Good self-dual quasi-cyclic codes exist

This document is downloaded from DR‑NTU (https://dr.ntu.edu.sg) Nanyang Technological University, Singapore. Good self‑dual quasi‑cyclic codes exist Ling, San; Sole, Patrick 2003 Ling, S., & Solé, P. (2003). Good self‑dual quasi‑cyclic codes exist. IEEE Transactions on Information Theory, 49(4), 1052‑1053. https://hdl.handle.net/10356/96309 https://doi.org/10.1109/TIT.2003.809501 © 2003 IEEE. This is the author created version of a work that has been peer reviewed and accepted for publication by IEEE Transactions on Information Theory, IEEE. It incorporates referee’s comments but changes resulting from the publishing process, such as copyediting, structural formatting, may not be reflected in this document. The published version is available at: [http://dx.doi.org/10.1109/TIT.2003.809501]. Downloaded on 30 Oct 2022 20:46:54 SGT 1052 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 4, APRIL 2003 Good Self-Dual Quasi-Cyclic Codes Exist iii) The number of self-dual F 4 -codes of length ` is given by San Ling and Patrick Solé, Member, IEEE Abstract—We show that there are long binary quasi-cyclic self-dual (either Type I or Type II) codes satisfying the Gilbert–Varshamov bound. Index Terms—Cubing construction, Gilbert–Varshamov bound, quasicyclic codes, self-dual codes. N (4; `) = II. KNOWN FACTS AND NOTATIONS A code is said to be quasi-cyclic of index ` or `-quasi-cyclic if and only if it is invariant under T ` , where T denotes the cyclic shift. If ` = 1, such a code is just a cyclic code. We assume that all binary codes are equipped with the Euclidean inner product and all the F 4 -codes are equipped with the Hermitian inner product. The latter condition is necessary, when using the cubing construction, to ensure that the resulting binary code is Euclidean self-dual. Self-duality in the following discussion is with respect to these respective inner products. A binary self-dual code is said to be of Type II if and only if all its weights are multiples of 4 and of Type I otherwise. We first recall some background material on mass formulas for self-dual binary and quaternary codes. M (4; `) = N (2; `) = 01 =1 i 02 =1 i 02 =0 i (22i+1 + 1): Proposition 2.2: Let ` be a positive integer divisible by 8. i) The number of Type II binary codes of length ` is given by T (2; `) = 2 02 (2i + 1): =1 i ii) Let v be a codeword of length ` and Hamming weight divisible by 4, other than 0 and 1. The number of Type II binary codes of length ` containing v is given by S (2; `) = 2 03 =1 i (2i + 1): Proof: i) is found in [7] and ii) is exactly [6, Corollary 2.4]. III. MAIN RESULT Let C1 denote a binary code of length ` and C2 a quaternary code of length `. We construct a binary code C of length 3` by the cubing construction [3]. Define a map 8: C1 2 C2 0! F 32` by the rule 8(x; a + b!) := (x + a; x + b; x + a + b) (2i + 1): ii) Let v be a codeword of length ` and even Hamming weight, other than 0 and 1. The number of self-dual binary codes of length ` containing v is given by M (2; `) = =0 i Proof: i) and iii) are well-known facts, cf. [7]. ii) is an immediate consequence of [6, Theorem 2.1] with s = 2. (Note that every self-dual binary code must contain the all-one vector 1.) iv) follows from [2, Theorem 1] with n1 = ` and k1 = 1. Proposition 2.1: Let ` be an even positive integer. i) The number of self-dual binary codes of length ` is given by (22i+1 + 1): iv) The number of self-dual F 4 -codes of length ` containing a given nonzero codeword of length ` and even Hamming weight is given by I. INTRODUCTION It has been known for 30 years that good long self-dual codes exist [6], and for more than a quarter century [1] that there are good long quasi-cyclic codes of rate 1=2. In this correspondence, we show that good long self-dual quasicyclic codes exist. Building on well-known mass formulas for self-dual binary and quaternary codes, we derive a Gilbert–Varshamov bound for long binary self-dual quasi-cyclic codes. The proof uses the cubing construction of [5], [3] and the proof technique of [6]. As suggested by one referee, it might have been possible to build on [1] to derive this asymptotic result. However, [1] uses quasi-cyclic codes of index 2 while we use quasi-cyclic codes of index n=3, where n denotes the length. In some sense, we provide information on a different asymptotic ensemble of codes than [1]. 01 where a; b are binary vectors of length `, and we F 4 = f0; 1; !; !2 g. Then we can define the code C as Im (8) write C := f8(x; a + b!) j x 2 C1 ; a + b! 2 C2 g: Now a direct calculation shows that (2i + 1): 8(x; !2 (a + b!)) = (x + a + b; x + a; x + b) Manuscript received March 14, 2001; revised November 11, 2002. The work of S. Ling was supported in part by MOE-ARF Research Grant R-146-000-029-112 and by DSTA Research Grant R-394-000-011-422. S. Ling is with the Department of Mathematics, National University of Singapore, Singapore 117543, Republic of Singapore (e-mail: [email protected]). P. Solé is with CNRS, I3S, ESSI, 06903 Sophia Antipolis, France (e-mail: [email protected]). Communicated by S. Litsyn, Associate Editor for Coding Theory. Digital Object Identifier 10.1109/TIT.2003.809501 is a shift of 8(x; a + b! ) by ` places. Therefore, C is `-quasi-cyclic. Furthermore, it is easy to check that C is self-dual if and only if both C1 and C2 are, and C is of Type II if and only if C1 is of Type II and C2 is self-dual. We assume henceforth that is a self-dual code constructed in the above way. Any codeword c in C must necessarily have even Hamming weight. Suppose that c corresponds to the pair (c1 ; c 2 ), where c1 2 C1 and c2 2 C2 . Since C1 and C2 are self-dual, it follows that 0018-9448/03$17.00 © 2003 IEEE IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 4, APRIL 2003 6 c1 and c2 must both have even Hamming weights. When c = are three possibilities for the pair (c1 ; c 2 ): 1) 2) 3) 6 c1 = c1 = c1 = 6 0, c2 0, c 2 0, c 2 6 0, there 0; 0. H (x) := We try to enumerate the number of words c in each of these categories for a given weight d (d even). For type 2, if the Hamming weight of c is d, then c 2 has Hamming weight d=2. Since c 2 has even Hamming weight, it follows that d is divisible by 4 in order for this case to occur. It is easy to see that the number A2 (`; d) of such words c is given by d=` 2 3d=2 (4jd). For d not divisible by 4, set A2 (`; d) = 0. The argument to obtain the number of words of type 3 is similar. It is easy to show that the number A3 (`; d) of such words is given by d=` 3 (6jd). When d is not divisible by 6, A3 (`; d) = 0. For A1 (`; d), the number of words of type 1, we simply give an upper bound. The total number of words in F 32` of weight d is 3d` , so A1 (`; d)  IV. ASYMPTOTIC ANALYSIS We will require the celebrated entropy function 6= 0; = = 1053 3` d 0 A (`; d) 0 A (`; d): 2 3 Combining the preceding observations and Proposition 2.1, the number of self-dual binary `-quasi-cyclic codes of length 3` whose minimum weight is < d is bounded above by (A1 (`; e)M (2; `)M (4; `) + A2 (`; e)N (2; `)M (4; `) e<d; e even + A3 (`; e)M (2; `)N (4; `)): e<d e 0mod2  e + e<d e 0mod4  + e<d  ` e=3 ` e=2 3 e=2 0 2 ` 1 2 Theorem 4.1: There exists an infinite family of self-dual quasicyclic binary codes Ci of length 3`i and of distance di such that the limit  of di =3`i for large i exists and is bounded below as   (2 01 + 1)(2`01 + 1): If we are interested only in Type II `-quasi-cyclic codes, using Proposition 2.2, we see easily that the number of Type II binary `-quasi-cyclic codes of length 3` whose minimum weight is <d is bounded above by (A1 (`; e)S (2; `)M (4; `) + A2 (`; e)T (2; `)M (4; `) + A3 (`; e)S (2; `)N (4; `)): Theorem 3.2: Let ` be divisible by 8 and let d be the largest multiple of 4 such that e<d e 0mod4  + e<d e 0mod4  +  e<d ` e=3 ` e=2 3 e=2 0 ` 1 2 1 Proof: The right-hand side (RHS) of the inequality of Theorem 3.1 is plainly of the order of 23`=2 for large `. We compare this in turn to each of the three summands in the left-hand side (at the price of a more stringent inequality, congruence conditions on the summation range are neglected). By [5, Ch. 10, Corollary 9], for large ` (with  =  and n = `), the first and third summands are of order 23`H ( ) and 2`+`H () , respectively. They both are of the order of the RHS for H ( ) = 1=2. By [5, Ch. 10, Lemma 7], for large ` (with  =  and n = `), the second summand is of order 2`f (3=2) for f (t) := 0:5 + t log2 (3) + H (t), which is of the order of the RHS for  = 0:1762 111 Theorem 4.2: There exists an infinite family of Type II quasi-cyclic binary codes Ci of length 3`i and of distance di such that the limit  of di =3`i for large i exists and is bounded below as   H 0 (1=2) = 0:110 1 1 1 : 1 Proof: Since we neglected the congruence conditions in the preceding analysis, the calculations are exactly the same. REFERENCES e<d; e 0mod4 e  H 0 (1=2) = 0:110 1 1 1 : Similarly, for doubly even codes, we have the following. e 0mod6 3` 2 a value > H 01 (1=2). 01 Then there exists a self-dual binary `-quasi-cyclic code of length 3` with minimum weight of at least d.  2 defined for x 2 (0; 1) and of constant use in estimating binomial coefficients of large arguments [5, pp. 309–310]. We are now in a position to state and prove the asymptotic versions of Theorems 3.1 and 3.2. Theorem 3.1: Let ` be an even integer and let d be the largest even integer such that 3` 0x log (x) 0 (1 0 x) log (1 0 x) 2 02  (2 0 2 0 ` 1 + 1)(2 + 1): e 0mod12 Then there exists a Type II binary `-quasi-cyclic code of length 3` with minimum weight of at least d. [1] T. Kasami, “A Gilbert–Varshamov bound for quasi-cyclic codes of rate 1/2,” IEEE Trans. Inform. Theory, vol. IT-20, p. 679, 1974. [2] J. H. Conway, V. Pless, and N. J. A. Sloane, “Self-dual codes over GF (3) and GF (4) of length not exceeding 16,” IEEE Trans. Inform. Theory, vol. IT–25, pp. 312–322, 1979. [3] G. D. Forney, “Coset codes II: Binary lattices and related codes,” IEEE Trans. Inform. Theory, vol. 34, pp. 1152–1187, 1988. [4] S. Ling and P. Solé, “On the algebraic structure of quasi-cyclic codes I: Finite fields,” IEEE Trans. Inform. Theory, vol. 47, pp. 2751–2760, Nov. 2001. [5] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977. [6] F. J. MacWilliams, N. J. A. Sloane, and J. G. Thompson, “Good self-dual codes exist,” Discr. Math., vol. 3, pp. 153–162, 1972. [7] E. M. Rains and N. J. A. Sloane, “Self-dual codes,” in Handbook of Coding Theory, V. S. Pless and W. C. Huffman, Eds. Amsterdam, The Netherlands: Elsevier Science, 1998, vol. I, pp. 177–294.