A class of linear codes with two weights or three weights from some planar functions

Abstract

Let $\mathbb F_{q}$ be a finite field with $q=p^{m}$ elements, where p is an odd prime and m is a positive integer. In this paper, let $D=\{(x_{1},x_{2},\ldots ,x_{n})\in \mathbb F_{q}^{n}\backslash \{(0,0,\ldots )\}: Tr(x_{1}^{p^{k_{1}}+1}+x_{2}^{p^{k_{2}}+1}+\cdots +x_{n}^{p^{k_{n}}+1})=c\}$, where $c\in \mathbb F_p$, Tr is the trace function from $\mathbb F_{q}$ to $\mathbb F_{p}$ and each $m/(m,k_{i})$ ( $1\le i\le n$ ) is odd. we define a p-ary linear code $C_{D}=\{c(a_{1},a_{2},\ldots ,a_{n}):(a_{1},a_{2},\ldots ,a_{n})\in \mathbb F_{q}^{n}\}$, where $c(a_{1},a_{2},\ldots ,a_{n})=(Tr(a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}))_{(x_{1},x_{2},\ldots ,x_{n})\in D}$. We present the weight distributions of the classes of linear codes which have at most three weights.