Codes from the Suzuki function field

We construct algebraic geometry (AG) codes from the function field F(2/sup 2n+1/)(x,y)/F(2/sup 2n+1/) defined by y(2/sup 2n+1/)-y=(x(2/sup 2n+/)-x) where n is a positive integer. These codes are supported by two places, and many have parameters that are better than those of any comparable code supported by one place of the same function field. To define such codes, we determine and exploit the structure of the Weierstrass gap set of an arbitrary pair of rational places of F(2/sup 2n+1/)(x,y)/F(2/sup 2n+1/). Moreover, we find some codes over F/sub 8/ with parameters that are better than any known code.

[1]  Cícero Carvalho,et al.  On Goppa Codes and Weierstrass Gaps at Several Points , 2005, Des. Codes Cryptogr..

[2]  Gretchen L. Matthews.  The Weierstrass Semigroup of an m-tuple of Collinear Points on a Hermitian Curve , 2003, International Conference on Finite Fields and Applications.

[3]  Seon Jeong Kim.  On the index of the Weierstrass semigroup of a pair of points on a curve , 1994 .

[4]  Henning Stichtenoth,et al.  Group codes on certain algebraic curves with many rational points , 1990, Applicable Algebra in Engineering, Communication and Computing.

[5]  Hans-Wolfgang Henn.  Funktionenkörper mit großer Automorphismengruppe. , 1978 .

[6]  Henning Stichtenoth,et al.  Algebraic function fields and codes , 1993, Universitext.

[7]  Masaaki Homma,et al.  Goppa codes with Weierstrass pairs , 2001 .

[8]  R. F. Lax,et al.  Consecutive Weierstrass gaps and minimum distance of Goppa codes , 1993 .

[9]  V. D. Goppa.  ALGEBRAICO-GEOMETRIC CODES , 1983 .

[10]  P. Griffiths,et al.  Geometry of algebraic curves , 1985 .

[11]  Iwan M. Duursma,et al.  Geometric Reed-Solomon codes of length 64 and 65 over F8 , 2003, IEEE Trans. Inf. Theory.

[12]  Ruud Pellikaan,et al.  The minimum distance of codes in an array coming from telescopic semigroups , 1995, IEEE Trans. Inf. Theory.

[13]  Gretchen L. Matthews,et al.  Riemann-Roch spaces of the Hermitian function field with applications to algebraic geometry codes and low-discrepancy sequences , 2005 .

[14]  Gretchen L. Matthews.  Weierstrass Pairs and Minimum Distance of Goppa Codes , 2001, Des. Codes Cryptogr..

[15]  V. D. Goppa.  Geometry and Codes , 1988 .