In this paper we study stopping sets of AG codes over a family of curves, denoted byXf,g, that includes several important curves with applications in coding theory. We present results concerning stopping sets of one-point and m-point codes overXf,g, generalizing some results for Hermitian codes presented by Anderson and Matthews.

In this paper, we construct some class of explicit subcovers of the curve X n,r defined over F q(n) by affine equation y q(n-1) + ··· + y q + y = x q(n-r)+1 - x q(n)+q(n-r) . These subcovers are defined over F q(n) by affine equation g s (y) = x q(n)+q(n-r) -x q(n-r) +1, where g s (y) is a q-polynomial of degree q s . The Weierstrass semigroup H(P ∞ ), where P ∞ is the only point at infinity on such subcovers, is determined for 1 ≤ s ≤ 2r - n + 1, and the corresponding one-point AG codes are investigated. Codes establishing new records on the parameters with respect to the previously known ones are discovered, and 108 improvements on MinT tables are obtained.

We determine the Weierstrass semigroup of a pair of certain rational points on the Giulietti-Korchmáros maximal curves. We use this semigroup to obtain two-point algebraic geometric (AG) codes with better parameters than comparable one-point AG codes arising from these curves. These parameters are new records in the MinT's tables.

We determine the full automorphism group of the Generalized Hermitian curve, denoted by GH, which generalizes the Hermitian curve. The automorphism group of a code is important in Coding Theory, and in this way, we determine completely the automorphism group of the one-point AG codes over the GH curve.