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Lower bounds on the minimum distance in Hermitian one-point differential codes

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Abstract

Korchmáros and Nagy [Hermitian codes from higher degree places. J Pure Appl Algebra, doi: 10. 1016/j.jpaa.2013.04.002, 2013] computed the Weierstrass gap sequence G(P) of the Hermitian function field

at any place P of degree 3, and obtained an explicit formula of the Matthews-Michel lower bound on the minimum distance in the associated differential Hermitian code C Ω(D, mP) where the divisor D is, as usual, the sum of all but one 1-degree \(\mathbb{F}_{q^2 }\)-rational places of

and m is a positive integer. For plenty of values of m depending on q, this provided improvements on the designed minimum distance of C Ω(D, mP). Further improvements from G(P) were obtained by Korchmáros and Nagy relying on algebraic geometry. Here slightly weaker improvements are obtained from G(P) with the usual function-field method depending on linear series, Riemann-Roch theorem and Weierstrass semigroups. We also survey the known results on this subject.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università della Basilicata, Potenza, 85100, Italy

    Gábor Korchmáros

  2. Bolyai Institute, University of Szeged, Szeged, 6725, Hungary

    Gábor Péter Nagy

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  1. Gábor Korchmáros
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  2. Gábor Péter Nagy
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Correspondence to Gábor Korchmáros.

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Korchmáros, G., Nagy, G.P. Lower bounds on the minimum distance in Hermitian one-point differential codes. Sci. China Math. 56, 1449–1455 (2013). https://doi.org/10.1007/s11425-013-4641-x

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