On the Constructions of Optical Orthogonal Codes from Combinatorial Design Theory

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Optical orthogonal codes are used in optical code-division multi-access communications so that multiple users can efficiently transmit and receive information in one single optic channel. It is known that an optimal optical orthogonal code is equivalent to a combinatorial structure called maximal cyclic t-difference packing. The main purpose of this research project is to construct optimal optical orthogonal codes from a combinatorial approach, that is, instead of constructing optimal optical orthogonal codes directly, we construct their corresponding maximal cyclic t-difference packings.Fuji-Hara, Miao and Mishima introduced a new concept of incomplete cyclic difference matrices, and used them to construct many cyclic t-difference packings with maximal number of blocks. Together with a computer search, many infinite families of optimal optical orthogonal codes were constructed. The existence of optimal optical orthogonal codes with weight 4, correlation constraint 1, and code length v … More with v=0 mod 24 was completely settled.Shinohara used special arcs and conies in finite geometries to construct optical orthogonal codes and obtained infinite families of asymptotic optimal optical orthogonal codes.Frequency hopping sequences are used in multi-access spread-spectrum communications which are closely related to optical orthogonal codes. Fuji-Hara, Miao and Mishima found an equivalence relationship between frequency hopping sequences and partition-type difference packings. By using this equivalence, Fuji-Hara, Miao and Mishima constructed many infinite families of optimal frequency hopping sequences.Miao also considered the security of multi-access communication systems. Topics covered include authentication codes, secret sharing schemes, ID-based signature schemes, and proxy cryptosystems.Fuji-Hara, Miao, Mishima and Shinohara also carried out theoretical research on several topics in combinatorial design theory which are related to optical orthogonal codes and frequency hopping sequences. Less