Goppa codes with Weierstrass pairs.

*(English)*Zbl 0991.94055Results 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. 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 \(\nu\). 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.

Reviewer: Robert F.Lax (Baton Rouge)

##### MSC:

94B27 | Geometric methods (including applications of algebraic geometry) applied to coding theory |

14H55 | Riemann surfaces; Weierstrass points; gap sequences |

94B65 | Bounds on codes |

##### Keywords:

Goppa codes; Weierstrass pair; pure gap; minimum distance; Hermitian curve; two-point codes
PDF
BibTeX
XML
Cite

\textit{M. Homma} and \textit{S. J. Kim}, J. Pure Appl. Algebra 162, No. 2--3, 273--290 (2001; Zbl 0991.94055)

Full Text:
DOI

##### References:

[1] | Feng, G.L.; Rao, T.R.N., Decoding algebraic-geometric codes up to the designed minimum distance, IEEE trans. inform. theory, 39, 37-45, (1993) · Zbl 0765.94021 |

[2] | Garcia, A.; Lax, R.F., Goppa codes and Weierstrass gaps, (), 33-42 · Zbl 0768.94023 |

[3] | Garcia, A.; Kim, S.J.; Lax, R.F., Consecutive Weierstrass gaps and minimum distance of Goppa codes, J. pure appl. algebra, 84, 199-207, (1993) · Zbl 0768.94014 |

[4] | Garcia, A.; Viana, P., Weierstrass points on certain non-classical curves, Arch. math., 46, 315-322, (1986) · Zbl 0575.14014 |

[5] | Goppa, V.D., Codes on algebraic curves, Soviet math. dokl., 24, 170-172, (1981) · Zbl 0489.94014 |

[6] | Homma, M., The Weierstrass semigroup of a pair of points on a curve, Arch. math., 67, 337-348, (1996) · Zbl 0869.14015 |

[7] | Høholdt, T.; van Lint, J.H.; Pellikaan, R., Algebraic geometry codes, (), 871-961 · Zbl 0922.94015 |

[8] | Kim, S.J., On the index of the Weierstrass semigroup of a pair of points on a curve, Arch. math., 62, 73-82, (1994) · Zbl 0815.14020 |

[9] | Lewittes, J., Automorphisms of compact Riemann surfaces, Amer. J. math., 85, 734-752, (1963) · Zbl 0146.10403 |

[10] | G.L. Matthews, Weierstrass pairs and minimum distance of Goppa codes, Preprint, Louisiana State University, 1999. · Zbl 0989.94032 |

[11] | Schoeneberg, B., Über die Weierstrass-punkte in den Körpern der elliptischen modulfunktionen, Abh. math. sem. Hamburg, 17, 104-111, (1951) · Zbl 0042.31902 |

[12] | Stichtenoth, H., Algebraic function fields and codes, (1992), Springer Berlin |

[13] | Yang, K.; Kumar, P.V., On the true minimum distance of Hermitian codes, (), 99-107 · Zbl 0763.94023 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.