Subfield Codes of Several Few-Weight Linear Codes Parametrized by Functions and Their Consequences

TL;DR

This paper investigates subfield codes derived from six families of linear codes over finite fields, determining their parameters, weight distributions, and duals, with some codes being optimal or best known, and explores their applications in combinatorial designs.

Contribution

It introduces new classes of subfield codes parametrized by functions, explicitly determines their parameters and weight distributions, and links these codes to combinatorial t-designs.

Findings

Some codes are optimal and meet bounds.

The first family is an optimal two-weight code meeting the Griesmer bound.

Derived codes give rise to infinite families of t-designs.

Abstract

Subfield codes of linear codes over finite fields have recently received much attention. Some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, the $q$-ary subfield codes $C_{f,g}^{(q)}$ of six different families of linear codes $C_{f,g}$ parametrized by two functions $f, g$ over a finite field $F_{q^m}$ are considered and studied, respectively. The parameters and (Hamming) weight distribution of $C_{f,g}^{(q)}$ and their punctured codes $\bar{C}_{f,g}^{(q)}$ are explicitly determined. The parameters of the duals of these codes are also analyzed. Some of the resultant $q$-ary codes $C_{f,g}^{(q)},$ $\bar{C}_{f,g}^{(q)}$ and their dual codes are optimal and some have the best known parameters. The parameters and weight enumerators of the first two families of linear codes $C_{f,g}$ are also settled, among which the first family is an optimal two-weight linear code meeting the Griesmer bound, and the dual codes of these two families are almost MDS codes. As a byproduct of this paper, a family of $[2^{4m-2},2m+1,2^{4m-3}]$ quaternary Hermitian self-dual code are obtained with $m \geq 2$. As an application, we show that three families of the derived linear codes give rise to several infinite families of $t$-designs ($t \in \{2, 3\}$).