Let $ \mathbb K$ be an algebraically closed field. In this paper, we consider the class of smooth plane curves of degree $n+1>3$ over $ \mathbb K$ , containing three points, $P_{1},P_{2}$ , and $P_{3}$ , such that $nP_{1}+P_{2}$ , $nP_{2}+P_{3}$ , and $nP_{3}+P_{1}$ are divisors cut out by three distinct lines. For such curves, we determine the dimension of certain special divisors supported on $\{P_{1},P_{2},P_{3}\}$ , as well as an explicit description of all pure gaps at each nonempty subset of the distinguished points $P_{1},P_{2}$ , and $P_{3}$ . When $ \mathbb K=\overline { \mathbb F}_{q}$ , this class of curves, which includes the Hermitian curve, is used to construct algebraic geometry codes having minimum distance better than the Goppa bound.

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.