Linear Size Optimal q-ary Constant-Weight Codes and Constant-Composition Codes

An optimal constant-composition or constant-weight code of weight $w$ has linear size if and only if its distance $d$ is at least $2w-1$. When $d\geq 2w$, the determination of the exact size of such a constant-composition or constant-weight code is trivial, but the case of $d=2w-1$ has been solved previously only for binary and ternary constant-composition and constant-weight codes, and for some sporadic instances. This paper provides a construction for quasicyclic optimal constant-composition and constant-weight codes of weight $w$ and distance $2w-1$ based on a new generalization of difference triangle sets. As a result, the sizes of optimal constant-composition codes and optimal constant-weight codes of weight $w$ and distance $2w-1$ are determined for all such codes of sufficiently large lengths. This solves an open problem of Etzion. The sizes of optimal constant-composition codes of weight $w$ and distance $2w-1$ are also determined for all $w\leq 6$, except in two cases.Comment: 12 page

The hull of two classical propagation rules and their applications

Propagation rules are of great help in constructing good linear codes. Both Euclidean and Hermitian hulls of linear codes perform an important part in coding theory. In this paper, we consider these two aspects together and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the direct sum construction and the $(\mathbf{u},\mathbf{u+v})$-construction. Some new criteria for resulting codes derived from these two propagation rules being self-dual, self-orthogonal or linear complement dual (LCD) codes are given. As applications, we construct some linear codes with prescribed hull dimensions and many new binary, ternary Euclidean formally self-dual (FSD) LCD codes, quaternary Hermitian FSD LCD codes and good quaternary Hermitian LCD codes which are optimal or have best or almost best known parameters according to Datebase at $http://www.codetables.de$. Moreover, our methods contributes positively to improve the lower bounds on the minimum distance of known LCD codes.Comment: 16 pages, 5 table

New binary and ternary LCD codes

LCD codes are linear codes with important cryptographic applications. Recently, a method has been presented to transform any linear code into an LCD code with the same parameters when it is supported on a finite field with cardinality larger than 3. Hence, the study of LCD codes is mainly open for binary and ternary fields. Subfield-subcodes of $J$-affine variety codes are a generalization of BCH codes which have been successfully used for constructing good quantum codes. We describe binary and ternary LCD codes constructed as subfield-subcodes of $J$-affine variety codes and provide some new and good LCD codes coming from this construction

Error Correcting Coding for a Non-symmetric Ternary Channel

Ternary channels can be used to model the behavior of some memory devices, where information is stored in three different levels. In this paper, error correcting coding for a ternary channel where some of the error transitions are not allowed, is considered. The resulting channel is non-symmetric, therefore classical linear codes are not optimal for this channel. We define the maximum-likelihood (ML) decoding rule for ternary codes over this channel and show that it is complex to compute, since it depends on the channel error probability. A simpler alternative decoding rule which depends only on code properties, called \da-decoding, is then proposed. It is shown that \da-decoding and ML decoding are equivalent, i.e., \da-decoding is optimal, under certain conditions. Assuming \da-decoding, we characterize the error correcting capabilities of ternary codes over the non-symmetric ternary channel. We also derive an upper bound and a constructive lower bound on the size of codes, given the code length and the minimum distance. The results arising from the constructive lower bound are then compared, for short sizes, to optimal codes (in terms of code size) found by a clique-based search. It is shown that the proposed construction method gives good codes, and that in some cases the codes are optimal.Comment: Submitted to IEEE Transactions on Information Theory. Part of this work was presented at the Information Theory and Applications Workshop 200

Five Families of Three-Weight Ternary Cyclic Codes and Their Duals

As a subclass of linear codes, cyclic codes have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, five families of three-weight ternary cyclic codes whose duals have two zeros are presented. The weight distributions of the five families of cyclic codes are settled. The duals of two families of the cyclic codes are optimal