Quantum error-correcting codes from matrix-product codes related to quasi-orthogonal matrices and quasi-unitary matrices
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Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good q-ary quantum codes of large length. Specifically, it follows from CSS construction (resp. Hermitian construction) that a matrix-product code over 𝔽_q (resp. 𝔽_q^2) which is Euclidean dual-containing (resp. Hermitian dual-containing) can produce a q-ary quantum code. In order to obtain such matrix-product codes, a common way is to construct quasi-orthogonal matrices (resp. quasi-unitary matrices) as the defining matrices of matrix-product codes over 𝔽_q (resp. 𝔽_q^2). The usage of NSC quasi-orthogonal matrices or NSC quasi-unitary matrices in this process enables the minimum distance lower bound of the corresponding quantum codes to reach its optimum. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in <cit.>; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices), Euclidean dual-containing (resp. Hermitian dual-containing) matrix-product codes and q-ary quantum codes derived from these newly constructed matrix-product codes.
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