On the minimum distance of Castle codes
Let \(\mathcal{X}\) be an absolutely irreducible nonsingular curve defined over \(\mathbb{F}_{q}\) and \(P_1,P_2,\dots,P_n\) be \(n\) rational points of \(\mathcal{X}\). Set \(D=P_1+P_2+\dots+P_n\), and let \(G\) be a divisor whose support is disjoint from \(D\). The algebraic geometry (AG) code associated to \(D\) and \(G\) is denoted by \(C(D,G)\), and it is composed by the evaluations of the rational functions in the Riemann-Roch space of \(G\) on the rational points \(P_1,\dots,P_n\). When \(G=mQ\), with \(Q\neq P_i\) for all \(i\), then \(C(d,mQ)\) is called a one-point code. Computing the parameters, and in particular the minimum distance, of a \(C(D,mQ)\) code is in general a difficult problem. Special attention deserves the so-called order bound \(d^*\) on \(d(C(D,mq))\). In this paper the authors compute the values for \(d^*\) for the case of the Castle curves. In particular, when the Weierstrass semigroup of \(Q\), \(H(Q)\), is generated by two elements, the authors can determine the bound \(d^*\) for all \(m\). The case of telescopic groups is also considered, obtaining similar but incomplete results. In the last part the particular case of the Suzuki curve is taken into account, that is, a Castle curve with \(H(Q)\) telescopic. In this case, \(d^*\) is completely computed.
scientific article; zbMATH DE number 6146996