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

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