The Construction Of Quantum Codes From Matrix-product Codes

In coding theory, we can construct new codes from some shorter codes. Matrix-product codes are a family of codes constructed from some shorter ones, such as the classical (u|u+υ)-construction. Let q be a prime power, C be the field of complex numbers. A q-ary quantum code of length n and dimension K is a K-dimensional sub-space of Hilbert space Cqn= Cq(?)…(?)Cq. Among quantum codes, stabilizer codes are a class of codes which relate with classical codes via certain self-orthogonality properties. In this paper, we get quantum codes from matrix-product codes. For this purpose, we have the following theorem.Theorem:Let A be a non-singular by columns and quasi-orthogonal matrix of order M over Fq, and assume that Ci(i=1,2, …,M) is a family of linear codes with parameters [n, ki, di]over the field  such that CiT (?) Ci. Then, there exists a stabilizer code with parametersIn this paper, we firstly introduce some basic knowledge of quantum codes, matrix-product codes and Reed-Muller codes. Then we give the construction of quasi-orthogonal square matrices. Finally, we get the above method of constructing quantum stabilizer codes and get some quantum codes via this method.