On the classification of ideals over $R[X]/\langle f(X)^{p^{s}}\rangle$ when $R=\mathbb {F}_{p^{m}}+u\mathbb {F}_{p^{m}}+\ldots +u^{n}\mathbb {F}_{p^{m}}$

Abstract

In this paper, we give a general form of ideals over $R[X]/\langle f(X)^{p^{s}}\rangle$ when f is a monic irreducible polynomial in R[X] and $R= \mathbb {F}_{p^{m}}+u\mathbb {F}_{p^{m}}+\ldots +u^{n}\mathbb {F}_{p^{m}}$ with u^n+ 1 = 0. For f(X) = X − 1 this gives a general form of cyclic codes over R. Furthermore, we apply the result to classify completely the cyclic codes of length 5p^s over $\mathbb {F}_{p^{m}}+u\mathbb {F}_{p^{m}}+u^{2}\mathbb {F}_{p^{m}}$ when p ≡ 2 (mod 5) or p ≡ 3 (mod 5) and m is odd.