Minimum Distance of New Generalizations of the Punctured Binary Reed-Muller Codes

TL;DR

This paper investigates the minimum distance of new generalized cyclic codes related to punctured binary Reed-Muller codes, providing improved bounds and results relevant for coding theory applications.

Contribution

It presents new results on the minimum distance of generalized cyclic codes, extending and improving previous findings in the literature.

Findings

New bounds on the minimum distance of ho(q,m,h) and ho(q,m,h) codes

Generalizations of punctured binary Reed-Muller codes

Improved theoretical understanding of code properties

Abstract

Motivated by applications in combinatorial design theory and constructing LCD codes, C. Ding et al \cite{DLX} introduced cyclic codes $\mho(q,m,h)$ and $\bar\mho(q,m,h)$ over $\mathbb{F}_q$ as new generalization and version of the punctured binary Reed-Muller codes. In this paper, we show several new results on minimum distance of $\mho(q,m,h)$ and $\bar\mho(q,m,h)$ which are generalization or improvement of previous results given in \cite{DLX}.