Optical orthogonal codes and cyclic t-designs.

摘要

This thesis consists of two parts and two appendices. In Part I, we present a new recursive construction for (n, ω, λ _a, λ_c) optical orthogonal codes. For the case of λ_a = λ _c = λ, this recursive construction will enlarge the original family with λ unchanged, and is asymptotically optimal, in the sense that it will produce a new family of asymptotically optimal codes, if the original family is. We call a code asymptotically optimal, if as the length of code n goes to infinity, the ratio of the number of codewords to the corresponding Johnson Bound approaches unity.; The first objective of Part II is to establish a close relationship between optical orthogonal codes (OOC) and cyclic t-designs. The study of OOC's is motivated by their applications in optical code-division multiple access networks and they have been studied extensively for the past two decades. t-designs are an important topic in combinatorial design theory.; In Part I, we give a new recursive construction for OOC's. In Part II, based on the close relationship between OOC's and cyclic t-designs, we are able to apply the new recursive construction to cyclic Steiner Quadruple Systems (SQS), and gain a new perspective on the structure of cyclic SQS's. As a consequence, several new recursive constructions for cyclic SQS's are given, and many new infinite families of cyclic SQS's are constructed.; In Appendix A, motivated by their roles in the new recursive constructions for OOC's, we introduce r-simple matrices and discuss the interactions among r-simple matrices, difference matrices and orthogonal arrays. Furthermore we summarize some applications of r-simple matrices in coding of FO-CDMA systems for both one dimensional and two dimensional codes.; In Appendix B, we list some new results concerning optical orthogonal codes and their connections with multiple target radar and sonar arrays. Also one new construction for multiple target radar and sonar arrays is presented.