Abstract
AbstractNecessary and sufficient conditions are given for projective plane curves to have cusps isomorphic to the origin of an affine plane curve xa + yb. Based on them, a family of curves Cab having only one such cusp as a singular point and a family of curves rCab 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 xiyi only. Second, by giving concrete examples, it is shown that families of curves Cab and rCab contain many curves which attain Hasse‐Weil upper bound.
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