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

Abstract

Let 𝒞 be an [n,k,d] linear code over F_q² with parity-check matrix H. The rank r=Rank(HH^†) measures how far the Hermitian dual 𝒞^⊥_h is from being Hermitian self-orthogonal, and it is the least number of coordinates that must be adjoined to 𝒞^⊥_h to make it Hermitian self-orthogonal; equivalently, the Hermitian hull 𝒞∩𝒞^⊥_h has dimension (n−k)−r. In this paper, we develop a structure theory of the Hermitian hull of a cyclic code and apply it to the construction of quantum stabilizer codes. For a cyclic code with defining set T⊆ℤ_n, we show that r is additive over the orbits of the involution τ(ν)=−qν on the q²-cyclotomic cosets and satisfies a parity constraint. A parameter dictionary exhibits r as a gauge trading one-for-one against the code length, and a complementation duality shows that the complement of T realises the logical-qubit count of the quantum code. The extreme values of r are settled by closed subset-sum criteria, and the τ-invariant defining sets are shown to be the unique locus on which the Hermitian dual is again cyclic. On the distance side, we determine the Hermitian dual distance of an r-fold extension of an arbitrary linear code exactly, and we prove that the admissible extension vectors lie in the image of HH^†, which reduces the search space from q^2(n−k) to q^2r. Applied to cyclic, quasi-cyclic and BCH codes, the construction improves $23$ best-known parameters in Grassl's code tables, one of them provably optimal.

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BibTeX RIS

BibTeX

@misc{lu2026hermitian,
  title = {{Hermitian Hulls of Cyclic Codes and Exact Dual Distances of Extended Linear Codes}},
  author = {Liangdong Lu and Ruipan Yang and Yang Liu and Qiang Fu and Guanmin Guo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.12242},
  eprint = {2607.12242},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.12242},
}

RIS

TY  - UNPB
TI  - Hermitian Hulls of Cyclic Codes and Exact Dual Distances of Extended Linear Codes
AU  - Liangdong Lu
AU  - Ruipan Yang
AU  - Yang Liu
AU  - Qiang Fu
AU  - Guanmin Guo
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2607.12242
UR  - https://arxiv.org/abs/2607.12242
AB  - Let 𝒞 be an [n,k,d] linear code over F_q^2 with parity-check matrix H. The rank r=Rank(HH^†) measures how far the Hermitian dual 𝒞^⊥_h is from being Hermitian self-orthogonal, and it is the least number of coordinates that must be adjoined to 𝒞^⊥_h to make it Hermitian self-orthogonal; equivalently, the Hermitian hull 𝒞∩𝒞^⊥_h has dimension (n−k)−r. In this paper, we develop a structure theory of the Hermitian hull of a cyclic code and apply it to the construction of quantum stabilizer codes. For a cyclic code with defining set T⊆ℤ_n, we show that r is additive over the orbits of the involution τ(ν)=−qν on the q^2-cyclotomic cosets and satisfies a parity constraint. A parameter dictionary exhibits r as a gauge trading one-for-one against the code length, and a complementation duality shows that the complement of T realises the logical-qubit count of the quantum code. The extreme values of r are settled by closed subset-sum criteria, and the τ-invariant defining sets are shown to be the unique locus on which the Hermitian dual is again cyclic. On the distance side, we determine the Hermitian dual distance of an r-fold extension of an arbitrary linear code exactly, and we prove that the admissible extension vectors lie in the image of HH^†, which reduces the search space from q^2(n−k) to q^2r. Applied to cyclic, quasi-cyclic and BCH codes, the construction improves $23$ best-known parameters in Grassl's code tables, one of them provably optimal.
ER  -

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