Combinatorial Constructions Of Optimal Three-dimensional Optical Orthogonal Codes

As their own good auto and cross-correlation properties, OOCs are widely used in fiber optic channel CDMA systems. In order to further improve system performance, S. Kim et al.[36]proposed the concept of three-dimensional optical orthogonal codes (3-D OOCs).The main issue of this thesis is to investigate the combinatorial constructions of optimal3-D OOCs with the at most one optical pulse per wavelength/time plane (AM-OPP) restriction. As an application, the exact number of codewords of an optimal AM-OPP3-D (u×v×w,3,1)-OOC is determined completely. On the other hand, we found, when λ≥λc=2, only limited progress has been made on optimal (v, k, λa,2)-OOCs. So we try to make a preliminary study on the size of optimal (v,4,3,2)-OOCs.This thesis is organized as follows.In Chapter1, we give a brief introduction on the background of optimal optical orthogonal codes.In Chapter2, we study optimal AM-OPP3-D (u x v x w, k, l)-OOCs. We build an equivalence relation between such an OOC and a certain combinatorial subject, called a w-cyclic group divisible packing of type (vw)u. By this link, the upper bound of the number of codewords is improved and some new combinatorial constructions are presented. As an application, the exact number of codewords of an optimal AM-OPP3-D (u×v×w,3,1)-OOC is determined for any positive integers v, w and u≥3.In Chapter3, we are concerned about optimal (v,4,3,2)-OOCs. A tight upper bound on the exact number of codewords of optimal (v,4,3,2)-OOCs is given. By the introduction of the auxiliary designs, such as g-regular (v,4,3,2)-OOCs and semi-cyclic group divisible designs, some new recursive constructions of optimal (v,4,3,2)-OOCs are also presented. As a result, the exact number of codewords of an optimal (v,4,3,2)-OOC is determined for some infinite series.In Chapter4, we list some problems which need to be further studied.