LAUSR

Abstract Let H be the Hermitian curve defined over a finite field F q 2 . In this paper we complete the geometrical characterization of the supports of the minimum-weight codewords of the algebraic-geometry codes over H , started in [26] : if d is the distance of the code, the supports are all the sets of d distinct F q 2 -points on H complete intersection of two curves defined by polynomials with prescribed initial monomials w.r.t. DegRevLex . For most Hermitian codes, and especially for all those with distance d ≥ q 2 − q studied in [26] , one of the two curves is always the Hermitian curve H itself, while if d q the supports are complete intersection of two curves none of which can be H . Finally, for some special codes among those with intermediate distance between q and q 2 − q , both possibilities occur. We provide simple and explicit numerical criteria that allow to decide for each code what kind of supports its minimum-weight codewords have and to obtain a parametric description of the family (or the two families) of the supports.