Weierstrass semigroup and codes over the curve $y^q + y = x^{q^r + 1}$

Abstract

We compute the Weierstrass semigroup at a pair of rational points on the curve defined by the affine equation$y^q + y = x^{q^r + 1}$ over $\mathbb{F}_{q^{2r}}$, where $r$ is a positive odd integer and $q$ is a prime power. We then construct a two-point AG code on the curve whose relative parameters are better than comparable one-point AG code.