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Towards a Gröbner-free approach to coding

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Abstract

Recently, new studies on the decoding of cyclic codes have been developed. They place themselves under the umbrella of Cooper Philosophy and they consist in using (sparse) locator polynomials, which, once evaluated at the syndromes, return the error locations. In particular, it has been recently shown that it is not necessary to use Gröbner bases to compute such kind of polynomials, and that some sparse versions can be found (at least for error correction capability at most 2), using interpolation on the syndrome variety. In this paper, we study the combinatorial structure of the syndrome variety of a cyclic code and some of its variants, for error correction capability 2, by means of standard monomials. Such monomials can be found without computing a Gröbner basis of the syndrome ideal, neither performing any step of Buchberger reduction, that is, in a degröbnerized way.

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Notes

  1. The recent mood, not depending on the network technology of the Sixties, and thus not needing the key equation, prefers considering the plain error locator polynomial \(\prod _{j=1}^{\mu } (x-\alpha ^{\ell _j})\).

  2. Monomials’ representation given by the Bar Code has several applications. See for example [11, 13, 14].

  3. The interested reader can find them at https://sites.google.com/view/michela-ceria-home-page/links.

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Funding

This research is partially supported by GNSAGA- Istituto Nazionale di Alta Matematica “Francesco Severi”.

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Author notes

  1. Michela Ceria and Teo Mora have contributed equally to this work.

Authors and Affiliations

  1. Department of Mechanics, Mathematics and Management, Politecnico di Bari, Via Orabona 4, 70125, Bari, Italy

    Michela Ceria

  2. DIMA, Università di Genova, Via Dodecaneso 35, 16100, Genoa, Italy

    Teo Mora

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  1. Michela Ceria
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Correspondence to Michela Ceria.

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Ceria, M., Mora, T. Towards a Gröbner-free approach to coding. Des. Codes Cryptogr. 92, 179–204 (2024). https://doi.org/10.1007/s10623-023-01302-9

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