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

Abstract: We prove that there is a Hermitian self-orthogonal $k$-dimensional truncated generalised Reed-Solomon code of length $n \leqslant q<sup>2$ over ${\mathbb F}<em>{q<sup>2}$ if and only if there is a polynomial $g \in {\mathbb F}</em>{q<sup>2}$ of degree at most $(q-k)q-1$ such that $g+g<sup>q$ has $q<sup>2-n$ 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 ${\mathbb F}<em>{q<sup>2}$, verifying a conjecture of Grassl and R\"otteler. We also provide examples of Hermitian self-orthogonal $k$-dimensional generalised Reed-Solomon codes of length $q<sup>2+1$ over ${\mathbb F}</em>{q<sup>2}$, for $k=q-1$ and $q$ an odd power of two.