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

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 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.

Hermitian codes and complete intersections

“…We also point out that analogous results for codes of the other phases, namely codes C m with m < 2q 2 − 2q − 2, has been obtained by the authors in [27]. Finally, a generalization to codewords of small weight is in progress and we are confident that, from this strong geometric characterization, also the explicit computation of the weight distribution will follow.…”

“…In [27] we have tested our new geometrical approach for codes of phases 1-2, both from a theoretical and a computational point of view, while in the present paper we have tested it for codes of phases 3-4, thus developing the radically different approach sought-after in [26]. We are also confident that this could be a powerful tool also to study the weights distribution of Hermitian codes and to design an efficient algorithm of error detection and correction.…”

Intersections between the norm-trace curve and some low degree curves

“…The norm-trace curve is a natural generalization of the Hermitian curve to any extension field F q r . It has been widely studied for coding theoretical purposes; see [1,8,13,[19][20][21].…”