Sets Of Dierence Parkings And Their Applications

Difference packing is one of the most important combinatorial structures in design theory, and it is a generalization of the concept of difference family. Cyclic difference packings can be used to construct optical orthogonal codes with maximum collision parameter 1 and frequency hopping sequences (FHSs), and a lots of design theorists and code theorists have given a deep study on these topics. In this dissertation, we focus on two types of sets of cyclic difference packings:partition-type balanced nested cyclic difference packings(BNCDPs) and pairwise 2-compatible cyclic relative difference families(CRDFs) with special external difference properties. In 2009, Ge et al. revealed a connection between FHS sets and partition-type BNCDPs, but they did not use combinatorial methods to obtain new results. In 2015, Luo et al. gave an equivalent characterization between multilength optical orthogonal codes with maximum collision parameter A and pairwise A-compatible CDP sets, and they obtained a few infinite classes of optimal multilength optical orthogonal codes.We construct partition-type balanced nested cyclic relative difference packings (BNCRDPs) and pairwise 2-compatible CRDFs by using cyclotomic classes of finite fields, discrete logarithm, cyclic difference matrices and skew starters. As a result, we obtain some infinite classes of optimal FHS sets, strictly optimal FHS sets and optimal multilength optical orthogonal codes. We also obtain some infinite classes of optimal two dimensional optical orthogonal codes. The organization of this thesis is as follows:In Chapter 2, we obtain a number of series of new optimal FHS sets. This goal is achieved by constructing a number of series of partition-type BNCRDPs. We present combinatorial constructions for FHS sets and BNCRDPs, including direct constructions by using cyclotomic classes, recursive constructions based on cyclic difference matrices, merging blocks and discrete logarithm.In Chapter 3, we focus on FHSs and FHS sets with partial hamming correlation.The partial Hamming correlation of FHSs was introducted by Eun et al. in 2004. We first establish a correspondence between FHS sets with optimal partial Hamming correlation and multiple partition-type balanced nested cyclic difference packings with a special property. We also describe combinatorial constructions for FHS sets with partial Hamming correlation, including direct constructions by using cyclotomic classes and generalized cyclotomic classeses, recursive constructions based on cyclic difference matrices, generalized cyclotomic classeses and discrete logarithm. As a consequence, our constructions yield a number of series of new strictly optimal FHSs and FHS sets.In Chapter 4, we focus on multilength optical orthogonal codes. The multilength optical orthogonal codes was introducted by Kwong et al. in 2002, and they can be used to simultaneously support multimedia services with different signaling rates and quality-of-service requirements in optical CDMA networks. We use cyclotomic classes, cyclic difference matrices, skew starters to obtain some pairwise 2-compatible CRDFs with block sizes three or four. As a consequence, we obtain some infinite classes of optimal multilength optical orthogonal codes of weight three or four.In Chapter 5, we construct some infinite classes of strictly m-cyclic and semi-cyclic H(m, n,4,3), and use them to give new infinite classes of optimal two dimensional optical orthogonal codes with maximum collision parameter ?= 2 and optimal two dimensional optical orthogonal codes with maximum collision parameter ?= 2 and AM-OPPTS/AM-OPPW property.