On the hull of linearized polynomial codes
Pith reviewed 2026-05-08 07:16 UTC · model grok-4.3
The pith
Gram-matrix ranks give the hull dimension for two families of linearized polynomial codes.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For image codes C(α) = im Φ_α, dim Hull(C(α)) equals rank(Φ_α) minus rank(G(α)), where G(α) is the Gram matrix built from the standard dot product. Specializing to the two-parameter family im(λx + μ L(x)) produces a quadratic Gram pencil whose determinant locates the LCD codes. For F_{q^m}-linear rank-distance codes spanned by X, F_1, …, F_k with the Delsarte inner product, a k-by-k Gram matrix over F_{q^m} determines the hull dimension; when L(X) = X^{q^k} the matrices are circulant, admit closed-form discriminants, and classify the hull in three of four bijectivity cases, with the remaining case settled by an explicit trace-isotropy test.
What carries the argument
The unified Gram-matrix method that converts the hull-dimension problem into a rank computation over the base field.
If this is right
- The determinant of the quadratic Gram pencil locates all LCD image codes in P^1(F_q).
- For L(X) = X^{q^k} the circulant Gram matrices give a closed-form discriminant and classify hull size in three bijectivity regimes.
- In the remaining regime the hull dimension equals the dimension of the intersection of the image and kernel of a certain map, and the maximum value δ = d is characterized by a trace-isotropy condition.
- An exact count of LCD versus non-LCD points is obtained, with the LCD density tending to 1 as q tends to infinity.
Where Pith is reading between the lines
- The formulas supply an immediate way to count the minimal number of entangled pairs needed for any given entanglement-assisted code built from these families.
- The same Gram-matrix reduction may apply to other families of linearized codes once an appropriate inner product is fixed.
- The density result suggests that random choices of parameters will almost always produce LCD codes for large fields, which could be verified by Monte-Carlo sampling.
Load-bearing premise
The chosen inner products remain compatible with the Gram-matrix construction for all the codes and field extensions considered.
What would settle it
Pick a concrete small field, a specific α, compute the hull dimension by direct linear algebra on the code, and check whether it equals rank(Φ_α) minus rank(G(α)).
read the original abstract
Motivated by entanglement-assisted quantum error-correcting codes, where the hull dimension determines the number of required pre-shared entangled pairs, we study hulls of two families of $\mathbb{F}_q$-linear codes defined by $q$-polynomial operators over $\mathbb{F}_{q^m}$. Our main tool is a unified Gram-matrix method. For image codes $\mathcal{C}(\boldsymbol{\alpha})=\operatorname{im}\Phi_{\boldsymbol{\alpha}}$, with $\Phi_{\boldsymbol{\alpha}}=\sum_i\alpha_iF_i$, we prove the master hull--rank formula $\dim\operatorname{Hull}(\mathcal{C}(\boldsymbol{\alpha}))=\operatorname{rank}(\Phi_{\boldsymbol{\alpha}})-\operatorname{rank}(G(\boldsymbol{\alpha}))$, where $G(\boldsymbol{\alpha})$ is the associated Gram matrix over $\mathbb{F}_q$. Specializing to $C_{\lambda,\mu}=\operatorname{im}(\lambda x+\mu L(x))$, we obtain a quadratic Gram pencil $\lambda^2G_0+\lambda\mu G_1+\mu^2G_2$ whose determinant describes the LCD locus in $\mathbb{P}^1(\mathbb{F}_q)$. We also treat $\mathbb{F}_{q^m}$-linear rank-distance codes $\mathcal{C}=\langle X,F_1,\ldots,F_k\rangle_{\mathbb{F}_{q^m}}$ with the Delsarte inner product, where a $k\times k$ Gram matrix over $\mathbb{F}_{q^m}$ determines the hull dimension. For $L(X)=X^{q^k}$, with $d=\gcd(k,m)$, the resulting circulant Gram matrices yield a closed-form discriminant and a complete classification in three of the four bijectivity configurations over $\mathbb{P}^1(\mathbb{F}_{q^m})$. In the remaining case, the hull dimension equals $\delta=\dim_{\mathbb{F}_q}(\operatorname{im}\phi_{\lambda,\mu}\cap\ker\phi_{\lambda,\mu}^{\dagger})$, and the extremal condition $\delta=d$ is characterized by an explicit trace-isotropy criterion. We conclude with an exact count of LCD and non-LCD points, showing that the LCD density tends to $1$ as $q\to\infty$, together with a worked example over $\mathbb{F}_{64}$ and a SageMath verification.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper develops a unified Gram-matrix method to compute hull dimensions for two families of F_q-linear codes arising from q-polynomial operators: image codes C(α) = im Φ_α and F_{q^m}-linear rank-distance codes. It proves the master formula dim Hull(C(α)) = rank(Φ_α) − rank(G(α)), specializes the image-code case to a quadratic Gram pencil whose determinant locates the LCD locus in P^1(F_q), and for rank-distance codes with L(X) = X^{q^k} obtains closed-form discriminants for circulant Gram matrices, a complete classification in three bijectivity cases, a trace-isotropy criterion for the remaining case, and an exact count of LCD points whose density tends to 1 as q → ∞, all supported by a SageMath verification over F_64.
Significance. If the derivations hold, the work supplies explicit algebraic criteria and density results that directly quantify the number of pre-shared entangled pairs needed for entanglement-assisted quantum codes constructed from these families. The master hull-rank formula, the closed-form discriminants, the trace-isotropy characterization, and the reproducible SageMath check are concrete strengths that make the results usable for both theoretical classification and practical code design.
minor comments (3)
- The abstract states that the LCD density tends to 1 as q → ∞, but the precise asymptotic statement (including the rate) appears only in the final section; moving a short version of this limit into the abstract would improve immediate visibility of the main counting result.
- In the rank-distance section, the four bijectivity configurations over P^1(F_{q^m}) are referenced but not enumerated; a one-sentence list of the four cases (with the three that admit closed forms) would clarify the scope of the classification.
- The SageMath verification is cited for the F_64 example; including the explicit code snippet or a pointer to a public repository in the manuscript would strengthen reproducibility.
Simulated Author's Rebuttal
We thank the referee for the positive and accurate summary of our manuscript on hulls of linearized polynomial codes, as well as for the recommendation of minor revision. The referee correctly identifies the unified Gram-matrix method, the master hull-rank formula, the explicit classifications, and the density result as key contributions. No specific major comments appear in the report.
Circularity Check
No significant circularity; derivations are standard linear algebra
full rationale
The master hull-rank formula follows directly from the definition of the hull as the radical of the bilinear form restricted to the code: for C = im(Φ_α) the Gram matrix G(α) records the inner products on a basis of C, so its corank equals dim(C ∩ C^⊥) by the rank-nullity theorem on the associated map. This identity holds for any finite-field bilinear form and requires no additional assumptions or prior results from the authors. The quadratic Gram pencil, circulant-matrix discriminants, and trace-isotropy criterion are explicit algebraic computations over the stated fields; the LCD density limit and F_64 SageMath check are independent verifications. No step reduces to a self-definition, fitted parameter, or self-citation chain.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math q-polynomials induce F_q-linear maps on F_{q^m}
- standard math Gram matrix rank equals the dimension of the radical of the associated bilinear form
Reference graph
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discussion (0)
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