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arXiv · 2605.29204

Comparative monotonicity of linear codes by Hermitian and symplectic hull dimensions

Keita Ishizuka
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Abstract

Extending recent work on the Euclidean hull, we derive closed-form ratio decompositions for the number of linear codes with prescribed Hermitian and symplectic hull dimension. The Hermitian ratio admits a uniform lower bound of at least $2/3$, while the symplectic ratio decays to $1/q^2$ asymptotically; a comparative analysis traces this qualitative difference to the Witt classification of the corresponding classical groups. The results translate directly into monotonicity statements for the number of entanglement-assisted quantum codes obtainable from Hermitian-hull-graded $[n, k]_{q^2}$ and symplectic-hull-graded $[2n, k]_q$ classical codes via the Guenda-Jitman-Gulliver and Wilde-Brun constructions, respectively.

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Keita Ishizuka. 2026-05-28. Comparative monotonicity of linear codes by Hermitian and symplectic hull dimensions. https://arxiv.org/abs/2605.29204

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