Linear codes with few weights have been a subject of study for many years, as they have applications in secret sharing, authentication codes, association schemes, and strongly regular graphs. In this article, two distinct classes of $p$-ary linear codes are constructed through the selection of two specific defining sets. Their weight distributions are completely determined for each case by detailed calculations on certain Weil sums. The constructed codes are shown to have only two, four, six, eight, and nine nonzero weights under different cases. In particular, we obtained an infinite family of two-weight optimal linear codes with respect to the Griesmer bound. Moreover, we observe that some of our newly constructed codes are minimal under certain conditions.

翻译：具有少数重量的线性码多年来一直是研究课题，因其在秘密共享、认证码、结合方案及强正则图中的应用而备受关注。本文通过选取两类特定的定义集合，构造了两种不同的$p$元线性码。通过对特定Weil和的详细计算，完全确定了每种情形下的重量分布。研究表明，所构造的码在不同情形下仅具有两个、四个、六个、八个及九个非零重量。特别地，我们获得了一族关于Griesmer界的两重量最优线性码。此外，观察到部分新构造的码在特定条件下具有极小性。