Special matrices over finite fields and their applications to quantum error-correcting codes
<|reference_start|>Special matrices over finite fields and their applications to quantum error-correcting codes: The matrix-product (MP) code $\mathcal{C}_{A,k}:=[\mathcal{C}_{1},\mathcal{C}_{2},\ldots,\mathcal{C}_{k}]\cdot A$ with a non-singular by column (NSC) matrix $A$ plays an important role in constructing good quantum error-correcting codes. In this paper, we study the MP code when the defining matrix $A$ satisfies the condition that $AA^{\dag}$ is $(D,\tau)$-monomial. We give an explicit formula for calculating the dimension of the Hermitian hull of a MP code. We provide the necessary and sufficient conditions that a MP code is Hermitian dual-containing (HDC), almost Hermitian dual-containing (AHDC), Hermitian self-orthogonal (HSO), almost Hermitian self-orthogonal (AHSO), and Hermitian LCD, respectively. We theoretically determine the number of all possible ways involving the relationships among the constituent codes to yield a MP code with these properties, respectively. We give alternative necessary and sufficient conditions for a MP code to be AHDC and AHSO, respectively, and show several cases where a MP code is not AHDC or AHSO. We provide the construction methods of HDC and AHDC MP codes, including those with optimal minimum distance lower bounds.<|reference_end|>
@article{cao2024special,
    title={Special matrices over finite fields and their applications to quantum
  error-correcting codes},
    author={Meng Cao},
    journal={arXiv preprint arXiv:2405.02285},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.02285},
    primaryClass={cs.IT math.IT}
}