Zero Divisors And Units In Finite Group Ring And Their Applications In Coding Theory

In this paper,we mainly study the zero divisor and unit in finite group rings and its application in coding theory.This is a cross study of algebra theory,number theory and coding theory.Zero divisor and unit are important components of group ring,and studying them will be of great help to understand the structure of group ring.T.Hurley and P.Hurley systematically discussed the scheme of constructing codes with units or zero divisors in group ring,and called the constructed codes as group ring codes.We summarize the existing unit groups for determining finite group rings,and give the structure of unit groups in some specific finite group rings.The paper also introduces zero divisor and zero divisor code.Deduces several zero divisor by using different group ring structure,and then constructs zero divisor code.The main body of the paper includes the following two parts.In the first part,we discuss the structure of the unit group in the ring of finite group,which consists of the commutative groups of order 17,18,19,20,22 and 23.In the second part,zero divisor and zero divisor code are discussed,and two schemes to derive codes by zero divisor are introduced.The first scheme is to transform the zero divisor in the ring into a specific matrix,thus a group of cyclic codes are obtained.The second scheme is to give the concrete definition of zero divisor code.We construct the zero divisor code of group ring Z2C6 or Z2C7 by the definition of zero divisor code,and through the zero divisor of Z2C7,we get a right zero divisor code of Z2D14.