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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. [ABSTRACT FROM AUTHOR]

For any affine-variety code we show how to construct an ideal whose solutions correspond to codewords with any assigned weight. We are able to obtain geometric characterizations for small-weight codewords for some families of Hermitian codes over any F q 2 . From these geometric characterizations, we obtain explicit formulas. In particular, we determine the number of minimum-weight codewords for all Hermitian codes with d ≤ q and all second-weight codewords for distance- 3 , 4 codes. [ABSTRACT FROM AUTHOR]

In this paper we consider the Hermitian codes defined as the dual codes of one-point evaluation codes on the Hermitian curve H over the finite field F q 2 . We focus on those with distance d ≥ q 2 − q and give a geometric description of the support of their minimum-weight codewords. We consider the unique writing μ q + λ (q + 1) of the distance d with μ , λ non negative integers, and μ ≤ q , and consider all the curves X of the affine plane A F q 2 2 of degree μ + λ defined by polynomials with x μ y λ as leading monomial with respect to the DegRevLex term ordering (with y > x). We prove that a zero-dimensional subscheme Z of A F q 2 2 is the support of a minimum-weight codeword of the Hermitian code with distance d if and only if it is made of d simple F q 2 -points and there is a curve X such that Z coincides with the scheme theoretic intersection H ∩ X (namely, as a cycle, Z = H ⋅ X). Finally, exploiting this geometric characterization, we propose an algorithm to compute the number of minimum weight codewords and we present comparison tables between our algorithm and MAGMA command MinimumWords. [ABSTRACT FROM AUTHOR]