Combinatorial Constructions For Optical Orthogonal Signature Pattern Codes And Their Related Designs

Kitayama proposed a novel code-division multiple access (CDMA) fiber optic net-work based on spatial frequency spread spectrum for image transmissions, called spatial CDMA. The spatial CDMA network has fostered the development of high-speed mul-tiple access network applications, especially image applications such as supercomputer visualizations, medical image access, and distribution and digital video broadcasting. Comparing with the traditional CDMA network, the spatial CDMA network provides higher throughput.As signature patterns of the spatial CDMA network, optical orthogonal signature pattern codes (OOSPCs) have attracted wide attention. An (m,n,k,λa,λc)-OOSPC is a family of m x n (0,1)-matrices (called codewords) with Hamming weight k satis-fying two correlation properties. Let Θ(m,n,k,λa,λc) be the largest possible num-ber of codewords among all (m, n, k, λa,λc)-OOSPCs. An (m, n, k, λa,λc)-OOSPC with Θ(m, n, k, λa, λc) codewords is said to be optimal (or maximum). For the case of Aa=λc=λ, the notations (m, n, k, λa, λc)-OOSPC and0(m, n, k, λa, λc) can be briefly written as (m, n, k,λ)-OOSPC and Θ(m, n, k,λ). The research on OOSPCs in this paper mainly focuses on determining the exact value of Θ(m, n, k, λa, λc) and constructing an optimal (m, n, k, λa, λc)-OOSPC.The material in this paper is organized as follows.Chapter1gives a brief introduction to the background of OOSPCs.In Chapter2, we shall ignore the constructions of optimal OOSPCs. We focus our attention on two classes of OOSPCs and obtain the formula of Θ(m, n,k,λ,k-1) for any positive integers k, m, n and λ=k or k-1.In Chapter3, an equivalent relationship between an OOSPC and a strictly Zm×Zn-invariant packing is established and several recursive constructions for (m,n,k,1)-OOSPCs are presented by means of (incomplete) difference matrices and group divis-ible designs. Furthermore, we give some direct constructions for (3,n,4,1)-OOSPCs, which are based on skew starters and cyclotomic classes.Chapters4and5concentrate on the constructions of optimal (m, n,4,1)-OOSPCs and (m,n,3,1)-OOSPCs respectively. In Chapter4, based on the constructions which are given in Chapter3, some known results of (m, n,4,1)-OOSPCs are extended, and for the following three cases:(1) gcd(m,18)=3and n=0(mod12), or (2) mn=8,16(mod24) and gcd(m,n,2)=2, or (3) mn=0(mod24) and gcd(m,n,6)=2, several infinite families of optimal (m,n,4,1)-OOSPCs are constructed. In Chapter5, we completely solve the construction problem of optimal (m, n,3,1)-OOSPCs for any positive integers m and n by using new direct constructions and the known recursive constructions in Chapter3.In Chapter6, we concentrate on the existence of a (G,4, A)-DM no matter what structure of a finite abelian group G is. The study is motivated by some recursive con-structions for optical orthogonal signature pattern codes and relative difference families. Eventually for the following two cases:λ=1and G is non-cyclic, or λ>1is an odd integer, we prove that a (G,4,λ)-DM exists if and only if G has no cyclic Sylow2-subgroup. Moreover, we point out that (G,4,λ)-DMs always exist for any even integer λ≥2and any finite abelian group G.