On the hull of linearized polynomial codes

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 $\mathbb{F}<em>q$-linear codes defined by $q$-polynomial operators over $\mathbb{F}</em>{q<sup>m}$. Our main tool is a unified Gram-matrix method. For image codes $$\mathcal{C}(\boldsymbolα)=\operatorname{im}Φ<em>{\boldsymbolα}$$, with $Φ</em>{\boldsymbolα}=\sum_iα<em>iF_i$, we prove the master hull--rank formula $$\dim\operatorname{Hull}(\mathcal{C}(\boldsymbolα))=\operatorname{rank}(Φ</em>{\boldsymbolα})-\operatorname{rank}(G(\boldsymbolα))$$, where $G(\boldsymbolα)$ is the associated Gram matrix over $\mathbb{F}<em>q$. Specializing to $C</em>{λ,μ}=\operatorname{im}(λx+μL(x))$, we obtain a quadratic Gram pencil $λ<sup>2G_0+λμG_1+μ<sup>2G_2$ whose determinant describes the LCD locus in $\mathbb{P}<sup>1(\mathbb{F}_q)$. We also treat $\mathbb{F}<em>{q<sup>m}$-linear rank-distance codes $$\mathcal{C}=\langle X,F_1,\ldots,F_k\rangle</em>{\mathbb{F}<em>{q<sup>m}}$$ with the Delsarte inner product, where a $k\times k$ Gram matrix over $\mathbb{F}</em>{q<sup>m}$ determines the hull dimension. For $L(X)=X<sup>{q<sup>k}$, 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 $\mathbb{P}<sup>1(\mathbb{F}_{q<sup>m})$. In the remaining case, the hull dimension equals $$δ=\dim_{\mathbb{F}<em>q}(\operatorname{im}φ</em>{λ,μ}\cap\kerφ<em>{λ,μ}<sup>{\dagger})$$, and the extremal condition $δ=d$ 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 $\mathbb{F}</em>{64}$ and a SageMath verification.

• The paper presents a unified Gram-matrix framework that computes hull dimensions in linearized polynomial codes, crucial for both classical and quantum error correction.
• It derives explicit LCD and self-orthogonality criteria using algebraic invariants, circulant matrices, and discriminant analysis, especially in the Frobenius twist case.
• The study connects hull dimensions directly to entanglement costs in EA-QECCs, offering practical insights for designing codes with prescribed resource requirements.

Hulls of Linearized Polynomial Codes via Gram-Matrix Analysis

Introduction and Motivation

The theory of the hull of a linear code—defined as the intersection of a code with its dual—has substantial implications in both classical and quantum coding theory, particularly for the design of entanglement-assisted quantum error-correcting codes (EA-QECCs), where the hull dimension directly measures the entanglement cost. The study of codes defined by $q$-polynomials (linearized polynomials) over finite fields is central in areas such as network coding and cryptography due to their connection with rank-metric codes, notably Maximum Rank Distance (MRD) codes. This paper (2604.23097) develops a unified approach to computing hulls of two families of linear codes defined by $q$-polynomial operators, leveraging a Gram-matrix formulation to obtain structural and quantitative results about LCD, self-orthogonal, and self-dual codes in both the Hamming and rank metrics.

Algebraic and Structural Background

Consider a finite field $\mathbb{F}_{q^m}$ and its subfield $\mathbb{F}_q$. A $q$-polynomial $L(X) = \sum_{i=0}^s a_i X^{q^i}$ defines an $\mathbb{F}_q$-linear operator on $\mathbb{F}_{q^m}$. The hull of a linear code $C \subseteq \mathbb{F}_{q^m}$ with respect to a non-degenerate bilinear form (typically the field trace inner product) is $Hull(C) = C \cap C^\perp$, a quantity critical in both code equivalence and the resource analysis for EA-QECCs.

A foundational result connecting the Gram matrix $q$0 of an $q$1-linear operator $q$2 with the hull dimension is: $q$3 This identity underlies the analysis throughout the paper, enabling systematic exploration of hull dimensions in parametric code families defined by $q$4-polynomial operators.

Master Hull--Rank Theorem and General Gram-Matrix Paradigm

The central theoretical contribution is the Master Hull–Rank Theorem: for codes defined as images of $q$5-polynomial operators, $q$6 where $q$7, the hull dimension is given by: $q$8 where $q$9, with $\mathbb{F}_{q^m}$0 defined by inner products of code generators. The LCD (Linear Complementary Dual) and self-orthogonality conditions reduce to invertibility ($\mathbb{F}_{q^m}$1) and vanishing ($\mathbb{F}_{q^m}$2) of the Gram matrix, respectively.

Parameterizing the code family, hull strata are defined in terms of orbits in the projective parameter space, and their structure is tied to classical algebraic invariant theory through quadratic forms and discriminants.

Specialization to Rank-Distance Codes

For $\mathbb{F}_{q^m}$3-linear rank-metric codes $\mathbb{F}_{q^m}$4, an explicit criterion is provided. The hull dimension is governed by the $\mathbb{F}_{q^m}$5 generator matrix $\mathbb{F}_{q^m}$6, with entries $\mathbb{F}_{q^m}$7 (suppressing the zero term for codes with $\mathbb{F}_{q^m}$8). The hull dimension is $\mathbb{F}_{q^m}$9, with a direct LCD criterion $\mathbb{F}_q$0, and self-orthogonality corresponds to $\mathbb{F}_q$1. It is also proven that such codes can never be self-dual or self-orthogonal as a whole when $\mathbb{F}_q$2 is included.

The Two-Parameter Code Family and Discriminant Locus

The paper provides a detailed examination of the two-parameter code family $\mathbb{F}_q$3 where $\mathbb{F}_q$4. The Gram matrix is a quadratic pencil $\mathbb{F}_q$5, and the hull structure is governed by the vanishing locus of the discriminant $\mathbb{F}_q$6 in the relevant projective line, identifying the non-LCD locus. It is established that the density of LCD codes grows asymptotically to one as $\mathbb{F}_q$7.

For the important case of $\mathbb{F}_q$8 (Frobenius twist), closed-form expressions for the Gram matrix entries are obtained using circulant matrices under a normal basis. The roots of $\mathbb{F}_q$9, which classify the non-LCD locus, are computed via the eigenvalues of companion pencils in the circulant family.

Complete Classification in the Frobenius Twist Case

A detailed and explicit classification is given for the possible hull dimensions at each parameter value in the Frobenius family, using field-theoretic arguments tied to the action of $q$0-power maps on the field structure. In almost all cases, the hull dimension is either $q$1, $q$2, or $q$3 (where $q$4), depending on whether the relevant operators are bijective or not and on a total isotropy condition. In the residual non-bijective case, $q$5 is characterized explicitly, with the extremal value $q$6 equivalent to an explicit trace-isotropy criterion.

A concrete computational example over $q$7 ($q$8) is provided; the full spectrum of hull dimensions is computed over $q$9, confirming the sharpness of the theoretical results.

Quantum Coding Implications

The entanglement cost (the number of ebits) in EA-QECCs constructed from such classical codes is precisely the hull dimension, as discussed in [GJG18]. The analysis in this work thus directly yields the entanglement requirements for the entire Frobenius family, and for the rank-distance families, via computable Gram ranks. In particular, the paper demonstrates that, within the bijective locus, at most $L(X) = \sum_{i=0}^s a_i X^{q^i}$0 points (in $L(X) = \sum_{i=0}^s a_i X^{q^i}$1) have $L(X) = \sum_{i=0}^s a_i X^{q^i}$2, with the typical entanglement cost being $L(X) = \sum_{i=0}^s a_i X^{q^i}$3 as $L(X) = \sum_{i=0}^s a_i X^{q^i}$4.

Conclusion

This work presents a comprehensive algebraic and combinatorial analysis of the hull structure for codes defined by $L(X) = \sum_{i=0}^s a_i X^{q^i}$5-polynomial operators. The Gram-matrix approach offers a unified, conceptual, and computationally tractable framework for studying hulls in both the Hamming and rank metrics, generalizing previous results on LCD codes and facilitating the explicit design of codes with prescribed hull dimensions and entanglement cost.

Future research directions include extending the Gram-matrix analysis to broader classes of $L(X) = \sum_{i=0}^s a_i X^{q^i}$6-polynomial codes beyond the two-parameter and Frobenius-twist families, exploring the intersection with MRD property under hull constraints, and investigating the construction of EA-QECCs meeting Gilbert–Varshamov-type bounds for general hull dimensions. The explicit computational connection between hull structure and entanglement resources firmly positions these results at the intersection of classical and quantum coding theory.