Linear codes with few weights have applications in data storage systems, secret sharing schemes and authentication codes. In this paper, inspired by the works of Heng and Yue (2016) [14] and Tan, Zhou, Tang and Helleseth (2017) [25], we extend Tan, Zhou, Tang and Helleseth's work to obtain a class of optimal 1-weight binary linear codes, new classes of 2-weight and 3-weight p-ary linear codes and a class of 4-weight binary linear codes. The lengths and weight distributions of the t-weight linear codes, where $t=1,2,3$, are closed-form expressions of Kloosterman sums over finite prime fields, and are completely determined when $p=2$ and $p=3$.

权重小的线性代码可用于数据存储系统，秘密共享方案和身份验证代码。在本文中，受Heng和Yue（2016）[14]和Tan，Zhou，Tang和Helleseth（2017）[25]的启发，我们扩展了Tan，Zhou，Tang和Helleseth的工作以获得一类最优1加权二进制线性代码，新的2加权和3加权p线性代码以及一类4加权二进制线性代码。t权重线性代码的长度和权重分布，其中$Ť=\mathrm{1个}，2，3$是在有限素数域上的Kloosterman和的闭式表达式，并且完全确定何时 $p=2$ 和 $p=3$。