Special orthogonal matrices: existence, enumeration, and applications

特殊正交矩阵：存在性、枚举和应用

• 批准号: RGPIN-2019-05389
• 负责人: Kharaghani, Hadi
• 金额: $ 1.53万
• 依托单位: University of Lethbridge
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=678509
• 关键词: Special; orthogonal; matrices; existence; enumeration

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.

我们已经在正交设计（=OD's）上工作了30多年。为了表明这一领域的重要性，考虑到1999年由Tarokh等人发表在IEEE信息论学报上的论文“来自正交设计的时空块编码”已被证明对无线通信有帮助，该论文被引用超过4000次，是IEEE信息论学报上发表的所有论文中被引用次数第四多的论文。***通过在2000年引入一个阵列，我们第一次能够构建一些正交设计。17年后，在2017年，我们在初等阿贝尔群EA（2^n）上使用了一类已知的OD和一些已知的广义Hadamard矩阵，首次证明了一种导致无偏和拟无偏基渐近存在的新方法[23]。***我们的计划是继续在该地区进行研究，并希望证明一些关于存在一些最佳外径的杰出猜想。考虑的第一个OD为128阶，包含16个变量，每个变量重复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，扩展Tarokh等人的结果。***循环群上的广义Hadamard矩阵一直处于组合学许多领域的中心阶段，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寻找合适的表示组，并结合一些创造性的方法将它们与其他对象（如已知的相互正交拉丁方类）结合起来。***我们成功地证明了32阶的不等价Hadamard矩阵有13,710,027个，这是继续研究h矩阵分类的主要动机。***对公正的基础和联合计划进行深入和彻底的研究并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设计和图论中的一些对象，如MOLS，正交阵列，有限几何，可分解群可分设计和正则图。在构造最大纠错码时使用OD，特别是Hadamard矩阵，已经被证明是非常重要的，除了设计理论的工作人员外，任何寻找剩余未解决情况的新的和新颖的想法都将引起工程师和编码理论家的明显兴趣。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的构建感兴趣。新的关联方案的发现对于具有一定数量类的方案分类具有重要意义。Hadamard矩阵的分类虽然非常困难，但对统计学家来说也很重要和感兴趣。