Self-dual codes over F-2 x (F-2 + vF(2))

In this study we consider Euclidean and Hermitian self-dual codes over the direct product ring F-2 x (F-2 + vF(2)) where v(2) = v. We obtain some theoretical outcomes about self-dual codes via the generator matrices of free linear codes over F-2 x(F-2 + vF(2)). Also, we obtain upper bounds on the minimum distance of linear codes for both the Lee distance and the Gray distance. Moreover, we find some free Euclidean and free Hermitian self-dual codes over F-2 x (F-2 + vF(2)) via some useful construction methods

Linear codes over F-2 x (F-2 + vF(2)) and the MacWilliams identities

In this work, we study linear codes over the ring F-2 x (F-2 + vF(2)) and their weight enumerators, where v(2) = v. We first give the structure of the ring and investigate linear codes over this ring. We also define two weights called Lee weight and Gray weight for these codes. Then we introduce two Gray maps from F-2 x (F-2 + vF(2)) to F-2(3) and study the Gray images of linear codes over the ring. Moreover, we prove MacWilliams identities for the complete, the symmetrized and the Lee weight enumerators

A new shortening method and Hermitian self-dual codes over F-2 + vF(2)

In this paper, we investigate free Hermitian self-dual codes whose generator matrices are of the form [I, A + vB] over the ring F-2 + vF(2) = {0, 1, v, 1 + v} with v(2) = v. We use the double-circulant, the bordered double-circulant and the symmetric construction methods to obtain free Hermitian self-dual codes of even length. By describing a new shortening method over this ring, we are able to obtain Hermitian self-dual codes of odd length. Using these methods, we also obtain a number of extremal codes. We tabulate the Hermitian self-dual codes with the highest minimum weights of lengths up to 50. (C) 2019 Elsevier B.V. All rights reserved