Optical Orthogonal Codes And Super Single Strict Cycle Design

The study of optical orthogonal codes OOCs was first motivated by an appli-cation in a code division multiple access channel which requires binary sequenceswith good auto-correlation and cross-correlation properties. The close relation-ships between OOCs and cyclic t-designs have been investigated by Chung, Miao,Yin, et al since Brickell and Wei constructed some OOCs from cyclic block designsin 1987. It has been shown that any strictly cyclic t-(v, k, 1) packing is equivalentto a (v, k, t-1)-OOC for any t≥2.Cyclic difference packing or difference families is the main method used inthe construction of OOCs and cyclic t-designs, and fruitful results have been ob-tained from it. Most of the known results are related to (v, k, 1)-OOCs. Usually,even a (v, k, 2)-OOC which is not optimal will be better than an optimal (v, h, 1)-OOC for applications. Recently, Chu, Chen, et al studied some classes of super-simple strictly cyclic 2-(v, k,λ) balanced incomplete block designs which lead toa strictly cyclic 3-(v, k, 1) packings, then they got some new results on (v, k, 2)-OOCs. Difference matrices played an important role in their constructions.In this thesis, we shall generalize Chen and Wei's ideas and present somenew recursive constructions for optimal super-simple strictly cyclic 2-(v, k,λ) pack-ings, denoted by (v, k,λ)-OSCP. We also give some direct constructions based onfinite fields. (v, k,λ)-OSCPs are also strictly cyclic 3-(v, k, 1) packings indeed, thuswe can obtain some new infinite families of OOCs.In Chapter 1 and Chapter 2, we shall give a brief introduction for OOCs andsome necessary concepts of combinatorial designs, also some known results willbe given. In Chapter 3, some new recursive constructions for (v, k,λ)-OSCP aredeveloped. In Chapter 4, we shall prove our main results. We solved the existenceof a (v, 3,λ)-OSCP withλ=2, 3, 4 for any admissible value v. Some new (v, k,λ)- OSCPs for k≥4 are also constructed. In the last Chapter, we shall show that thereexists a (v, 4,λ)-OSCP for all admissibleλand 8≤v≤24 with three exception(v,λ)∈{(9, 3), (12, 5), (13,5)} and a possible exception (v,λ)=(24, 11).