Self-orthogonality Of A Class Of Constacyclic Codes Over A Special Chain Ring

Cyclic codes over finite fields or finite rings are a class of important linear codes.Such codes have good algebraic structures so that the complexity of the encoding and decoding algorithms is lower than that of the general linear block codes.It can reduce the error rate of various types of communication systems so that it can improve communication quality.Constacyclic codes are the natural generalization of cyclic codes which can be technically implemented by shift registers.They have the similar algebraic structures to cyclic codes so that they inherit most of the good properties of cyclic codes.The properties of constacyclic codes are easy to analyze.Thus,they can easily be encoded and decoded.Self-orthogonal codes over finite rings or finite fields are a class of important linear codes which are closely related to Combinatorial designs and modular lattices.Self-dual codes are a special class of self-orthogonal codes.A self-dual code has the same weight distribution as its dual code.A large number of good codes are self-dual codes.So they have been an important subject in the research of error-correcting codes.With the research of recent decades,the coding theory over finite fields is getting mature.So a large number of scholars have begun to study the coding theory over finite rings.Constacyclic codes on the finite commutative chain rings are a very important research object.If the characteristic of the finite commutative chain ring is relatively prime to the code length of a constacyclic code,we call this code a simple-root code;otherwise it is called a repeated-root code.Some researchers have classified all the A-constacyclic codes of length ps and 2ps over the finite commutative chain ring Fpm + uFpm where u2 = 0,and obtained the algebraic structures of them.In this thesis,we study the self-orthogonality of the above repeated-root constacyclic codes over Fpm + uFpm.First,we obtain the annihilator of the ideal with associated a ?-constacyclic code by the relationship between the number of codewords of every A-constacyclic codes and its dual.Then we obtain the structures of the dual codes of the ?-constacyclic codes of length ps over Fpm + uFpm by the reciprocal polynomials,which generalizes the result about the cyclic codes over the chain ring Fpm + uFpm.Secondly,we compare the structures of every ?-constacyclic code and its dual code and obtain the relation between the polynomials of the generating sets of the A-constacyclic codes and their dual codes.Thus,we can get the necessary and sufficient conditions for every A-constacyclic code to be self-orthogonal.In particular,since self-dual codes are a special kind of self-orthogonal code,we determine the self dual constacyclic codes over Fpm + uFpm.