Special orthogonal matrices: existence, enumeration, and applications

我们从事正交设计已有30多年的历史了。为了表明这一领域的重要性，请考虑Tarokh等人1999年的论文《来自正交设计的空时分组编码》。[32]这篇发表在IEEE信息理论会刊上的论文被证明对无线通信很有帮助，这篇论文被引用了4000多篇，是IEEE信息理论会刊上发表的所有论文中被引用次数第四多的论文。 通过在2000年引入一个数组，我们第一次能够构造一些正交设计。17年后的2017年，我们在初等交换群EA(2^n)上使用了一类已知的OD和一些已知的广义Hadamard矩阵，证明了一种新的方法，首次导致了无偏和拟无偏基的渐近存在性[23]。 该计划将继续这一领域的研究，并有望证明一些关于某些最佳OD存在的突出猜想。正在考虑的第一个OD是128阶的，每个变量有16个变量，重复8次。我们非常接近于证明这一点，并对找到一些新的OD寄予厚望，更重要的是，推广了Tarokh等人的结果。[32][老外谈]。 循环群上的广义Hadamard矩阵一直处于许多组合学领域的中心阶段，我们的研究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搜索适当的表示群，以及一些创造性的方法，以将它们与其他对象组合在一起，例如已知的相互正交的拉丁方(=MOLS)类。 我们成功地证明了恰好存在13,710,027个32阶不等价Hadamard矩阵，这是继续研究H-矩阵分类的主要动机。 深入和彻底地研究不偏不倚的基础和联合计划，并在这两个领域取得丰硕成果，需要时间和艰苦工作。所提出的主要方法将包括使用设计和图论中的一些对象，如MOLS、正交表、有限几何、可分解组可分设计和正则图。在构造最大纠错码中使用OD矩阵，特别是Hadamard矩阵，已被证明是相当重要的，并且在寻找剩余未解决案例方面的任何新的和新的想法将是工程师和编码理论家以及那些从事设计理论工作的人明显感兴趣的。一些理论物理学家和从事量子信息理论工作的人会对新的无偏和准无偏基团的构建感兴趣。新关联模式的发现将对具有一定类别数的模式进行分类具有重要意义。Hadamard矩阵的分类虽然非常困难，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.