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arxiv: 2605.29204 · v1 · pith:NEA66DPFnew · submitted 2026-05-28 · 🧮 math.CO

Comparative monotonicity of linear codes by Hermitian and symplectic hull dimensions

Keita Ishizuka This is my paper

Pith reviewed 2026-06-29 07:03 UTC · model grok-4.3

classification 🧮 math.CO
keywords linear codesHermitian hullsymplectic hullhull dimensionentanglement-assisted quantum codesmonotonicityWitt classification
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The pith

Hermitian hull-dimension ratios for linear codes stay at least 2/3 while symplectic ratios decay to 1/q².

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper finds closed-form expressions that decompose the proportion of linear codes having any fixed Hermitian or symplectic hull dimension. These expressions show that the Hermitian proportion never drops below two thirds for any field size, whereas the symplectic proportion shrinks toward one over q squared as the field grows larger. The contrast is attributed to the way the Witt classification organizes the underlying classical groups. The bounds immediately yield statements that the number of entanglement-assisted quantum codes produced by two standard constructions increases monotonically when the input classical codes are ordered by increasing hull dimension.

Core claim

Closed-form ratio decompositions are derived 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/q² 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]_{q²} and symplectic-hull-graded [2n, k]_q classical codes via the Guenda-Jitman-Gulliver and Wilde-Brun constructions, respectively.

What carries the argument

Closed-form ratio decompositions of the count of linear codes having a prescribed hull dimension, interpreted through the Witt classification of the associated classical groups.

If this is right

  • The number of entanglement-assisted quantum codes obtained from Hermitian-hull-graded classical codes via the Guenda-Jitman-Gulliver construction is monotonic in the hull dimension.
  • The number of entanglement-assisted quantum codes obtained from symplectic-hull-graded classical codes via the Wilde-Brun construction is monotonic in the hull dimension.
  • The Hermitian monotonicity statements hold uniformly across all field sizes and lengths, while the symplectic statements hold asymptotically.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Hermitian-based constructions may remain preferable for quantum coding when field size is large because their hull ratios do not vanish.
  • The same ratio technique could be applied to other hull variants such as Euclidean or alternating to compare their monotonicity properties.
  • Explicit formulas for the ratios might allow direct computation of the exact number of quantum codes for moderate parameters without enumeration.

Load-bearing premise

The Witt classification of the corresponding classical groups is the right way to account for why the Hermitian and symplectic ratio behaviors differ.

What would settle it

A concrete sequence of codes over some finite field where the Hermitian hull ratio falls below 2/3 for infinitely many lengths would disprove the uniform lower bound.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript extends prior results on Euclidean hull dimensions to the Hermitian and symplectic settings. It derives closed-form ratio decompositions for the proportion of linear codes with a prescribed hull dimension, proves that the Hermitian ratio is bounded below by 2/3 uniformly in the parameters, and shows that the symplectic ratio tends to 1/q² as q grows. The qualitative contrast is interpreted via the Witt classification of the associated classical groups. These bounds are then used to obtain monotonicity statements on the number of entanglement-assisted quantum codes arising from the Guenda–Jitman–Gulliver and Wilde–Brun constructions applied to Hermitian-hull-graded [n,k]_{q²} and symplectic-hull-graded [2n,k]_q classical codes, respectively.

Significance. If the counting arguments and resulting closed-form ratios are correct, the work supplies explicit, parameter-dependent bounds that quantify the prevalence of codes with given hull dimensions in two non-Euclidean geometries. The monotonicity corollaries furnish concrete, falsifiable predictions about the growth of entanglement-assisted quantum code families, which may be directly useful for enumeration and construction algorithms. The comparative geometric analysis, while interpretive, is not required for the validity of the quantitative statements.

minor comments (2)
  1. [Abstract] Abstract, final sentence: the phrase “traces this qualitative difference to the Witt classification” could be rephrased to make explicit that the classification supplies an explanatory lens rather than a necessary step in the counting proofs.
  2. The manuscript would benefit from a short table or remark comparing the three geometries (Euclidean, Hermitian, symplectic) side-by-side with their respective ratio behaviors and the underlying group orders.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript and the recommendation to accept. The report accurately captures the main contributions regarding the closed-form ratios for Hermitian and symplectic hull dimensions and their implications for entanglement-assisted quantum codes.

Circularity Check

0 steps flagged

Minor self-citation on Euclidean hull; central ratio bounds from independent counting

full rationale

The abstract notes extension of recent Euclidean hull work, indicating possible self-citation, but the Hermitian (≥2/3) and symplectic (→1/q²) ratio claims are presented as direct consequences of closed-form decompositions from finite-field counting formulas. No quoted step shows a fitted parameter renamed as prediction, self-definition, or load-bearing reliance on prior self-work for the bounds or monotonicity statements. Witt classification is offered only as interpretive lens. This is self-contained against external benchmarks and matches the expected 0-2 range for non-circular papers with incidental self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on abstract; no explicit free parameters, ad-hoc axioms, or invented entities are mentioned. Standard finite-field and group-theoretic background is assumed but not itemized.

pith-pipeline@v0.9.1-grok · 5639 in / 1093 out tokens · 20967 ms · 2026-06-29T07:03:46.733444+00:00 · methodology

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Reference graph

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13 extracted references · 2 canonical work pages

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