Weierstrass semigroups on the third function field in a tower attaining the Drinfeld-Vlăduţ bound (Q6584308)

For coding theory, it is necessary to find an explicit description of bases of the Riemann-Roch spaces over function fields. In the construction of AG codes, that is, error-correcting linear codes from algebraic geometry, there are several problems in order to have codes with good parameters: a function field \(F\) with many rational places and small genus; an explicit description of the rational places of \(F\); a basis for the Riemann-Roch space of a given divisor; the determination of the minimum distance of an AG code.\N\NThe main goal of this paper is to investigate the third function field \(F^{(3)}\) in the tower of Artin-Schreier extensions described by \textit{A. Garcia} and \textit{H. Stichtenoth} [Invent. Math. 121, No. 1, 211--222 (1995; Zbl 0822.11078)]. This family reaches the Drinfeld-Vlăduţ bound. The family is given as follows. Let \(F^{(1)}:={\mathcal K} (x_1)\) be the rational function field over \({\mathcal K}:= {\mathbb F}_{q^2}\). For \(i\geq 1\), let \(F^{(i+1)}:=F^{(i)}( z_{i+1})\), where \(z_{i+1}^q+z_{i+1}=x_i^{q+1}\) with \(x_{i+1} :=\frac{z_{i+1}}{x_i}\in F^{(i+1)}\). In [IEEE Trans. Inf. Theory 43, No. 1, 128--135 (1997; Zbl 0876.94045)] \textit{C. Voss} and \textit{T. Høholdt} introduced two rational places \( P^{(3)}\) and \(Q^{(3)}\) and two divisors \(S_0^{(3)}\) and \(S_1^{(3)}\) in \(F^{(3)}\) and described a basis of \({\mathfrak L}(tS_1^{(3)}+uP^{(3)})\), where \(t\) and \(u\) are integers.\N\NThe main results of this paper, Propositions 2.4 and 3.3, motivate and generalize the results of Voss and Høholdt. The authors give a natural construction of basis for various Riemann-Roch spaces \({\mathfrak L}(rQ^{(3)}+sS_0^{(3)} +tS_1^{(3)}+ uP^{(3)})\) and \({\mathfrak L}\big(rQ^{(3)}+\sum_{ \mu=1}^{q-1}s_{\mu}S_{0,\mu}^{(3)}+\sum_{\nu=1}^{q-1} t_{\nu}S_{1,\nu}^{(3)}+uP^{(3)}\big)\), where \(r, s, t, u, s_{\mu}\) and \(t_{\nu}\) are integers. From these bases, the authors compute the Weierstrass semigroups and pure gaps at several places of \(F^{(3)}\).