The weight distributions of some linear codes derived from Kloosterman sums
Mengzhen Zhao, Yanxun Chang
TL;DR
The paper addresses constructing $p$-ary linear codes with few weights from a defining set and determining their weight distributions via Kloosterman sums. It defines $D=\{(x,y)\in (F_q\times F_q)\setminus\{(0,0)\}:Tr_1^m(x+y^{p^t-1})=0\}$ with $q=p^m$, $m=2t$, and shows a four-weight code $C_D$ of length $n=p^{2m-1}-1$, along with subcodes $C_{D1}$ and $C_{D2}$ whose weights are governed by sums like $S=\sum_{z\in F_p^*}\sum_{x\in \Delta}\chi(zx)$ and by $K_1$. The main contributions are the complete weight distributions for $C_D$, $C_{D1}$, and $C_{D2}$, explicit weight formulas in terms of $S$ and $K_1$, and minimality results for the subcodes; the results rely on additive character sums and the properties of Kloosterman sums, enabling applications to secret sharing, data storage, and authentication schemes. These findings illustrate how Kloosterman-sum techniques can classify and construct minimal, few-weight linear codes with practical impact.
Abstract
Linear codes with few weights have applications in data storage systems, secret sharing schemes, and authentication codes. In this paper, some kinds of p-ary linear codes with few weights are constructed by use of the given de ning set, where p is a prime. Their weight distributions are determined based on Kloosterman sums over nite elds. In addition, some linear codes we given is minimal.