Grassmannian Codes from Paired Difference Sets
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.
Designs, Codes and Cryptography
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]