Quaternary quasi-cyclic codes

Quasi-cyclic codes of length mn over Z4 are shown to be equivalent to A-submodules of A^n, where A = Z4[x]/（x^m - 1）. In the case of m being odd, all quasi-cyclic codes are shown to be decomposable into the direct sum of a fixed number of cyclic irreducible A-submodules. Finally the distinct quasi-cyclic codes as well as some specific subclasses are enumerated.     

Construction of MDS self-dual codes over Galois rings

The purpose of this paper is to construct nontrivial MDS self-dual codes over Galois rings. We consider a building-up construction of self-dual codes over Galois rings as a GF(q)-analogue of (Kim and Lee, J Combin Theory ser A, 105:79–95). We give a necessary and sufficient condition on which the building-up construction holds. We construct MDS self-dual codes of lengths up to 8 over GR(3²,2), GR(3³,2) and GR(3⁴,2), and near-MDS self-dual codes of length 10 over these rings. In a similar manner, over GR(5²,2), GR(5³,2) and GR(7²,2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self-dual codes of length 12. Furthermore, over GR(11²,2) we have MDS self-dual codes of lengths up to 12.      

Optimal ternary quasi-cyclic codes

Quasi-cyclic codes have provided a rich source of good linear codes. Previous constructions of quasi-cyclic codes have been confined mainly to codes whose length is a multiple of the dimension. In this paper it is shown how searches may be extended to codes whose length is a multiple of some integer which is greater than the dimension. The particular case of 5-dimensional codes over GF(3) is considered and a number of optimal codes (i.e., [n, k, d]-codes having largest possible minimum distance d for given length n and dimension k) are constructed. These include ternary codes with parameters [45, 5, 28], [36, 5, 22], [42, 5, 26], [48, 5, 30] and [72, 5, 46], all of which improve on the previously best known bounds.This research has been supported by the British SERC.     

Quaternary 1-generator quasi-cyclic codes

Quaternary 1-generator quasi-cyclic codes are considered in the paper. Under the conditions that n is odd and gcd(|2| _n , m) = 1, where |2| _n denotes the order of 2 modulo n, we give the enumeration of quaternary 1-generator quasi-cyclic codes of length mn, and describe an algorithm which will obtain one, and only one, generator for each quaternary 1-generator quasi-cyclic code.     

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.     

Asymptotically good quasi-cyclic codes of fractional index

Generalizing the quasi-cyclic codes of index $1\frac{1}{3}$ introduced by Fan et al., we study a more general class of quasi-cyclic codes of fractional index generated by pairs of polynomials. The parity check polynomial and encoder of these codes are obtained. The asymptotic behaviours of the rates and relative distances of this class of codes are studied by using a probabilistic method. We prove that, for any positive real number $\delta$ such that the asymptotic GV-bound at $\frac{k+l}{2}\delta$ is greater than $\frac{1}{2}$, the relative distance of the code is convergent to $\delta$, while the rate is convergent to $\frac{1}{k+l}$. As a result, quasi-cyclic codes of fractional index are asymptotically good.     

Skew quasi-cyclic codes over Galois rings

Recently there has been a lot of interest on algebraic codes in the setting of skew polynomial rings. In this paper we have studied skew quasi-cyclic (QC) codes over Galois rings. We have given a necessary and sufficient condition for skew cyclic codes over Galois rings to be free, and determined a distance bound for free skew cyclic codes. A sufficient condition for 1-generator skew QC codes to be free is determined. Some distance bounds for free 1-generator skew QC codes are discussed. A canonical decomposition of skew QC codes is presented.     

Constructing quasi-cyclic codes from linear algebra theory

Let F _q be a finite field of cardinality q, l and m be positive integers and M _l (F _q ) the F _q -algebra of all l × l matrices over F _q . We investigate the relationship between monic factors of X ^m ? 1 in the polynomial ring M _l (F _q )[X] and quasi-cyclic (QC) codes of length lm and index l over F _q . Then we consider the idea of constructing QC codes from monic factors of X ^m ? 1 in polynomial rings over F _q -subalgebras of M _l (F _q ). This idea includes ideas of constructing QC codes of length lm and index l over F _q from cyclic codes of length m over a finite field F _q l, the finite chain ring F _q  + uF _q  + · · · + u ^{l ? 1} F _q (u ^l  = 0) and other type of finite chain rings.     

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.     

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.     

A relation between quasi-cyclic codes and 2-D cyclic codes

We consider a q-ary quasi-cyclic code C of length m? and index ?, where both m and ? are relatively prime to q. If the constituents of C are cyclic codes, we show that C can also be viewed as a 2-D cyclic code of size $m\times ?$ over ${\mathbb{F}}_q$. In case m and ? are also coprime to each other, we easily observe that the code C must be equivalent to a cyclic code, which was proved earlier by Lim.     

The classification of self-dual modular codes

A classification method of self-dual codes over Zm is given. If m=rs with relatively prime integers r and s, then the classification can be accomplished by double coset decompositions of Sn by automorphism groups of self-dual codes over Zr and Zs. We classify self-dual codes of length 4 over Zp for all primes p in terms of their automorphism groups and then apply our method to classify self-dual codes over Zm for arbitrary integer m. Self-dual codes of length 8 are also classified over Zpq for p,q=2,3,5,7.     

On the enumeration of self-dual codes

We give the complete classification of all binary, self-dual, doubly-even (32, 16) codes. There are 85 non-equivalent, self-dual, doubly-even (32, 16) codes. Five of these have minimum weight 8, namely, a quadratic residue code and a Reed-Muller code, and three new codes. A set of generators is given for a code in each equivalence class together with its entire weight distribution and the order of its entire group with other information facilitating the computation of permutation generators. From this list it is possible to identify all self-dual codes of length less than 32 and the numbers of these are included.     

On the additive cyclic structure of quasi-cyclic codes

An index $?$, length $m?$ quasi-cyclic code can be viewed as a cyclic code of length $m$ over the field ${\mathbb{F}}_{q^{?}}$ via a basis of the extension ${\mathbb{F}}_{q^{?}}∕{\mathbb{F}}_q$. However, this cyclic code is only linear over ${\mathbb{F}}_q$, making it an additive cyclic code, or an ${\mathbb{F}}_q$-linear cyclic code, over the alphabet ${\mathbb{F}}_{q^{?}}$. This approach was recently used in Shi et al. (2017) [16] to study a class of quasi-cyclic codes, and more importantly in Shi et al. (2017) [17] to settle a long-standing question on the asymptotic performance of cyclic codes. Here, we answer one of the problems posed in these two articles, and characterize those quasi-cyclic codes which have ${\mathbb{F}}_{q^{?}}$-linear cyclic images under a basis of the extension ${\mathbb{F}}_{q^{?}}∕{\mathbb{F}}_q$. Our characterizations are based on the module structure of quasi-cyclic codes, as well as on their CRT decompositions into constituents. In the case of a polynomial basis, we characterize the constituents by using the theory of invariant subspaces of operators. We also observe that analogous results extend to the case of quasi-twisted codes.