| The quantum computer has many characteristics in information processing,such as its fast speed,powerful computing power and lower energy consumption,which shows great advantage than the classical computer.However,affected by the quantum noise,the quantum system is more likely to cause the decoherence in information processing,which weakens its advantage.To solve this problem,the researchers invented the technology of quantum error correction.In this technology,the good quantum codes is an effective manner to reduce the frequency of decoherence,which provides a necessary protection for quantum error correction and fault-tolerant quantum computation.In recent years,a great deal of research is devoted to the construction of good quantum codes,which leads it to be a hot topic in quantum computation and quantum communication.In this paper,by using three classes of methods,i.e.CSS construction,Hermitian construction and preshared entanglement,we will construct many good and new quantum codes.The idea of CSS construction and Hermitian construction are the utilization of q-ary dual-containing classical linear codes and q2-ary dual-containing classical linear codes,respectively,to obtain q-ary quantum codes.The essence of pre-shared entanglement is the construction of q-ary quantum codes from any q-ary classical linear codes with the help of maximally entangled states.The main innovation of this paper is summarized as follows.First,for the finite field Fq,we propose a general method to construct quantum codes from q-ary matrix-product codes.To construct quantum codes with optimal minimum distance in the sense of matrix-product codes,we provide a constructive method of NSC quasi-orthogonal matrices for the first time.Based on this method,we construct four new NSC quasi-orthogonal matrices.By CSS construction,we obtain some good quantum codes,some of which improve the previous ones in literature.Besides,for the defining matrix A such that AAT is a special class of monomial matrices,we obtain some good quantum codes with new parameters.Second,for the finite field Fq2,we provide a general method to construct quantum codes from q2-ary matrix-product codes.Accordingly,we propose a constructive method of NSC quasi-unitary matrices for the first time.Combining this method with some properties of the polynomial ring Fq[x1,…,xk],we prove that there always exist 3 × 3 NSC quasi-unitary matrices and k × k NSC quasi-unitary matrices for each k<q for the first time,improving and generalizing some known results.In terms of Hermitian construction,we acquire many good quantum codes with parameters either new or better than those in literature.Moreover,by giving a class of defining matrices A such that AA(?)is a special class of monomial matrices,we acquire some good quantum codes which improve the ones in literature or add new parameters.Third,to obtain more good quantum codes from pre-shared entanglement,i.e.entanglement-assisted quantum codes,we explore the Galois hulls of GRS codes and extended GRS codes.We give the necessary and sufficient condition under which an arbitrary codeword of these codes is contained in their Galois dual codes,respectively,which generalize the Euclidean case and the Hermitian case obtained in literature.By investigating the additive subgroups of the finite field Fq,the multiplicative subgroups of Fq*and their cosets,we construct four families of GRS codes and extended GRS codes with Galois hulls of arbitrary dimensions for the first time.As a consequence,we present four families of MDS entanglement-assisted quantum codes with flexible parameters,most of which are obtained by us for the first time. |
