Research On Some Problems Of Linear Codes Over Two Classes Of Finite Non-chain Rings

Before the 1990s, coding theory mostly took place in the setting of vector spaces over finite fields, and after that codes over finite rings have been given more attention. As the generalized cyclic codes the skew cyclic codes and constacyclic codes over some finite rings which are not finite chain rings also have been considered by some authors. The details are given as follows:1. The skew cyclic codes over ring R= Z4+vZ4 are defined. Based on the structural and properties of skew polynomial ring R[x;0], we proved that the skew cyclic code C over R of length n is a left R[x;θ]- submodule of R[x;θ]/(xn-1). The generator polynomials and definitions of the duals of skew codes with respect to Euclidean and Hermitian inner products are gived.At the end, we dicussed the presence of self-dual skew codes.2. The (1+2v)-constacyclic codes over the ring Z4+vZ4 are studied.We introduce a Gray map based on the homogeneous distance.It is proved that the image of (1+2v)-constacyclic codes is a quasi-cyclic code of index 4.The structure of such constacyclic codes for odd length is determined, and their dual codes are dicussed.3. We instroduce a non-trivial ring automorphism over F2+uF2+vF2+uvF2 and a Gray map from F2+uF2+vF2+uvF2 to F2+vF2. The generators of skew cyclic codes over F2+uF2+vF2+uvF2 are determined.We dicusse the properties of the image of their dual codes and self-dual skew codes.