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

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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 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.

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Acknowledgements

The authors are very grateful to an anonymous reviewer for his/her comments and suggestions, which helped to improve the quality of the note.

Funding

The second named author is supported by Early Career Research Award (ECR/2016/000649) by the Department of Science & Technology (DST), Government of India.

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Patanker, N., Singh, S. On Geometric Goppa Codes from Elementary Abelian p-Extensions of \({{\mathbb{F}}}_{{p}^{s}}(x)\). Probl Inf Transm 56, 253–269 (2020). https://doi.org/10.1134/S0032946020030035

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