Negacyclic and Constacyclic Codes Over Finite Chain Rings

Abstract

The negashift ν of Z4n is defined as the permeation of Z4n such that ν(a0, a1, ...., ai, ...., an-1)= (-an-1, a0, ...., ai, ....an-2) and negacyclic code of length n over Z4 is defined as a subset C of Z4n such that ν(C) = C. We prove that the Gray image of a linear negacyclic code over Z4 of length n is a binary distance invariant cyclic code. We introduce (1 + u)-constacyclic and cyclic codes over the ring R2 = F2 + uF2, and study them by analogy with the Z4 case. We prove that the Gray image of a linear (1 + u)-constacyclic code over R2 of length n is a binary distance invariant cyclic code. Also we define a Gray map between codes over R3 = F2+uF2+u2F2 and codes over F2. We prove that the Gray image of a linear (1 - u2)-constacyclic code over R3 of length n is a binary distance invariant linear quasi-cyclic code. Finally we define a new Gray map between Rk = F2 + uF2 + u2F2 + . . . + uk-1F2 and F22(k-1). We prove that the Gray image of a linear (1 - uk-1)-constacyclic code over Rk of length n is a binary distance invariant quasi-cyclic code of order k - 1.