Projective two-weight codes with small parameters and their corresponding graphs

Projective two-weight codes with relatively small parameters are enumerated up to equivalence. Some properties of codes and related strongly-regular graphs are presented. I. Bouyukliev was partially supported by the Bulgarian National Science Fund under Contract MM1304/2003. J.Winne thanks the Fund for Scientific Research—Flanders (Belgium) for a Research grant.     

A q-polynomial approach to constacyclic codes

As a generalization of cyclic codes, constacyclic codes is an important and interesting class of codes due to their nice algebraic structures and various applications in engineering. This paper is devoted to the study of the q-polynomial approach to constacyclic codes. Fundamental theory of this approach will be developed, and will be employed to construct some families of optimal and almost optimal codes in this paper.     

Several classes of optimal ternary cyclic codes with minimal distance four

In this paper, by analyzing the solutions of certain equations over ${\mathbb{F}}_{3^m}$, we present four classes of optimal ternary cyclic codes with parameters $[3^m-1,3^m-1-2m,4]$. It is shown that some recent work on this class of optimal ternary cyclic codes are special cases of our results.     

Linear transformations on codes

This paper studies and classifies linear transformations that connect Hamming distances of codes. These include irreducible linear transformations and their concatenations. Their effect on the Hamming weights of codewords is investigated. Both linear and non-linear codes over fields are considered. We construct optimal linear codes and a family of pure binary quantum codes using these transformations.     

An efficient algorithm for constructing reversible quasi-cyclic codes via Chinese remainder theorem

Regarding quasi-cyclic codes as certain polynomial matrices, we show that all reversible quasi-cyclic codes are decomposed into reversible linear codes of shorter lengths corresponding to the coprime divisors of the polynomials with the form of one minus x to the power of m. This decomposition brings us an efficient method to construct reversible quasi-cyclic codes. We also investigate the reversibility and the self-duality of the linear codes corresponding to the coprime divisors of the polynomials. Specializing to the cases where the number of cyclic sections is not more than two, we give necessary and sufficient conditions for the divisors of the polynomials for which the self-dual codes are reversible and the reversible codes of half-length-dimension are self-dual. Our theorems are utilized to search reversible self-dual quasi-cyclic codes with two cyclic sections over binary and quaternary fields of lengths up to seventy and thirty-six, respectively, together with the maximums of their minimum weights.     

Classification of Griesmer codes and dual transform

In this work, we consider a classification of infinite families of linear codes which achieve the Griesmer bound, using the projective dual transform. We investigate the correspondence between families of linear codes with given properties via dual transform.     

An intrinsical description of group codes

A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism which maps G to the standard basis of . Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion to decide when a linear code is a group code in terms of its intrinsical properties in the ambient space , which does not assume an “a priori” group algebra structure on . As an application we provide a family of groups (including metacyclic groups) for which every two-sided group code is an abelian group code. It is well known that Reed–Solomon codes are cyclic and its parity check extensions are elementary abelian group codes. These two classes of codes are included in the class of Cauchy codes. Using our criterion we classify the Cauchy codes of some lengths which are left group codes and the possible group code structures on these codes. Research supported by D.G.I. of Spain and Fundación Séneca of Murcia.     

On the binary projective codes with dimension 6

All binary projective codes of dimension up to 6 are classified. Information about the number of the codes with different minimum distances and automorphism group orders is given.     

Complete weight enumerators of a class of linear codes with two weights

In this paper, a class of $p$-ary linear codes with two weights is constructed by using the properties of cyclotomic classes of ${\mathbb{F}}_{p^2}^1$. The complete weight enumerators of these linear codes are also determined. In some cases, they are optimal and can be employed to obtain secret sharing schemes with interesting access structures and asymptotically optimal systematic authentication codes.     

On Hermitian LCD codes and their Gray image

We provide methods and algorithms to construct Hermitian linear complementary dual (LCD) codes over finite fields. We study existence of self-dual basis with respect to Hermitian inner product, and as an application, we construct Euclidean LCD codes by projecting the Hermitian codes over such a basis. Many optimal quaternary Hermitian and ternary Euclidean LCD codes are obtained. Comparisons with classical constructions are made.