Cyclic codes over a non-commutative finite chain ring

Abstract

In this study, we consider the finite (not necessary commutative) chain ring R:=Fpm[u,θ]/u2, where θ is an automorphism of Fpm, and completely explore the structure of left and right cyclic codes of any length N over R, that is, left and right ideals of the ring S: = R[ x] / ;xN− 1. For a left (right) cyclic code, we determine the structure of its right (left) dual. Using the fact that self-dual codes are bimodules, we discuss on self-dual cyclic codes over R. Finally, we study Gray images of cyclic codes over R and as some examples, three linear codes over F4 with the parameters of the best known ones, but with different weight distributions, are obtained as the Gray images of cyclic codes over R. © 2017, Springer Science+Business Media, LLC.