Monomial-like codes

Abstract. As a generalization of cyclic codes of length p over Fpa , we study n-dimensional cyclic codes of length p1 ×· · ·×pn over Fpa generated by a single “monomial”. Namely, we study multi-variable cyclic codes of the form 〈(x1 − 1) i1 · · · (xn − 1) n〉 ⊂ Fq [x1,...,xn] 〈x ps1 1 −1,...,x psn n −1〉 . We call such codes monomial-like codes. We show that these codes arise from the product of certain single variable codes and we determine their minimum Hamming distance. We determine the dual of monomial-like codes yielding a parity check matrix. We also present an alternative way of constructing a parity check matrix using the Hasse derivative. We study the weight hierarchy of certain monomial like codes. We simplify an expression that gives us the weight hierarchy of these codes.