Hermitian Hulls of Cyclic Codes and Exact Dual Distances of Extended Linear Codes

The gist The Jordan-canonical-form framework allows for constructing q-ary quantum stabilizer codes from arbitrary classical linear codes over Fq2 without requiring the classical code to satisfy Hermitian self-orthogonality, thereby relaxing a significant restriction in previous constructions.

Obstruction and Framework

The construction addresses the requirement that a classical ingredient code C must be Hermitian self-orthogonal, which is necessary for obtaining quantum stabilizer codes via the standard Hermitian construction (Page 1). The obstruction to this condition is measured by the rank r = (n − k) − dimFq2(C⊥h ∩ C), where C = [n, k, d]q2 with parity-check matrix H (Page 1). The framework systematically reduces the rank of the Hermitian inner-product matrix A = HH† through rank-one perturbations along the Jordan basis W = P−1 of the decomposition A = P JAP−1 (Page 2).

Construction Steps

The process involves systematic reduction and extension to create a Hermitian self-orthogonal code Cso = [n+r, n−k]q2 (Page 1). The construction relies on the following key steps:

1. Decomposition of the Hermitian inner-product matrix A = HH† into its Jordan canonical form A = P JAP−1 to obtain the Jordan basis W = P−1 (Page 2).

2. Successive rank reduction via rank-one perturbations along this basis, utilizing the Wedderburn rank-one reduction formula or Lemma III.3 to ensure Rank(A + uu†) = Rank(A) − 1 (Page 4).

3. The iterative process continues until the rank of the Hermitian product matrix is reduced to zero, resulting in a Hermitian self-orthogonal code Cso = [n + r, n − k]q2 (Page 6).

Distance Preservation Criterion

A crucial aspect of this framework is guaranteeing distance preservation for the resulting quantum code. The exact criterion for preserving the dual distance d(C⊥h ext) is given by Theorem III.9, which states that it holds if and only if daff(wj) ≥ d − 1, where daff(wj) is the affine coset-leader weight (Page 7). This condition is necessary and sufficient for distance preservation (Page 7).

Quantum Code Parameters

Applying the Hermitian construction to the resulting Hermitian self-orthogonal code Cso yields a q-ary quantum code with parameters [[n + r, 2k − n + r, ≥ d]]q (Page 1). This approach systematically produces record quantum codes that improve or supplement existing best-known parameters in Grassl’s tables (Page 1).

Theoretical Advancements

The paper highlights theoretical advantages over prior methods. The Jordan basis W = P−1 provides an explicit, deterministic supply of candidate extension vectors, removing reliance on special algebraic structures of the underlying classical code (Page 7). Furthermore, Theorem III.9 provides a basis-free criterion—the active condition (C1) together with the affine coset-leader condition (10)—that decides precisely when the dual distance is preserved at each step (Page 7). The framework also clarifies the nilpotent case, where direct checking of the active condition Awj ≠ 0 on the Jordan basis is required because Im(A) ⊆ Ker(A) (Page 7).

Practical Applications

The methodology has been applied to various classical codes, including cyclic, quasi-cyclic, and BCH codes over Fq2 (Page 14). The results yield several record quantum codes listed in Table I and Table IV, demonstrating that the Jordan extension procedure successfully generates new quantum codes that improve the best-known parameters in Grassl’s tables (Page 14). The empirical evidence supports the sufficiency of the rank-reduction and distance-preservation criteria for attaining lower bounds on minimum distance (Page 14).

Conclusion

The proposed Jordan extension framework provides a systematic, explicit construction for quantum codes from arbitrary linear codes over Fq2 by explicitly extending the parity-check matrix H with r = Rank(HH†) additional columns (Page 1). This method successfully bypasses the Hermitian self-orthogonality requirement and guarantees distance preservation through the application of rank reduction theorems guided by the Jordan canonical form (Page 1). The resulting quantum code parameters are [[n + r, 2k − n + r, ≥ d]]q (Page 1).

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The gist The Jordan-canonical-form framework allows for constructing q-ary quantum stabilizer codes from arbitrary classical linear codes over Fq2 without requiring the classical code to satisfy Hermitian self-orthogonality.

Practical Applications

The methodology has been applied to various classical codes, including cyclic, quasi-cyclic, and BCH codes over Fq2 (Page 14). The results yield several record quantum codes listed in Table I and Table IV, demonstrating that the Jordan extension procedure successfully generates new quantum codes that improve or supplement existing best-known parameters in Grassl’s tables (Page 14).

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The gist The Jordan-canonical-form framework allows for constructing q-ary quantum stabilizer codes from arbitrary classical linear codes over Fq2 without requiring the classical code to satisfy Hermitian self-orthogonality.

1. Bold header: Jordan-canonical-form framework for arbitrary classical codes

The system can construct an explicit Hermitian self-orthogonal code Cso = [n + r, n − k]q 2 from any r-nearly self-orthogonal code C = [n, k, d]q 2 by systematically reducing the rank of the Hermitian inner-product matrix A through rank-one perturbations along the Jordan basis W = P−1.

2. Bold header: Systematic selection of extension vectors

The framework utilizes the columns of P−1, which form a Jordan basis of A, and selects a column wj from this basis that satisfies the active condition (C1) Awj ≠ 0, thereby guaranteeing that the rank drops by one at each step.

3. Bold header: Exact dual distance preservation criterion

The system ensures distance preservation by verifying the affine coset-leader weight daff(wj):= min wt(x): x ∈ Fnq2, HxT = wj is ≥ d − 1, which is stated as the necessary and sufficient condition for the dual distance to be preserved.