Let $p$ be an odd prime number. In this paper, we construct $2(2p-3)$ classes of codes over the ring $R=\Bbb F_p+u\Bbb F_p,u^2=0$, which are associated with down sets. We compute the Lee weight distributions of the $2(2p-3)$ classes of codes when the down sets are generated by a single maximal element. Moreover, by using the Gray map of the linear codes over $R$, we find out $2(p-1)$ classes of $p$-ary distance optimal linear codes. Two of them meet the Griesmer bound.

中文翻译：

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$\Bbb F_p+u\Bbb F_p$ 上与向下集相关的少权重代码及其距离最优灰度图像

令 $p$ 为奇素数。在本文中，我们在环 $R=\Bbb F_p+u\Bbb F_p,u^2=0$ 上构造了 $2(2p-3)$ 类代码，这些代码与下集相关。当向下集由单个最大元素生成时，我们计算 $2(2p-3)$ 类代码的 Lee 权重分布。此外，通过使用 $R$ 上线性代码的格雷图，我们找出 $2(p-1)$ 类 $p$-ary 距离最优线性代码。他们中的两个人遇到了格里斯默的束缚。