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

TL;DR

This paper characterizes the supports of minimum-weight codewords in Hermitian algebraic-geometry codes, showing they are supported on complete intersections of curves, with explicit criteria depending on code parameters.

Contribution

It completes the geometric description of minimum-weight codeword supports for Hermitian codes, detailing when these are supported on Hermitian curves or other curves, with explicit criteria.

Findings

Supports are on complete intersections of two curves.

For most codes with large distance, one curve is the Hermitian curve.

Explicit criteria determine the support structure for each code.

Abstract

Let $\mathcal{H}$ be the Hermitian curve defined over a finite field $\mathbb{F}_{q^2}$. In this paper we complete the geometrical characterization of the supports of the minimum-weight codewords of the algebraic-geometry codes over $\mathcal{H}$, started in [1]: if $d$ is the distance of the code, the supports are all the sets of $d$ distinct $\mathbb{F}_{q^2}$-points on $\mathcal{H}$ complete intersection of two curves defined by polynomials with prescribed initial monomials w.r.t. \texttt{DegRevLex}. For most Hermitian codes, and especially for all those with distance $d\geq q^2-q$ studied in [1], one of the two curves is always the Hermitian curve $\mathcal{H}$ itself, while if $d<q$ the supports are complete intersection of two curves none of which can be $\mathcal{H}$. 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. [1] C. Marcolla and M. Roggero, Hermitian codes and complete intersections, arXiv preprint arXiv:1510.03670 (2015).