Multi-point codes over Kummer extensions (Q1692162)

The authors study algebraic geometric codes defined from Kummer extensions. They extend results of \textit{A. S. Castellanos} et al. [IEEE Trans. Inf. Theory 62, No. 9, 4867--4872 (2016; Zbl 1359.94825)], from one- and two-point codes over Kummer extensions to multi-point codes. Let \(\mathcal{F}_{\mathcal{K}}/\mathbb{F}_q(X)\) be a Kummer extension given by \[ y^m = f(x)^{\lambda} = \prod_{i=1}^r (x-\alpha_i)^{\lambda}, \] where \(\alpha_1,\ldots, \alpha_r\) are pairwise distinct elements of \(\mathbb{F}_q\) and \(\gcd(m,r\lambda) = 1\). Let \(P_1,\ldots, P_r\) be the places of the rational function field \(\mathcal{F}_{\mathcal{K}}\) associated to the zeros of \(x-\alpha_1,\ldots, x - \alpha_r\) and let \(P_{\infty}\) denote the unique place at infinity. Consider the divisor \[ G = s_1 P_1 + \cdots + s_r P_r + t P_\infty. \] The authors give an explicit basis for the Riemannn-Roch space \(\mathcal{L}(G)\) and compute the `floor of the divisor' \(G\), the divisor of minimum degree with the same Riemann-Roch space as \(G\). Let \(Q_1,\ldots, Q_l\) be distinct rational places of a function field \(F\) over \(\mathbb{F}_q\). The Weierstrass semigroup \(H(Q_1,\ldots, Q_l)\) is defined by \[ \left\{ (s_1,\ldots, s_l) \in \mathbb{N}_0^l\;\mid \exists f \in F \text{ with } (f)_\infty = \sum_{i=1}^l s_i Q_i \right\}. \] The authors compute Weierstrass semigroups and the set of `pure gaps' corresponding to several totally ramified places of \(\mathcal{F}_{\mathcal{K}}\). Building on work of \textit{H. Maharaj} et al. [J. Pure Appl. Algebra 195, No. 3, 261--280 (2005; Zbl 1087.14024)], the authors use the results described above to give several examples of codes with better parameters than those that were previously known.