Construction of Quantum Codes From Matrix-Product Codes

Abstract

Recently, the construction of matrix-product codes with certain self-orthogonality is an effective way to obtain good quantum codes. One of the most frequently used means in this process is to construct quasi-unitary matrices as the defining matrices of matrix-product codes. Further, if the defining matrix is NSC, then the corresponding matrix-product code has maximal minimum distance. Inspired by these idea, we propose a constructive method about NSC quasi-unitary matrices over $\mathbb {F}_{q^{2}}$ with $q$ being a prime power, and construct $k\times k$ NSC quasi-unitary matrices for any $k by certain properties of the polynomial ring $\mathbb {F}_{q}[x_{1},\ldots,x_{k}]$ , extending or improving the known results. Through such defining matrices, we obtain new classes of quantum codes from matrix-product codes, many of which show better parameters than previously known.