Two classes of $p$-ary linear codes and their duals

TL;DR

This paper introduces new classes of $p$-ary linear codes derived from subfield codes, analyzes their weight distributions, and demonstrates that some of these codes and their duals are optimal or near-optimal with respect to classical bounds.

Contribution

The paper generalizes previous results on subfield codes of conic codes, providing explicit weight distributions and establishing optimality of the codes and their duals in certain cases.

Findings

Some codes are optimal or almost optimal.

Dual codes meet Sphere Packing bound for $p>3$.

Dual of $ar{ ext{C}}_k$ is optimal for specific parameters when $p=3$.

Abstract

Let $\mathbb{F}_{p^m}$ be the finite field of order $p^m$, where $p$ is an odd prime and $m$ is a positive integer. In this paper, we investigate a class of subfield codes of linear codes and obtain the weight distribution of \begin{equation*} \begin{split} \mathcal{C}_k=\left\{\left(\left( {\rm Tr}_1^m\left(ax^{p^k+1}+bx\right)+c\right)_{x \in \mathbb{F}_{p^m}}, {\rm Tr}_1^m(a)\right) : \, a,b \in \mathbb{F}_{p^m}, c \in \mathbb{F}_p\right\}, \end{split} \end{equation*} where $k$ is a nonnegative integer. Our results generalize the results of the subfield codes of the conic codes in \cite{Hengar}. Among other results, we study the punctured code of $\mathcal{C}_k$, which is defined as $$\mathcal{\bar{C}}_k=\left\{\left( {\rm Tr}_1^m\left(a x^{{p^k}+1}+bx\right)+c\right)_{x \in \mathbb{F}_{p^m}} : \, a,b \in \mathbb{F}_{p^m}, \,\,c \in \mathbb{F}_p\right\}.$$ The parameters of these linear codes are new in some cases. Some of the presented codes are optimal or almost optimal. Moreover, let $v_2(\cdot)$ denote the 2-adic order function and $v_2(0)=\infty$, the duals of $\mathcal{C}_k$ and $\mathcal{\bar{C}}_k$ are optimal with respect to the Sphere Packing bound if $p>3$, and the dual of $\mathcal{\bar{C}}_k$ is an optimal ternary linear code for the case $v_2(m)\leq v_2(k)$ if $p=3$ and $m>1$.