A Class of Linear Codes and Their Complete Weight Enumerators

Abstract

Linear codes may have a few weights if their defining sets are chosen properly. Let $s $ and $t $ be positive integers. For an odd prime $p $ and an even integer $m $ , let $q=p^{m} $ , $m=2s $ and Trm (resp. Trs) be the absolute trace function from $\mathbb {F}_{q}$ (resp. $\mathbb {F}_{p^{s}}$ ) to $\mathbb {F}_{p}$ . In this paper, we define $D_{b} =\{ (x_{1},\ldots,x_{t})\in \mathbb {F}_{q}^{t} \backslash \{(0,\ldots,0)\}: \mathrm {Tr}_{m} (x_{1}+\cdots +x_{t})=b\}$ , where $b \in \mathbb {F}_{p} $ . By employing exponential sums, we demonstrate the complete weight enumerators of a class of $p$ -ary linear codes given by $C_{D_{b} }=\{\mathsf {c}(a_{1}, \ldots, a_{t}): a_{1},\ldots,a_{t}\in \mathbb {F}_{p^{s}}\}$ , where $\mathsf {c}(a_{1}, \ldots, a_{t})=(\mathrm {Tr}_{s}(a_{1}x_{1}^{p^{s}+1}+\cdots +a_{t}x_{t}^{p^{s}+1}))_{(x_{1},\ldots,x_{t})\in D_{b} }$ . Then we get their weight enumerators explicitly, which will give us several linear codes with a few weights. The presented codes are suitable with applications in secret sharing schemes.

Keywords

• Linear code
• complete weight enumerator
• Gauss sum
• cyclotomic number