On Multidimensional Quasi-cyclic Codes Over Finite Chain Rings

As the generalization of quasi-cyclic codes,the multi-dimensional quasi-cyclic code is an important class of linear codes.It not only has a good algebraic structure,but also has important applications in engineering fields,such as the information recognition,the shift keying system of urban multi-path channels,the multi-user optical fiber wireless communication and so on.This thesis studies the structural properties of multi-dimensional quasi-cyclic codes over finite chain rings,mainly including:the algebraic structure of multi-dimensional quasi-cyclic codes over finite chain rings,the equivalence of this class codes among other linear codes over finite chain rings,the minimal Hamming distance bound of multi-dimensional quasi-cyclic codes;the linear complementary pair of multi-dimensional quasi-cyclic codes;the relationship between multi-dimensional quasi-cyclic codes and multi-dimensional convolutional codes over finite chain rings.The specific research contents are as follows.In Chapter 3,we study the algebraic structure of multi-dimensional quasi-cyclic codes over finite chain rings,give a decomposition of this class code,prove the algebraic structural equivalence of such class codes with direct product codes and quasi-abelian codes over finite chain rings,analysis of the relationship between n-dimensional quasi-cyclic codes and（n-1）-dimensional quasi-cyclic codes.In Chapter 4,we study the minimum Hamming distance bound of multi-dimensional quasi-cyclic codes over finite chain rings,give the Lally bound and Jensen bound of minimum Hamming distance of multi-dimensional quasi-cyclic codes over finite chain rings.In Chapter 5,we study the linear complementary pairs of multi-dimensional quasi-cyclic codes on finite chain rings,prove the equivalence of C and D^?in the linear complementary pair（C,D）of a class of multi-dimensional quasi-cyclic codes over finite chain rings,give the security parameter of the linear complementary pair（C,D）of this class of multi-dimensional quasi-cyclic codes.In Chapter 6,we study the relationship between multi-dimensional quasi-cyclic codes and multi-dimensional convolutional codes over finite chain rings,compare the minimum hamming distance of multi-dimensional quasi-cyclic codes with the free distance of corresponding multi-dimensional convolutional codes,give the upper and lower bound of the free distance of a class of multi-dimensional convolutional codes.In Chapter 7,we summarize the main results of this thesis,propose several urgent and open problems of multi-dimensional quasi-cyclic codes over finite chain rings.