Hermitian codes and complete intersections

摘要

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-q2. 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 mu q + lambda(q + 1) of the distance d with mu, lambda non negative integers, and mu = q, and consider all the curves chi of the affine plane A(F)(q2)(2) of degree mu +lambda defined by polynomials with x(mu)y(lambda) as leading monomial with respect to the DegRevLex term ordering (with y x). We prove that a zero-dimensional subscheme Z of A(F)(q2)(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-q2-points and there is a curve chi such that Z coincides with the scheme theoretic intersection H boolean AND chi (namely, as a cycle, Z = H.chi). 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. (C) 2019 Elsevier Inc. All rights reserved.