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Decreasing norm-trace codes

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Abstract

The decreasing norm-trace codes are evaluation codes defined by a set of monomials closed under divisibility and the rational points of the extended norm-trace curve. In particular, the decreasing norm-trace codes contain the one-point algebraic geometry (AG) codes over the extended norm-trace curve. We use Gröbner basis theory and find the indicator functions on the rational points of the curve to determine the basic parameters of the decreasing norm-trace codes: length, dimension, and minimum distance. We also obtain their dual codes. We give conditions for a decreasing norm-trace code to be a self-orthogonal or a self-dual code. We provide a linear exact repair scheme to correct single erasures for decreasing norm-trace codes, which applies to higher rate codes than the scheme developed by Jin et al. (IEEE Trans Inf Theory 64(2):900–908, 2018) when applied to the one-point AG codes over the extended norm-trace curve.

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Authors and Affiliations

  1. Faculdade de Matemática Universidade Federal de Uberlândia, Uberlândia, MG, Brazil

    Cícero Carvalho

  2. Department of Mathematics, Virginia Tech, Blacksburg, VA, USA

    Hiram H. López & Gretchen L. Matthews

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  1. Cícero Carvalho
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  2. Hiram H. López
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  3. Gretchen L. Matthews
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Correspondence to Cícero Carvalho.

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Communicated by L. Storme.

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Cícero Carvalho was partially supported by FAPEMIG APQ-00864-21. Hiram H. López was partially supported by NSF DMS-2201094 and NSF DMS-2401558. Gretchen L. Matthews was partially supported by NSF DMS-2201075 and the Commonwealth Cyber Initiative.

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Carvalho, C., López, H.H. & Matthews, G.L. Decreasing norm-trace codes. Des. Codes Cryptogr. 92, 1143–1161 (2024). https://doi.org/10.1007/s10623-023-01334-1

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  • DOI: https://doi.org/10.1007/s10623-023-01334-1

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