Special orthogonal matrices: existence, enumeration, and applications

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We have been working on Orthogonal Designs (=OD's) now for over 30 years. To indicate the significance of this area consider that the 1999 paper "Space-Time Block Coding from Orthogonal Designs," by Tarokh et al. [32] that appeared in IEEE Transactions on Information Theory has proven to be instrumental in wireless communications, the paper has over 4000 citations and is the fourth most cited paper among all papers published in the IEEE Transactions on Information Theory ever. By introducing an Array in 2000, we were able to construct some orthogonal designs for the first time. Seventeen years later, in 2017, we used a known class of OD's together with some known Generalized Hadamard matrices over the Elementary Abelian group EA(2^n), demonstrating a new approach that led to an asymptotic existence for the unbiased and quasi--unbiased Bases for the first time [23]. The plan is to continue with research in the area and hopefully prove some outstanding conjectures regarding the existence of some optimal OD's. The first OD under consideration is of order 128 with 16 variables each repeated eight times. We are quite close in showing this and have high hopes of finding some new OD's, and more importantly, extend the result of Tarokh et al. [32]. Generalized Hadamard matrices over cyclic groups have been at the center stage of many areas of combinatorics, and our study has led us to some interesting combinatorial results. The primary method of applying these tools have been to search for appropriate representing groups together with some creative methods to combine them with other objects such as the known classes of Mutually Orthogonal Latin Squares (=MOLS). Our success in showing that there are precisely 13,710,027 inequivalent Hadamard matrices of order 32 is the primary motivation to continue with research on the classification of H-matrices. An in-depth and thorough study of unbiased bases and association schemes and success in achieving fruitful results in both areas require time and hard work. The proposed main approach will include the use of some objects from Design and Graph Theory, such as MOLS, Orthogonal Arrays, Finite Geometries, Resolvable Group Divisible Designs, and Regular Graphs. The use of OD's, particularly Hadamard matrices, in the construction of maximally error-correcting codes has proven to be quite significant, and any new and novel ideas in finding the remaining unresolved cases would be of obvious interest to engineers and coding theorists, in addition to those working in design theory. The construction of new unbiased and quasi--unbiased bases would be of interest to some theoretical physicists and people working in quantum information theory. The discovery of new association schemes will be of significance in classifying schemes with a certain number of classes. The classification of Hadamard matrices, though very difficult, is also important and of interest to statisticians too.