In \cite{KoNa} the authors computed the Weierstrass gap sequence $G(P)$ of the Hermitian function field $\mathbb{F}_{q^2}(\HC)$ 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 $C_{\Omega}(D,mP)$ where the divisor $D$ is, as usual, the sum of all but one $1$-degree $\mathbb{F}_{q^2}$-rational places of $\mathbb{F}_{q^2}(\HC)$ and $m$ is a positive integer. For plenty of values of $m$ depending on $q$, this provided improvements on the designed minimum distance of $C_{\Omega}(D,mP)$. Further improvements from $G(P)$ were obtained in \cite{KoNa} 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.

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

KORCHMAROS, Gabor;
2013

Abstract

In \cite{KoNa} the authors computed the Weierstrass gap sequence $G(P)$ of the Hermitian function field $\mathbb{F}_{q^2}(\HC)$ 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 $C_{\Omega}(D,mP)$ where the divisor $D$ is, as usual, the sum of all but one $1$-degree $\mathbb{F}_{q^2}$-rational places of $\mathbb{F}_{q^2}(\HC)$ and $m$ is a positive integer. For plenty of values of $m$ depending on $q$, this provided improvements on the designed minimum distance of $C_{\Omega}(D,mP)$. Further improvements from $G(P)$ were obtained in \cite{KoNa} 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.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11563/50243
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 6
  • ???jsp.display-item.citation.isi??? 6
social impact