Grassmannian codes from paired difference sets
Grassmannian codes from paired difference sets
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DOI：
10.1007/s10623-021-00937-w
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发表时间：
2020-10
期刊：
影响因子：
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通讯作者：
M. Fickus;Joseph W. Iverson;J. Jasper;E. King
中科院分区：
文献类型：
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作者：
M. Fickus;Joseph W. Iverson;J. Jasper;E. King
等角紧框架（ETF）是希尔伯特空间中的向量序列，它在Welch界中达到相等，因此具有最小相干性。更一般地，等弦紧融合框架（ECTFF）是希尔伯特空间的等维子空间序列，其在康威、哈丁和斯隆的单纯形界中实现相等。每个ECTFF都是一类最优格拉斯曼码，即希尔伯特空间的等维子空间的最优填充。我们通过利用已知ETF之间的新关系来构建ECTFF。调和ETF等同于有限阿贝尔群的差集。我们说这样一个群的差集与它的庞特里亚金对偶的差集“配对”，当它的调和ETF的相应子序列恰好是它的跨度的ETF时。我们表明，每个这样的对产生一个ECTFF。此外，我们构造了一个无限家庭的配对差集使用二次型在两个元素的领域。这一起产生了两个无限族的真实的ECTFF。
An equiangular tight frame (ETF) is a sequence of vectors in a Hilbert space that achieves equality in the Welch bound and so has minimal coherence. More generally, an equichordal tight fusion frame (ECTFF) is a sequence of equi-dimensional subspaces of a Hilbert space that achieves equality in Conway, Hardin and Sloane’s simplex bound. Every ECTFF is a type of optimal Grassmannian code, that is, an optimal packing of equi-dimensional subspaces of a Hilbert space. We construct ECTFFs by exploiting new relationships between known ETFs. Harmonic ETFs equate to difference sets for finite abelian groups. We say that a difference set for such a group is “paired” with a difference set for its Pontryagin dual when the corresponding subsequence of its harmonic ETF happens to be an ETF for its span. We show that every such pair yields an ECTFF. We moreover construct an infinite family of paired difference sets using quadratic forms over the field of two elements. Together this yields two infinite families of real ECTFFs.