Generalized Cyclotomy Over Residue Class Ring And The Construction Of 1(1/2)-designs

The theory of cyclotomy can date back to Gauss. Classical cyclotomy was first considered in detail by Gauss in connection with the rule-and-compass construction of the regular polygon of n sides. He introduced Gaussian periods and cyclotomic numbers in his book " Disquisitiones Arithmeticae" in 1801. Both Gaussian periods and cyclotomic numbers are closely related to some cyclic codes. The theory of cyclotomy has a lot of applications in number theory. Recently, it has been proved useful in more applied fields such as coding theory and cryptography and so on. The realm of Combinatorics has also benefited from the use of cyclotomy, which can be applied for example to the construction of difference sets and then to obtain new designs.In the first part of this paper, we introduce a generalized cyclotomy over residue class ring, which includes classical cyclotomy, Whiteman’s and Ding-Helleseth’s cyclotomy as special cases. We discuss how to calculate cyclotomic numbers of order 6 and order 8. By using of the cyclotomic numbers we have calculated, we construct two kinds of difference system of sets, whose resultant are all asymptotically optimal relative to Levenshtein bound.The definition of 1(1/2)design was firstly introduced by Bose et al. in 1976. Neumaier defined t(1/2)designs in 1980. When t was equal or greater than two, t(1/2)designs were con-sidered clearly. Duval firstly defined the directed strongly regular graphs in 1988, which is a generalization of strongly regular graphs. Soon afterwards, Brouwer has exhibited the close relationship between 1(1/2)designs and directed strongly regular graphs.1(1/2)signs have become one of effective instruments for the construction of directed strongly regular graphs.In the second part of this paper, we introduce some properties of 1(1/2)designs and construct some new 1(1/2)designs. Particularly, We consider all the parameters and prove the necessary and sufficient conditions for the existence of 1(1/2)designs with every block having three elements. At the same time, we have proved some necessary conditions for the existence of 1(1/2)designs with every block having four elements. More importantly, we constructed some 1(1/2)designs and proved the nonexistence of some 1(1/2)designs with specific parameters when the blocks have four elements.