Cyclic and Skew Cyclic Codes Over the F-q

Abstract

In this paper, we study the structure of cyclic and skew cyclic codes over the finite ring D-k = F-q +v(1)F(q) + ... v(k)F(q), v(i)(2) = v(i), v(i)v(j) = v(j)v(i) = 0, 1 <= i, j <= k, q = p(m), p is a prime for k >= 1 which contains the ring F-q + v(1)F(q), v(1)(2) = v(1). We define a new Gray map from D-k to F-q(k+1). The algebraic structures of cyclic codes and duality properties are investigated. A linear code over D-k is represented by means of k+1 q-ary codes. The non trivial automorphism over D-k is given and the skew cyclic codes over D-k are introduced. The algebraic structure of skew cyclic codes and duality properties are investigated. The Gray images of both cyclic and skew cyclic codes over D-k are obtained.