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a b s t r a c tWe investigate the weights of a family of numerical semigroups by means of even gaps and the Weierstrass property for such a family.

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 give an algorithm to compute such bases for one point divisors, and Weierstrass semigroups over an optimal tower of function fields. We also explicitly compute Weierstrass semigroups till level eight.where the Riemann-Roch space L(sP j ∞ ) is defined by:The knowledge of explicit bases of the Riemann-Roch spaces allows to construct the matrices of such codes. The main result of this paper, Theorem 3.1, is an algorithm to compute such bases. The central idea is to apply results of [6] (see Theorem 2.9 in Section 2 below) to decompose the vector space L(sP j ∞ ) in T j as a direct sum of Riemann-Roch spaces of divisors at the lower level T j−1 , and continue this way till the rational function field T 0 , where the bases can be easily computed. In order this process to be performed, the divisors we get at each level k < j should be invariant for the action of the Galois group of T k /T k−1 . Unfortunately this condition is not always satisfied and the procedure has to be suitably modified, as done in Sections 3.1 and 3.2.As a consequence of the main result we get an algorithm to compute the Weierstrass semigroups:at the totally ramified points P j ∞ , Theorem 3.2. As an application, we also explicitly present the semigroups till level eight, Section 3.3.The ramification structure of the tower and the computation of the genus is presented in Section 2. PreliminariesThe object of study is an asymptotically optimal tower of functions fields defined over the finite field K = F p 2 with p 2 elements, where p is an odd prime. This tower is recursively defined by: T 0 = K(x 0 ) and, for j ≥ 0, T j+1 = T j (x j+1 ), where the function x j+1 satisfies the relation:

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