Cyclic codes over a non-commutative finite chain ring

Abstract

In this study, we consider the finite (not necessary commutative) chain ring \(\mathcal {R}:=\mathbb {F}_{p^{m}}[u,\theta ]/{\left < u^{2} \right >}\) , where θ is an automorphism of \(\mathbb {F}_{p^{m}}\) , and completely explore the structure of left and right cyclic codes of any length N over \(\mathcal {R}\) , that is, left and right ideals of the ring \(\mathcal {S}:=\mathcal {R}[x]/{\left < x^{N}-1 \right >}\) . 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 \(\mathcal {R}\) . Finally, we study Gray images of cyclic codes over \(\mathcal {R}\) and as some examples, three linear codes over \(\mathbb {F}_{4}\) with the parameters of the best known ones, but with different weight distributions, are obtained as the Gray images of cyclic codes over \(\mathcal {R}\) .