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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.

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