Theory And Application Of Several Kinds Of Linear Codes On Galois Rings

Based on coding theory of finite rings,this paper focused on double cir-culant codes over finite chain rings.On the same time,additive perfect codes in Doob graph have been improved.In particular,the existence of LCD maximum rank metric codes has been investigated in this paper.(1)We investigate self-dual double circulant and LCD double circulant codes over the Galois ring which characteristic is p2 and order is p4.By special factorization of xn-1,we then get a precise enumeration of self-dual double circulant codes and LCD double circulant codes.Finally,by Gray map and using random coding,we obtain families of asymptotically good Gray image codes.In particular,the image of LCD double circulant codes is asymptotically good.(2)We investigate additive perfect codes in Doob graph.We mainly solve an open problem in[23].By some Galois theory and its relation with perfect codes in Doob graph,we construct additive perfect with given parameters.Consequently,we obtain sufficient and necessary condition of additive perfect codes in Doob graph based on known results.At the same time,we obtain some quasi-cyclic perfect codes by some special construction.(3)We investigate a sufficient condition of LCD maximum rank metric code by self-dual basis of finite fields.