On the classification and enumeration of self-dual codes

A complete classification is given of all [22, 11] and [24, 12] binary self-dual codes. For each code we give the order of its group, the number of codes equivalent to it, and its weight distribution. There is a unique [24, 12, 6] self-dual code. Several theorems on the enumeration of self-dual codes are used, including formulas for the number of such codes with minimum distance ? 4, and for the sum of the weight enumerators of all such codes of length n. Selforthogonal codes which are generated by code words of weight 4 are completely characterized.     

On self-dual double circulant codes

Self-dual double circulant codes of odd dimension are shown to be dihedral in even characteristic and consta-dihedral in odd characteristic. Exact counting formulae are derived for them, generalizing some old results of MacWilliams on the enumeration of circulant orthogonal matrices. These formulae, in turn, are instrumental in deriving a Varshamov–Gilbert bound on the relative minimum distance of this family of codes.     

On designs and formally self-dual codes

Binary formally self-dual (f.s.d.) even codes are the one type of divisible [2n, n] codes which need not be self-dual. We examine such codes in this paper. On occasion a f.s.d. even [2n, n] code can have a larger minimum distance than a [2n, n] self-dual code. We give many examples of interesting f.s.d even codes. We also obtain a strengthening of the Assmus-Mattson theore. IfC is a f.s.d. extremal code of lengthn2 (mol 8) [n 6 (mod 8)], then the words of a fixed weight inC C hold a 3-design [1-design]. Finally, we show that the extremal f.s.d. codes of lengths 10 and 18 are unique.The author thanks the University of Illinois at Chicago for their hospitality while this work was in progress.This work was supported in part by NSA Grant MDA 904-91-H-0003.     

On circulant self-dual codes over small fields

We construct self-dual codes over small fields with q = 3, 4, 5, 7, 8, 9 of moderate length with long cycles in the automorphism group. With few exceptions, the codes achieve or improve the known lower bounds on the minimum distance of self-dual codes.      

On the asymptotic number of inequivalent binary self-dual codes

Let Ψn be the number of inequivalent self-dual codes in . We prove that , where . Let Δn be the number of inequivalent doubly even self-dual codes in . We also prove that .     

On the constructions of MDS self-dual codes via cyclotomy

MDS self-dual codes over finite fields have attracted a lot of attention in recent years by their theoretical interests in coding theory and applications in cryptography and combinatorics. In this paper we present a series of MDS self-dual codes with new length by using generalized Reed-Solomon codes and extended generalized Reed-Solomon codes as the candidates of MDS codes and taking their evaluation sets as a union of cyclotomic classes. The conditions on such MDS codes being self-dual are expressed in terms of cyclotomic numbers.     

On the existence of self-dual permutation codes of finite groups

Motivated by a research on self-dual extended group codes, we consider permutation codes obtained from submodules of a permutation module of a finite group of odd order over a finite field, and demonstrate that the condition “the extension degree of the finite field extended by n’th roots of unity is odd” is sufficient but not necessary for the existence of self-dual extended transitive permutation codes of length n + 1. It exhibits that the permutation code is a proper generalization of the group code, and has more delicate structure than the group code.     

On self-dual cyclic codes over finite chain rings

In this paper, we give necessary and sufficient conditions for the existence of non-trivial cyclic self-dual codes over finite chain rings. We prove that there are no free cyclic self-dual codes over finite chain rings with odd characteristic. It is also proven that a self-dual code over a finite chain ring cannot be the lift of a binary cyclic self-dual code. The number of cyclic self-dual codes over chain rings is also investigated as an extension of the number of cyclic self-dual codes over finite fields given recently by Jia et al.     

Galois self-dual constacyclic codes

Generalizing the Euclidean inner product and the Hermitian inner product, we introduce Galois inner products, and study Galois self-dual constacyclic codes in a very general setting by a uniform method. The conditions for existence of Galois self-dual and isometrically Galois self-dual constacyclic codes are obtained. As consequences, results on self-dual, iso-dual and Hermitian self-dual constacyclic codes are derived.     

On self-dual negacirculant codes of index two and four

We study the asymptotic performance of quasi-twisted codes viewed as modules in the ring \(R=\mathbb {F}_q[x]/\langle x^n+1\rangle , \) when they are self-dual and of length 2n or 4n. In particular, in order for the decomposition to be amenable to analysis, we study factorizations of \(x^n+1\) over \(\mathbb {F}_q, \) with n twice an odd prime, containing only three irreducible factors, all self-reciprocal. We give arithmetic conditions bearing on n and q for this to happen. Given a fixed q,  we show these conditions are met for infinitely many n’s, provided a refinement of Artin primitive root conjecture holds. This number theory conjecture is known to hold under generalized Riemann hypothesis (GRH). We derive a modified Varshamov–Gilbert bound on the relative distance of the codes considered, building on exact enumeration results for given n and q.     

On the support designs of extremal binary doubly even self-dual codes

Let \(D\) be the support design of the minimum weight of an extremal binary doubly even self-dual \([24m,12m,4m+4]\) code. In this note, we consider the case when \(D\) becomes a \(t\) -design with \(t \ge 6\) .     

Construction of quasi-cyclic self-dual codes

There is a one-to-one correspondence between ?-quasi-cyclic codes over a finite field ${\mathbb{F}}_q$ and linear codes over a ring $R={\mathbb{F}}_q[Y]/(Y^m?1)$. Using this correspondence, we prove that every ?-quasi-cyclic self-dual code of length m? over a finite field ${\mathbb{F}}_q$ can be obtained by the building-up construction, provided that $\mathrm{char}({\mathbb{F}}_q)=2$ or $q\equiv 1(\mathrm{mod}4)$, m is a prime p, and q is a primitive element of ${\mathbb{F}}_p$. We determine possible weight enumerators of a binary ?-quasi-cyclic self-dual code of length p? (with p a prime) in terms of divisibility by p. We improve the result of Bonnecaze et al. (2003) [3] by constructing new binary cubic (i.e., ?-quasi-cyclic codes of length 3?) optimal self-dual codes of lengths $30,36,42,48$ (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When $m=5$, we obtain a new 8-quasi-cyclic self-dual $[40,20,12]$ code over ${\mathbb{F}}_3$ and a new 6-quasi-cyclic self-dual $[30,15,10]$ code over ${\mathbb{F}}_4$. When $m=7$, we find a new 4-quasi-cyclic self-dual $[28,14,9]$ code over ${\mathbb{F}}_4$ and a new 6-quasi-cyclic self-dual $[42,21,12]$ code over ${\mathbb{F}}_4$.     

Binary formally self-dual odd codes

Most of the research on formally self-dual (f.s.d.) codes has been developed for binary f.s.d. even codes, but only limited research has been done for binary f.s.d. odd codes. In this article we complete the classification of binary f.s.d. odd codes of lengths up to 14. We also classify optimal binary f.s.d. odd codes of length 18 and 24, so our result completes the classification of binary optimal f.s.d. odd codes of lengths up to 26. For this classification we first find a relation between binary f.s.d. odd codes and binary f.s.d. even codes, and then we use this relation and the known classification results on binary f.s.d. even codes. We also classify (possibly) optimal binary double circulant f.s.d. odd codes of lengths up to 40.     

Construction of long MDS self-dual codes from short codes

In this paper, we propose a mechanism on how to construct long MDS self-dual codes from short ones. These codes are special types of generalized Reed-Solomon (GRS) codes or extended generalized Reed-Solomon codes. The main tool is utilizing additive structure or multiplicative structure on finite fields. By applying this method, more MDS self-dual codes can be constructed.