{"title":"Naimark-spatial families of equichordal tight fusion frames","authors":"Matthew Fickus, Benjamin R. Mayo, Cody E. Watson","doi":"10.1016/j.acha.2024.101720","DOIUrl":null,"url":null,"abstract":"<div><div>An equichordal tight fusion frame (<figure><img></figure>) is a finite sequence of equi-dimensional subspaces of a Euclidean space that achieves equality in Conway, Hardin and Sloane's simplex bound. Every <figure><img></figure> is a type of optimal Grassmannian code, being a way to arrange a given number of members of a Grassmannian so that the minimal chordal distance between any pair of them is as large as possible. Any nontrivial <figure><img></figure> has both a Naimark complement and spatial complement which themselves are <figure><img></figure>s. We show that taking iterated alternating Naimark and spatial complements of any <figure><img></figure> of at least five subspaces yields an infinite family of <figure><img></figure>s with pairwise distinct parameters. Generalizing a method by King, we then construct <figure><img></figure>s from difference families for finite abelian groups, and use our Naimark-spatial theory to gauge their novelty.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"74 ","pages":"Article 101720"},"PeriodicalIF":2.6000,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied and Computational Harmonic Analysis","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1063520324000976","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
An equichordal tight fusion frame () is a finite sequence of equi-dimensional subspaces of a Euclidean space that achieves equality in Conway, Hardin and Sloane's simplex bound. Every is a type of optimal Grassmannian code, being a way to arrange a given number of members of a Grassmannian so that the minimal chordal distance between any pair of them is as large as possible. Any nontrivial has both a Naimark complement and spatial complement which themselves are s. We show that taking iterated alternating Naimark and spatial complements of any of at least five subspaces yields an infinite family of s with pairwise distinct parameters. Generalizing a method by King, we then construct s from difference families for finite abelian groups, and use our Naimark-spatial theory to gauge their novelty.
等弦密融合框（）是欧几里得空间的等维子空间的有限序列，它在康威、哈丁和斯隆的单数约束中达到相等。每一个都是一种最优格拉斯曼编码，是一种排列给定数量的格拉斯曼成员，使任意一对成员之间的最小弦距尽可能大的方法。我们的研究表明，对至少五个子空间中的任意子空间进行迭代交替的奈马克补集和空间补集，就能得到一个具有成对不同参数的无穷 s 族。然后，我们推广了金的一种方法，从有限无性群的差分族中构造出 s，并利用我们的奈马克空间理论来衡量它们的新颖性。
期刊介绍：
Applied and Computational Harmonic Analysis (ACHA) is an interdisciplinary journal that publishes high-quality papers in all areas of mathematical sciences related to the applied and computational aspects of harmonic analysis, with special emphasis on innovative theoretical development, methods, and algorithms, for information processing, manipulation, understanding, and so forth. The objectives of the journal are to chronicle the important publications in the rapidly growing field of data representation and analysis, to stimulate research in relevant interdisciplinary areas, and to provide a common link among mathematical, physical, and life scientists, as well as engineers.
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