Determining when a truncated generalised Reed-Solomon code is Hermitian self-orthogonal

TL;DR

This paper characterizes when truncated generalized Reed-Solomon codes are Hermitian self-orthogonal over finite fields, determines the minimal length for such codes, and provides explicit examples, confirming a conjecture.

Contribution

It establishes a necessary and sufficient condition for Hermitian self-orthogonality of truncated GRS codes and verifies a conjecture regarding their minimal length.

Findings

Characterization of Hermitian self-orthogonality using polynomial conditions.

Determination of the smallest length for such codes over finite fields.

Explicit examples of codes with length q^2+1 confirming the conjecture.

Abstract

We prove that there is a Hermitian self-orthogonal $k$-dimensional truncated generalised Reed-Solomon code of length $n \leqslant q^2$ over ${\mathbb F}_{q^2}$ if and only if there is a polynomial $g \in {\mathbb F}_{q^2}$ of degree at most $(q-k)q-1$ such that $g+g^q$ has $q^2-n$ distinct zeros. This allows us to determine the smallest $n$ for which there is a Hermitian self-orthogonal $k$-dimensional truncated generalised Reed-Solomon code of length $n$ over ${\mathbb F}_{q^2}$, verifying a conjecture of Grassl and R\"otteler. We also provide examples of Hermitian self-orthogonal $k$-dimensional generalised Reed-Solomon codes of length $q^2+1$ over ${\mathbb F}_{q^2}$, for $k=q-1$ and $q$ an odd power of two.