On Matrix Product Codes And Constacyclic Codes Over Finite Rings

The study of coding over finite rings began in the 1960s.In the 1990s,some scholars proved that a few important families of binary non-linear codes are in fact images under a Gray map of linear codes over Z4,which motivates the systematic and in-depth study of coding over finite rings.Coding over finite rings mainly studies structures and distances of codes and their fitness for a specific application.The Hamming distance and the homogeneous distance are very important distances over finite rings.In particular,the study of the construction of codes and linear codes with rich algebraic structures is an important research topic in coding theory over finite rings.Matrix product codes are a class of codes with longer length constructed by some codes with a shorter length.Constacyclic codes are a class of codes with special algebraic structures.In this thesis,we mainly study structures and distances of matrix product codes and constacyclic codes over finite commutative principal ideal rings(PIRs).The specific works are as follows:In Chapter 3,we study the homogeneous metrics on arbitrary finite PIRs.In general,the homogeneous distance induced by the homogeneous weight over a finite ring is not necessarily a metric.Since a finite commutative PIR can be viewed as a product of finite chain rings,we classify the finite PIRs according to the cardinality of the residual fields of these finite chain rings,and give a necessary and sufficient condition for the homogeneous distance on an arbitrary finite commutative PIRs to be a metric,which generalizes the result for a homogeneous distance on the residue ring of integers to be a metric.In Chapter 4,we further apply this result to the characterization of minimum homogeneous distances of matrix product codes over finite commutative PIRs.In Chapter 4,we study the lower bound of homogeneous distances of matrix product codes constructed by non-singular by columns matrices over arbitrary finite PIR and their duals.It has been pointed out that the lower bound of the minimum distances of such matrix product codes over a class of special finite PIRs satisfies the inequality du(C)?min{ldh(C1),(l-1)dh(C2),...,(l-m+1)dh(Cm)}.For a finite commutative PIR satisfying that the homogeneous distance is a metric,we provide a necessary and sufficient condition for the above inequality to hold for any matrix product codes and show that there exist matrix product codes such that the equality holds in the above inequality.Furthermore,we give the lower bound of the minimum distances of the duals of the matrix product codes constructed by non-singular by columns matrices and two kinds of matrix product codes whose minimum distances can reach the lower bound.In Chapter 5,we generalize the concept of constacyclic codes over finite field-s and finite rings,introduce the non-invertible-element constacyclic codes(NIE-constacyclic codes)over finite rings and study the algebraic structures and minimum Hamming distances of the NIE-constacyclic codes over finite PIRs.We determine the algebraic structures of all NIE-constacyclic codes over finite chain rings,give the unique form of the sets of the defining polynomials and obtain their minimum distances.A general form of the duals of NIE-constacyclic codes over finite chain rings is also provided.In particular,we give a necessary and sufficient condition for the dual of an NIE-constacyclic code to be a constacyclic code.Using the Chi-nese Remainder Theorem,we determine the algebraic structures and minimum dis-tances of NIE-constacyclic codes over finite PIRs.Furthermore,we construct some NIE-constacyclic codes over finite PIRs which can achieve the maximum possible minimum distances for some given lengths and cardinalities.In Chapter 6,we give a method to construct cyclic codes over Galois rings of composite length from cyclic codes of shorter length.Using the minimum Hamming distances of cyclic codes of shorter length,we estimate the minimum distance of cyclic codes of composition length.We also generalize the cyclotomic constructions of cyclic codes over finite fields given by Ding and Xiong to the cyclic codes over Galois rings.In particular,we use the cyclic codes over Galois rings under the cyclotomic constructions to construct the cyclic codes of composition length,and study the properties of these codes.