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

• URL: http://arxiv.org/abs/2106.10180v3
• Date: Thu, 23 Dec 2021 15:49:47 GMT
• Title: Determining when a truncated generalised Reed-Solomon code is Hermitian self-orthogonal
• Authors: Simeon Ball and Ricard Vilar
• Abstract summary: We prove that there is a Hermitian self-orthogonal $k$-dimensional truncated generalised Reed-Solomon code of length $n$ over $mathbb F_q2$. We also provide examples of Hermitian self-orthogonal $k$-dimensional Reed-Solomon codes of length $q2+1$ over $mathbb F_q2$, for $k=q-1$ and $q$ an odd power of two.
• Score: 0.7614628596146599
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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