On the minimum distance and the minimum weight of Goppa codes from a quotient of the Hermitian curve

In this paper we study evaluation codes arising from plane quotients of the Hermitian curve, defined by affine equations of the form yq + y = xm, q being a prime power and m a positive integer which divides q+ 1. The dual minimum distance and minimum weight of such codes are studied from a geometric point of view. In many cases we completely describe the minimum-weight codewords of their dual codes through a geometric characterization of the supports, and provide their number. Finally, we apply our results to describe Goppa codes of classical interest on such curves.