The Construction Of Q-Cyclic Codes

The theory of rank distance codes and the maximum rank distance codes was introduced by E.M.Gabidulin in 1985,and the method of determining the minimum rank distance of a given code and some kinds of maximum rank distance codes were also given.Then Du Weizhang and Wang Xinmei have proposed the rank distance BCH codes and rank distance Reed-Solomon codes since 2001.When the given rank distance BCH code is maximum rank distance BCH code,the condition that roots of generator polynomial must satisfy was discussed.When the generator polynomial of a given linear rank distance code possesses generalized continuous roots set, the sufficient condition that makes the given linear rank distance code forming rank distance BCH code is discussed. The problems of the minimum rank distance of the linear rank distance codes and the conformation of the maximum rank distance codes are not solved when the set of the roots of the generator polynomial is not continuous roots set.In this thesis,we discuss the construction of the q-cyclic codes and get the method of determining the rank distance of the linear rank distance code.As a result the problems raised by Du Weizhang and Wang Xinmei are solved partly.Firstly,we review the definition and some important characteristics of a linearized polynomial.Then we give a simple algorithm of finding roots of a linearized polynomial in any finite fields .They are closely related to the q-cyclic codes.Secondly,we discuss the construction of q-cyclic codes which are defined on the finite field GF(q~N) and get the generator matrix,parity check matrix and dimensionality.After that we study the rank distance defined on this kind of codes and propose the way to determine the rank distance.Next, the concept of rank distance generalized BCH code on the finite field GF(q~N) is given and it is shown that the minimum rank distance is related to nonzero of the determinant of any k-submatrix of its parity check matrix.Finally,two examples of deciding the minimum rank distance of rank distance generalized BCH code are given.