The Order Bound on the Minimum Distance of the One-Point Codes Associated to the Garcia–Stichtenoth Tower

Garcia and Stichtenoth discovered a tower of function fields that meets the Drinfeld-Vladut bound on the ratio of the number of points to the genus. For this tower, Pellikaan, Stichtenoth, and Torres derived a recursive description of the Weierstrass semigroups associated to a tower of points on the associated curves. In this correspondence, a nonrecursive description of the semigroups is given and from this the enumeration of each of the semigroups is derived as well as its inverse. This enables us to find an explicit formula for the order (Feng-Rao) bound on the minimum distance of the associated one-point codes.