Goppa codes with Weierstrass pairs (Q5946455)

Results about the Weierstrass semigroup \(H(Q_1,Q_2)\) of a pair of points on an algebraic curve \(X\) have previously been obtained by both the first author [Arch. Math. 67, 337-348 (1996; Zbl 0869.14015)] and the second author [Arch. Math. 62, 73-82 (1994; Zbl 0815.14020)]. Here, the authors apply these results and some new ones to so-called ``two-point'' Goppa codes of the form \(C_\Omega(D,\nu_1Q_1+\nu_2Q_2)\), where \(X\) is a nonsingular, absolutely irreducible projective curve defined over a finite field. \textit{A. Garcia}, the second author, and the reviewer [J. Pure Appl. Algebra 84, 199-207 (1993; Zbl 0768.94014)] showed that the presence of consecutive Weierstrass gaps at a point \(P\) could allow one to conclude that codes of the form \(C_\Omega(D,\nu P)\) have greater minimum distance than the usual Goppa bound for certain values of \(\nu\). \textit{G. L. Matthews} [Des. Codes Cryptography 22, No. 2, 107-121 (2001; Zbl 0989.94032)] showed how knowledge of \(H(Q_1,Q_2)\) could allow one to conclude that certain two-point codes have minimum distance one greater than the usual lower bound. Here, the authors define a pair \((\alpha_1,\alpha_2)\) to be a pure gap at \((Q_1,Q_2)\) if \(l(\alpha_1Q_1+\alpha_2Q_2)=l((\alpha_1-1)Q_1+\alpha_2Q_2)=l(\alpha_1Q_1+(\alpha_2-1)Q_2)\). They then show that the presence of ``blocks'' of adjacent pure gaps allow one to conclude that certain two-point codes have mimimum distance significantly greater than the Goppa lower bound. The authors apply these results to Hermitian curves, using Matthews' computation of the semigroup of a pair on such curves, and to cyclic quotients of Hermitian curves. They give examples of two-point codes with better parameters than the analogous one-point codes.