Two classes of subfield codes of linear codes

TL;DR

This paper generalizes existing subfield codes of linear codes over GF(q) to p-ary cases with odd p, determining their parameters, weight distributions, and dual code properties, including near-MDS and dimension-optimality.

Contribution

It extends previous results on subfield codes to p-ary cases, providing new parameters, weight distributions, and dual code analyses for these generalized codes.

Findings

Dual codes when m=1 are almost MDS.

Dual codes when m>1 are dimension-optimal.

Parameters and weight distributions of the new subfield codes are explicitly determined.

Abstract

Recently, subfiled codes of linear code over GF$ (q) $ with good parameters were studied, and many optimal subfield codes were obtained. In this paper, Our mainly motivation is to generlize the results of the subfield codes of hyperoval in Ding and Heng (Finite Fields Their Appl. 56, 308-331 (2019)), and generlize the results of two families of subfield codes in Xiang and Yin (Cryptogr. Commun. 13(1), 117-127 (2021)) to $ p $-ary where $ p $ is odd. We get the parameters and weight distribution of these subfield codes. At the same time, the parameters of their dual codes are also studied. When $ m=1 $, The dual codes of these subfield codes are almost MDS code, when $ m>1 $ and $ p $ odd, these dual codes are dimension-optimal with respect to the sphere-backing bound.