Research On Matrix-Product Codes Over Finite Fields And Finite Rings

Matrix-product codes are one of specially important codes in coding the-ory and error correcting codes theory.Matrix-product codes over finite field were first considered by T.Blackmore and G.H.Norton in the 1990s.In the early 21st century,many investigators concentrated their focus on matrix-product codes over some kinds of finite rings and finite field.In this paper,we mainly discuss the sufficient and necessary conditions of matrix-product self-dual codes over finite field for different inner products,and properties for cyclic matrix-product codes over finite chain rings and matrix-product codes over finite non-chain rings as well.We also define NSR matrix-product codes and investigate its properties.A kind of new matrix-product codes of the form D=A[C1,C2,…,CM]B are also discussed.The contents are as follows:1、Sufficient and necessary conditions for self-dual matrix-product codes over Fq are discussed(for Euclidean inner product、Hermitian inner product and Galois inner product).An example is given to show the validity of the mentioned sufficient and necessary conditions.Moreover,we check that it’s also a class of asymptotically good algebraic codes.2、Cyclic matrix-product codes over finite chain rings and matrix-product codes over the ring R=Fp+uFP+vFp+uvFp(where p is a prime,u2=u,v2=v,uv=vu)are studied,and an important conclusion of the new generalized N-quasi-cyclic codes over the ring S is given.3、We define a kind of NSR matrix and study the properties of NSR matrix-product codes.A kind of matrix-product codes of the form D =A[C1,C2,…,Cm]B are also discussed.