# On geometric Goppa codes from Elementary Abelian $p$-Extensions of $\mathbb{F}_{p^{s}}(x)$

## TL;DR

This paper studies one-point geometric Goppa codes from elementary abelian p-extensions over finite fields, determining their parameters and constructing related quantum, convolutional, and locally recoverable codes.

## Contribution

It provides explicit dimension, minimum distance, and self-duality criteria for these codes, and constructs new quantum and locally recoverable codes from the function field.

## Key findings

- Exact dimension and minimum distance in specific cases
- Criteria for self-duality and quasi-self-duality
- Construction of quantum, convolutional, and locally recoverable codes

## Abstract

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 the exact minimum distance in a few cases. These codes are a special case of weak Castle codes. We also list the exact values of the second generalized Hamming weight of these codes in a few cases. Simple criteria for the self-duality and the 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.