Special matrices over finite fields and their applications to quantum error-correcting codes

The matrix-product (MP) code C₀, ₊: =C₁, C₂, , C₊ 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^ is (D, ) -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.