On geometric Goppa codes from elementary abelian $p$-extensions of $\mathbb{F}_{p^s}(x)$

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.