Algebraic geometric codes on certain plane curves

Abstract

Abstract Necessary and sufficient conditions are given for projective plane curves to have cusps isomorphic to the origin of an affine plane curve x a + y b . Based on them, a family of curves C a b having only one such cusp as a singular point and a family of curves r C a b having only two such cusps as singular points are formulated. Also, the structures of algebraic geometric codes generated on curves contained in these families are shown, where a and b are relatively prime. Then the genus is ( a − 1)/( b − 1)/2 and the basis of linear space L ( m · P ) is given by rational functions x i y i only. Second, by giving concrete examples, it is shown that families of curves C a b and r C a b contain many curves which attain Hasse‐Weil upper bound.