Combinatotial Constructions Of Optical Orthogonal Signature Pattern Codes

Spatial optical code division multiple access(OCDMA),an extension of OCDMA to a two-dimensional(2-D)space coding for image transmission and multiple access,can exploit the inherent parallelism of optics.The spatial OCDMA provides higher throughput comparing with the traditional OCDMA.One of the keys to spatial OCD-MA is the methodology in the construction of optical orthogonal signature pattern codes(OOSPCs).There is a one-to-one correspondence between an(m,n,w,?)-OOSPC and a(?+1)-(mn,w,1)packing design admitting a point-regular automorphism group isomorphic to Zm × Zn.To our knowledge,Pan and Chang constructed some infinite classes of(m,n,4,1)-OOSPCs,and Sawa constructed an infinite class of(m,n,4,2)-OOSPCs.In this thesis,various 3-designs admitting a point-regular automorphism group are used to present constructions of(m,n,w,2)-OOSPCs,and cyclotomy is used to present relative difference families which lead to(m,n,w,1)-OOSPCs.Many new infinite classes of optimal OOSPCs are then obtained.The organization of this thesis is as follows:In Chapter 2,inversive planes,G-designs,1-fan designs and Steiner quadruple systems(SQSs)are used to construct 3-(mn,w,1)packing designs admitting a point-regular automorphism group isomorphic to Zm × Zn.By applying our recursive con-structions with the known S-cyclic SQSs and rotational SQSs,we obtain many new infinite classes of optimal Zm × Zn-invariant 3-(mn,w,1)packing designs,which cor-responds to optimal(m,n,w,2)-OOSPCs.In Chapter 3,it is proved that there is a one-factor in the Kohler graph of Z2?p × Z2?'relative to the Sylow 2-subgroup if there is an S-cyclic SQS(2p),where p?5(mod 12)is a prime and 1 ? ?,? 2.Using this one-factor,we construct a strictly Z2?p × G2?'-invariant regular G*(p,2?+?',4,3)celative vo 1he Sylow 2-subgroup.By using the known S-cyclic SQS(2p),we construct two new infinite classes of optimal(m,n,4,2)-OOSPCs.In Chapter 4,holey perfect bases and holey quadruple systems over abelian groups are introduced in order to construct relative difference families with block size 4 over abelian groups.Cyclotomy is used to present directions of holey perfect bases,holey quadruple systems and relative difference families.Four new infinite classes of optimal(m,n,4,1)-OOSPCs are then obtained.In Chapter 5,symmetric holey quadruple systems over abelian groups are intro-duced in order to construct relative difference families with block size 5 over abelian groups.Moreover,each obtained symmetric holey quadruple system over an abelian group G is also an initial round of a strictly G-invariant special directed-ordered whist tournament frame.One infinite class of optimal(m,n,5,)1OOSPCs is obtained.