Hermitian codes and complete intersections

Abstract

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 µq+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 2 F q 2 of degree µ + λ defined by polynomials with x µ y λ as leading monomial w.r.t. the DegRevLex term ordering (with y > x). We prove that a zero-dimensional subscheme Z of A 2 F q 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.

Minimum-weight codewords of the Hermitian codes are supported on complete intersections

“…A proof for Theorem A applied to the codes with m ≥ 2q 2 − 2q − 2 (called codes of the III and IV phase) is given in [26]. In this paper we prove this same result in the case m ≤ 2q 2 − 2q − 3 (I and II phase).…”

“…In this work we continue the geometrical description of the support of the minimum weight codewords of the Hermitian codes started in [26]. We also refer to that paper for a more detailed historical introduction and bibliography.…”

“…In the previous paper [26] the authors prove that the distance d(m) of the Hermitian codes C m with m ≥ 2q 2 −2q −2 is given by the integer δ(m) := m−(q −2)(q +1). This formula agrees with that proved by [13,31,32], taking in account our convention on the choice of the integer m labeling the code C m given in Remark 2.5 (2).…”

“…In this paper we complete the geometrical characterization of the supports of the minimumweight 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.2010 Mathematics Subject Classification. 11G20,11T71.…”

“…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.…”