Let p be a prime number and s > 0 an integer. In this short note, we investigate one-point geometric Goppa codes associated with an elementary abelian p-extension of \({{\mathbb{F}}}_{{p}^{s}}(x)\). We determine their dimension and exact minimum distance in a few cases. These codes are a special case of weak Castle codes. We also list exact values of the second generalized Hamming weight of these codes in a few cases. Simple criteria for self-duality and quasi-self-duality of these codes are also provided. Furthermore, we construct examples of quantum codes, convolutional codes, and locally recoverable codes on the function field.

令p为质数，s  > 0为整数。在这篇短文中，我们调查与基本阿贝尔相关的一个点几何Goppa码p的-extension \（{{\ mathbb {F}}} _ {{P} ^ {S}}（X）\） 。在某些情况下，我们会确定其尺寸和确切的最小距离。这些代码是弱城堡代码的特例。在少数情况下，我们还将列出这些代码的第二个广义汉明权重的精确值。还提供了这些代码的自我对偶性和准自我对偶性的简单标准。此外，我们在函数域上构造了量子码，卷积码和局部可恢复码的示例。