A construction of several classes of two-weight and three-weight linear codes

Abstract

Linear codes constructed from defining sets have been extensively studied and may have a few nonzero weights if the defining sets are well chosen. Let ${\mathbb {F}}_q$ be a finite field with $q=p^m$ elements, where p is a prime and m is a positive integer. Motivated by Ding and Ding’s recent work (IEEE Trans Inf Theory 61(11):5835–5842, 2015), we construct p-ary linear codes ${\mathcal {C}}_D$ by

$$\begin{aligned} {\mathcal {C}}_D=\{{\mathbf {c}}(a,b)=\big (\text {Tr}_m(ax+by)\big )_{(x,y)\in D}: a, b \in {\mathbb {F}}_q\}, \end{aligned}$$

where $D \subset {\mathbb {F}}_q^2$ and $\text {Tr}_m$ is the trace function from ${\mathbb {F}}_q$ onto ${\mathbb {F}}_p$. In this paper, we will employ exponential sums to investigate the weight enumerators of the linear codes ${\mathcal {C}}_D$, where $D=\{(x, y) \in {\mathbb {F}}_q^2 \setminus \{(0,0)\}: \text {Tr}_m(x^{N_1}+y^{N_2})=0\}$ for two positive integers $N_1$ and $N_2$. Several classes of two-weight and three-weight linear codes and their explicit weight enumerators are presented if $N_1, N_2 \in \{1, 2, p^{\frac{m}{2}}+1\}$. By deleting some coordinates, more punctured two-weight and three-weight linear codes ${\mathcal {C}}_{\overline{D}}$ which include some optimal codes are derived from ${\mathcal {C}}_D$.