Construction of some new entanglement-assisted quantum error-correcting codes of large lengths

Abstract

The study of entanglement-assisted quantum error-correcting codes (EAQECCs) is currently a hot topic in coding theory and quantum information theory. As far as we know, there are few constructions of q-ary EAQECCs of large lengths, and most of the q-ary EAQECCs constructed in the literature have lengths less than or equal to q2+1. As a result, it will be interesting to construct q-ary EAQECCs of large lengths. The goal of this paper is to construct some good and new q-ary EAQECCs of large lengths from q2-ary matrix-product codes associated with the so-called non-singular by columns (NSC) quasi-unitary matrices. To this end, we first propose an effective method and a corresponding algorithm for constructing NSC quasi-unitary matrices. We then present an effective method for constructing EAQECCs of large lengths from matrix-product codes associated with NSC quasi-unitary matrices. Subsequently, we give the explicit construction of q-ary EAQECCs of lengths kn with n≤q2+1, k=k0 for q≥k0, where k0=3,4,5,7,8,9,11,13,16,17,19, and k=q+2, q2−1, q2 for 3≤q≤19. Finally, some new q-ary EAQECCs of large lengths such as lengths 3(q2−1), 3q2, 4(q2−1), 4q2, q3, q(q2+1), q2(q+2), (q2+1)(q+2), q2(q2−1), q4−1, q4, q2(q2+1) for some q are constructed, which are not covered by previously known ones.