Determining when a truncated generalised Reed-Solomon code is Hermitian self-orthogonal
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We prove that there is a Hermitian self-orthogonal k-dimensional truncated generalised Reed-Solomon of length n ⩽ q^2 over 𝔽_q^2 if and only if there is a polynomial g ∈𝔽_q^2 of degree at most (q-k)q-1 such that g+g^q has q^2-n zeros. This allows us to determine the smallest n for which there is a Hermitian self-orthogonal k-dimensional truncated generalised Reed-Solomon of length n over 𝔽_q^2, verifying a conjecture of Grassl and Rötteler. We also provide examples of Hermitian self-orthogonal k-dimensional generalised Reed-Solomon codes of length q^2+1 over 𝔽_q^2, for k=q-1 and q an odd power of two.
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