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.