Spherical codes and designs

Later this month, Hans Parshall will participate in a summer school on “Sphere Packings and Optimal Configurations.” In preparation for this event, Hans was assigned the task of writing lecture notes that summarize the main results of the following paper:

P. Delsarte, J. M. Goethals, J. J. Seidel,

Spherical codes and designs,

Geometriae Dedicata 6 (1977) 363–388.

I found Hans’ notes to be particularly helpful, so I’m posting them here with his permission. I’ve lightly edited his notes for formatting and hyperlinks.

Without further ado:

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Game of Sloanes

Emily King recently launched an online competition to find the best packings of points in complex projective space. The so-called Game of Sloanes is concerned with packing n points in \mathbf{CP}^{d-1} for d\in\{2,\ldots,7\} and for n\in\{d+2,\ldots,49\}. John Jasper, Emily King and I collaborated to make the baseline for this competition by curating various packings from the literature and then numerically optimizing sub-optimal packings. See our paper for more information:

J. Jasper, E. J. King, D. G. Mixon, Game of Sloanes: Best known packings in complex projective space

If you have a packing that improves upon the current leader board, you can submit your packing to the following email address:

asongofvectorsandangles [at] gmail [dot] com

In this competition, you can win money if you find a new packing that achieves equality in the Welch bound; see this paper for a survey of these so-called equiangular tight frames (ETFs).

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Some news regarding the Paley graph

Let \mathbb{F}_p denote the field with p elements, and let Q_p denote the multiplicative subgroup of quadratic residues. For each prime p\equiv 1\bmod 4, we consider the Paley graph G_p with vertex set \mathbb{F}_p, where two vertices are adjacent whenever their difference resides in Q_p. For example, the following illustration from Wikipedia depicts G_{13}:

800px-Paley13.svg

The purpose of this blog entry is to discuss recent observations regarding the Paley graph.

Continue reading Some news regarding the Paley graph

Algebra, geometry, and combinatorics of subspace packings

Last week, the SIAM Conference on Applied Algebraic Geometry hosted a session on “Algebra, geometry, and combinatorics of subspace packings,” organized by Emily King and myself. Sadly, I wasn’t able to attend, but thankfully, most of the speakers gave me permission to post their slides on my blog. Here’s the lineup:

Emily KingAlgebra, Geometry, and Combinatorics of Subspace Packings

Romanos MalikiosisGroup frames, full spark, and other topics

John Jasper Equiangular tight frames from nonabelian groups

Gene KoppSIC-POVM existence and the Stark conjectures

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Recent Advances in Packing

Last week, I co-organized (with Joey Iverson and John Jasper) a special session on “Recent Advances in Packing” for the AMS Spring Central Sectional Meeting at the Ohio State University. All told, our session had 13 talks that covered various aspects of packing, such as sphere packing, packing points in projective space, applications to quantum physics, and connections with combinatorics. It was a great time! And after the talks, we learned how to throw axes!

What follows is the list of speakers and links to their slides. (I anticipate referencing these slides quite a bit in the near future.) Thanks to all who participated!

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Tight Frames and Approximation 2018

I just returned from an amazing workshop in New Zealand organized by Shayne Waldron. The talks and activities were both phenomenal! Here’s a photo by Emily King that accurately conveys the juxtaposition:

28059469_10112541778108014_4963753337247640734_n

A few of the talks gave me a lot to think about, and I wanted to take a moment to record some of these ideas.

Continue reading Tight Frames and Approximation 2018

Optimal line packings from finite group actions

Joey Iverson recently posted our latest paper with John Jasper on the arXiv. This paper can be viewed as a sequel of sorts to our previous paper, in which we introduced the idea of hunting for Gram matrices of equiangular tight frames (ETFs) in the adjacency algebras of association schemes, specifically group schemes. In this new paper, we focus on the so-called Schurian schemes. This proved to be a particularly fruitful restriction: We found an alternate construction of Hoggar’s lines, we found an explicit representation of the “elusive” 7\times 36 packing from the real packings paper (based on a private tip from Henry Cohn), we found an 11\times 66 packing involving the Mathieu group M_{11} (this one beating the corresponding packing in Sloane’s database), we found some low-dimensional mutually unbiased bases, and we recovered nearly all small sized ETFs. In addition, we constructed the first known infinite family of ETFs with Heisenberg symmetry; while these aren’t SIC-POVMs, we suspect they are related to the objects of interest in Zauner’s conjecture (as in this paper, for example). This blog entry briefly describes the main ideas in the paper.

Continue reading Optimal line packings from finite group actions