 8, g 3 x = x 3 +x 5 +x 6 +x 8, g 4 x = 1+x 2 +x 4 +x 5 +x 6, g 5 x = 1+x 2 +x 4 +x 6, g 6 x = 1+x 2 +x 6. Taking k = 3, from 2 of Theorem 8, the matrix GD F 2 [D] 3 7 given by g 0 D g 1 D g 2 D g 3 D g 4 D g 5 D g 6 D GD = Dg 6 D g 0 D g 1 D g 2 D g 3 D g 4 D g 5 D Dg 5 D Dg 6 D g 0 D g 1 D g 2 D g 3 D g 4 D is a minimal basic encoder which generates a rate 3/7, degree 24 convolutional code C over F 2 with free distance d free 12. VI. CONCLUSION In this paper, we develop a general method to study cyclic codes of composite length and to estimate the minimal distance Theorem 1, which can be used to study the cyclic codes from Ding s three constructions Theorems 4, 5, 6. We also give a general construction of cyclic codes of composite length which are related to quadratic residue codes of prime length and study some of their properties Theorems 2 and 3. The cyclic codes in this paper can be used to construct convolutional codes with large free distance. While we have obtained an almost square-root bound on the minimal distance of all these cyclic codes that we are interested in, numerical data shows that the minimal distance of these codes should be much larger. So we list the following as open questions: 1 For the code C given in 5, is it possible to improve the lower bound on dc in Theorem 3 when 0,...,0 or 1,...,1? 2 For the codes C i i = 1,2,3 from Ding s three constructions, is it possible to improve the lower bound on d C i from Theorems 4, 5 and 6? In particular is it true that d C 1 for any {±1} 3 and d for any i = 2,3 and = 1, 1, 1? C i

24 24 Acknowledgments The author is grateful to Professor Cunsheng Ding for many useful suggestions of the paper. In particular the numerical computation by the end of Section III can not be done so efficiently without Professor Ding s generous sharing of his original Magma codes from the work [1]. REFERENCES [1] C. Ding, Cyclotomic constructions of cyclic codes with length being the product of two primes, IEEE Trans. Inform. Theory , no. 4, [2] C. Ding, Codes from difference sets, World Scientific Publishing, [3] C. Ding, V. Pless, Cyclotomy and Duadic codes of prime lengths, IEEE Trans. Inform. Theory , no. 2, [4] P. Elias, Coding for noisy channels, IRE Conv. Rep., Pt. 4, pp , [5] H. Gluesing-Luerssen, J. Rosenthal, and R. Smarandache, Strongly-MDS convolutional codes, IEEE Trans. Inform. Theory, , no. 2, [6] W. C. Huffman, V. Pless, Fundamentals of Error-Correcting Codes. Cambridge University Press, [7] R. Johannesson and K. S. Zigangirov, Fundamentals of Convolutional Coding. Piscataway, NJ: IEEE Press, [8] J. Justesen, New convolutional code constructions and a class of asymptotically good time-varying codes, IEEE Trans. Inform. Theory bf , no. 2, [9] R. J. McEliece, The algebraic theory of convolutional codes, in Handbook of Coding Theory, V. Pless and W. C. Huffman, Eds. Amsterdam, The Netherlands: Elsevier, 1998, vol. 1, pp [10] P. Piret, Convolutional Codes, an Algebraic Approach. Cambridge, MA: MIT Press, [11] R. Smarandache, H. Gluesing-Luerssen, and J. Rosenthal, Constructions for MDS-convolutional codes, IEEE Trans. Inform. Theory, vol. 47, no. 5, pp , [12] A. J. Viterbi, Error bounds for convolutional codes and an asymptotically optimum decoding algorithm, IEEE Trans. Inform. Theory, vol. 13, pp , 1967.