The Weight Hierarchy of Some Reducible Cyclic Codes

TL;DR

This paper determines the complete generalized Hamming weights for a family of reducible cyclic codes, providing insights into their structure and potential applications in coding theory.

Contribution

It extends previous methods to higher dimensions and employs combinatorial arguments to find the weight hierarchy of reducible cyclic codes with arbitrary nonzeroes.

Findings

Complete weight hierarchy for several cases of reducible cyclic codes

Extension of existing methods to higher dimensions

Use of combinatorial techniques to analyze code parameters

Abstract

The generalized Hamming weights (GHWs) of linear codes are fundamental parameters, the knowledge of which is of great interest in many applications. However, to determine the GHWs of linear codes is difficult in general. In this paper, we study the GHWs for a family of reducible cyclic codes and obtain the complete weight hierarchy in several cases. This is achieved by extending the idea of \cite{YLFL} into higher dimension and by employing some interesting combinatorial arguments. It shall be noted that these cyclic codes may have arbitrary number of nonzeroes.