Recently, linear codes with a few weights have been constructed and extensively studied due to their applications in secret sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, we construct several classes of linear codes with a few weights over ${\mathbb{F}}_p$, where $p$ is an odd prime. The weight distributions of these constructed codes are also settled by applications of the theory of quadratic forms and Gauss sums over finite fields. Some of the linear codes obtained are optimal or almost optimal. The parameters of these linear codes are new in most cases. Moreover, two classes of MDS codes are obtained.

中文翻译：

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具有权重的几类线性代码的构造

近来，由于它们在秘密共享，认证码，关联方案和强规则图中的应用，已经构造并广泛研究了权重较小的线性码。在本文中，我们构造了几类权重为${\mathbb{F}}_p$，在哪里 $p$是一个奇怪的素数。这些构造代码的权重分布也通过应用二次形式理论和有限域上的高斯和来确定。获得的一些线性代码是最优的或几乎最优的。这些线性代码的参数在大多数情况下是新的。此外，获得了两类MDS代码。