Special orthogonal matrices: existence, enumeration, and applications

我们一直致力于正交设计（=OD的）现在超过30年。为了表明该领域的重要性，请考虑Tarokh等人于1999年发表在《IEEE信息论汇刊》上的论文“来自正交设计的空时块编码”。[32]已被证明在无线通信中发挥重要作用，该论文已超过4000次引用，是IEEE信息论汇刊有史以来发表的所有论文中引用次数第四多的论文。 通过在2000年引入Array，我们第一次能够构造一些正交设计。17年后，在2017年，我们使用了一类已知的OD和一些已知的初等阿贝尔群EA（2^n）上的广义Hadamard矩阵，证明了一种新的方法，首次证明了无偏和准无偏基的渐近存在性[23]。 该计划是继续在该地区的研究，并希望证明一些突出的成果，关于存在一些最佳OD的。所考虑的第一个OD是128阶，16个变量，每个变量重复8次。我们在显示这一点上非常接近，并且很有希望找到一些新的OD，更重要的是，扩展Tarokh等人的结果。 循环群上的广义Hadamard矩阵一直是组合数学许多领域的中心问题，我们的研究也使我们得到了一些有趣的组合结果。应用这些工具的主要方法是寻找合适的表示群，并采用一些创造性的方法将它们与其他对象（例如已知的相互正交拉丁方（=MOLS）类）联合收割机结合起来。 我们成功地证明了32阶不等价的Hadamard矩阵有13，710，027个，这是我们继续研究H-矩阵分类的主要动机。 深入和彻底研究无偏见的基地和协会计划，并成功地在这两个领域取得丰硕成果，需要时间和艰苦的工作。建议的主要方法将包括使用一些对象从设计和图论，如MOLS，正交阵列，有限几何，可分解群可分设计，和正则图。OD的使用，特别是阿达玛矩阵，在最大纠错码的建设已被证明是相当重要的，任何新的和新颖的想法，在寻找剩余的未解决的情况下，将是明显的兴趣，工程师和编码理论家，除了那些工作在设计理论。新的无偏基和准无偏基的构造将引起一些理论物理学家和量子信息理论工作者的兴趣。发现新的关联方案对于分类具有一定类别数的方案具有重要意义。阿达玛矩阵的分类虽然很困难，但也很重要，也是统计学家感兴趣的。

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.