Group codes on certain algebraic curves with many rational points

Abstract

We construct a series of algebraic geometric codes using a class of curves which have many rational points. We obtain codes of lengthq 2 over $$\mathbb{F}$$ q , whereq = 2q 0 2 andq 0 = 2 n , such that dimension + minimal distance ≧q 2 + 1 − q 0 (q − 1). The codes are ideals in the group algebra $$\mathbb{F}$$ q [S], whereS is a Sylow-2-subgroup of orderq 2 of the Suzuki-group of orderq 2 (q 2 + 1)(q − 1). The curves used for construction have in relation to their genera the maximal number of $$\mathbb{F}$$ GF q -rational points. This maximal number is determined by the explicit formulas of Weil and is effectively smaller than the Hasse—Weil bound.