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valbert4 committed Nov 25, 2023
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An \([[n,k,d]]_q\) true Galois-qudit stabilizer code constructed from a Hermitian self-orthogonal linear code over \(GF(q^2)\) using the one-to-one correspondence between the Galois-qudit Pauli matrices and elements of the Galois field \(GF(q^2)\).
Galois-qudit stabilizer codes are in one-to-one correspondence with trace-alternating self-orthogonal additive codes of length \(n\) over \(GF(q^2)\) via the \hyperref[topic:gfqsq-representation]{\(GF(q^2)\) representation}.
Hermitian self-orthogonal linear codes over \(GF(q^2)\) are automatically trace-alternating self-orthogonal, and applying this construction to such codes yields Hermitian-construction codes \cite[Corr. 19]{arxiv:quant-ph/0508070
Hermitian self-orthogonal linear codes over \(GF(q^2)\) are automatically trace-alternating self-orthogonal, and applying this construction to such codes yields Hermitian-construction codes \cite[Corr. 19]{arxiv:quant-ph/050807}.
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A Hermitian self-orthogonal linear \([n,k,d]_{q^2}\) code yields an \([[n,n-2k]]_q\) true stabilizer code with distance no less than \(d\); this is called the \textit{Hermitian construction}.
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