Grassmannian Codes from Paired Difference Sets
Document Type
Article
Publication Date
9-7-2021
Abstract
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.
DOI
Version of record:10.1007/s10623-021-00937-w
Source Publication
Designs, Codes and Cryptography (ISSN 0925-1022 | e-ISSN 1573-7586)
Recommended Citation
Fickus, M., Iverson, J.W., Jasper, J. et al. Grassmannian codes from paired difference sets. Des. Codes Cryptogr. 89, 2553–2576 (2021). https://doi.org/10.1007/s10623-021-00937-w arXiv:2010.06639 [math.FA], https://doi.org/10.48550/arXiv.2010.06639
Comments
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The published version of record for this article appears in volume 89 of the journal Designs, Codes & Cryptography, published by Springer Nature. The published version is available to subscribers of that journal via the DOI link below.