On the Linear Codes Over the Ring Z(4)

Abstract

Some results on linear codes over the ring Z(4) + uZ(4) + vZ(4); u(2) = u, v(2) = v; uv = vu = 0 in [6,7] are generalized to the ring D-t = Z(4) + v(1)Z(4) + ... + v(t)Z(4); v(i)(2) = v(i),v(i)v(j) = v(j)v(i) = 0 for i not equal j; 1 <= i, j <= t. A Gray map Phi(t) from D-t(n) to Z(4)((t+1)n) is defined. The Gray images of the cyclic, constacyclic and quasi-cyclic codes over Dt are determined. The cyclic DNA codes over D-t are introduced. The binary images of them are determined. The nontrivial automorphism on D-i for i = 2; 3; ..., t is given. The skew cyclic, skew constacyclic and skew quasi-cyclic codes over D-t are introduced. The Gray images of them are determined. The skew cyclic DNA codes over D-t are introduced. Moreover, some properties of MDS codes over D-t are discussed.