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Pré-Publication, Document De Travail Année : 2019

Residues of skew rational functions and linearized Goppa codes

Résumé

This paper constitutes a first attempt to do analysis with skew polynomials. Precisely, our main objective is to develop a theory of residues for skew rational functions (which are, by definition, the quotients of two skew polynomials). We prove in particular a skew analogue of the residue formula and a skew analogue of the classical formula of change of variables for residues. We then use our theory to define and study a linearized version of Goppa codes. We show that these codes meet the Singleton bound (for the sum-rank metric) and are the duals of the linearized Reed-Solomon codes defined recently by Martínez-Peñas. We also design efficient encoding and decoding algorithms for them.
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Dates et versions

hal-02268790 , version 1 (21-08-2019)
hal-02268790 , version 2 (16-06-2021)

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Xavier Caruso. Residues of skew rational functions and linearized Goppa codes. 2019. ⟨hal-02268790v1⟩
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