On Goppa codes and Weierstrass gaps at several points (Q1781001)

One-point geometric Goppa codes are defined as the algebraic geometric codes \(C(D,G)\) where \(G=\mu Q\) is a multiple of a single point. In [J. Pure Appl. Algebra 84, 199--207 (1993; Zbl 0768.94014)] \textit{A. Garcia}, \textit{S. J. Kim} and \textit{R. Lax} showed that the true minimum distance of these codes can be greater than the Goppa bound for certain values of \(\mu\), and this fact is related to the structure of gaps in the Weierstrass semigroup at \(Q\). In [J. Pure Appl. Algebra 162, 273--290 (2001; Zbl 0991.94055)], \textit{M. Homma} and \textit{S. J. Kim} proved similar results for two-point geometric Goppa codes. The present article generalizes these results for arbitrary \(m\)-point codes, that is for codes \(C(D,G)\) where \(G=\mu_1 Q_1+\dots+\mu_m Q_m\). Some interesting supplementary results concerning the structure of Weierstrass semigroups at \(m\) points are also included.