Weierstrass Semigroups From a Tower of Function Fields Attaining the Drinfeld-Vladut Bound

Abstract

For applications in algebraic geometric codes, an explicit description of bases of Riemann-Roch spaces of divisors on function fields over finite fields is needed. We investigate the third function field F (3) in a tower of Artin-Schreier extensions described by Garcia and Stichtenoth reaching the Drinfeld-Vlȃdut ¸bound. We construct bases for the related Riemann-Roch spaces on F (3) and present some basic properties of divisors on a line. From the bases, we explicitly calculate the Weierstrass semigroups and pure gaps at several places on F (3) . All of these results can be viewed as a generalization of the previous work done by Voss and Høholdt (1997).

A Modular Interpretation of BBGS Towers

“…Good towers that are recursive play important roles in studies of Ihara's quantity, coding theory, and cryptography [1,7,31,32,46,49]. A tower T is called recursive by an absolutely irreducible polynomial f (x, y) ∈ F q (x)[y] (see [42,Sections 3.6 and 7.2]), if…”