Monomial-like codes

TL;DR

This paper introduces monomial-like codes, a class of multi-variable cyclic codes generated by monomials, and analyzes their structure, duals, minimum distance, and weight hierarchy, extending cyclic code theory to higher dimensions.

Contribution

It generalizes cyclic codes to multi-variable cases, providing explicit constructions, duals, and weight hierarchy analysis for monomial-like codes.

Findings

Monomial-like codes are products of single-variable codes.

The minimum Hamming distance of these codes is determined.

A method using Hasse derivatives for constructing parity check matrices is presented.

Abstract

As a generalization of cyclic codes of length p^s over F_{p^a}, we study n-dimensional cyclic codes of length p^{s_1} X ... X p^{s_n} over F_{p^a} generated by a single "monomial". Namely, we study multi-variable cyclic codes of the form <(x_1 - 1)^{i_1} ... (x_n - 1)^{i_n}> in F_{p^a}[x_1...x_n] / < x_1^{p^{s_1}}-1, ..., x_n^{p^{s_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.