Comparative monotonicity of linear codes by Hermitian and symplectic hull dimensions

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.