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We report on a search for CAZAC sequences by using nonlinear sum of squares optimization. Up to equivalence, we found all length 7 CAZAC sequences. We obtained evidence suggesting there are finitely many length 10 CAZAC sequences with a total of 3040 sequences. Last, we compute longer sequences and compare their aperiodic autocorrelation properties to known sequences. The code and results of this search are publicly available through GitHub., Comment: 6 pages, 4 figures, submitted to IEEE Radar Conference 2023

The Lov\'{a}sz theta number is a semidefinite programming bound on the clique number of (the complement of) a given graph. Given a vertex-transitive graph, every vertex belongs to a maximal clique, and so one can instead apply this semidefinite programming bound to the local graph. In the case of the Paley graph, the local graph is circulant, and so this bound reduces to a linear programming bound, allowing for fast computations. Impressively, the value of this program with Schrijver's nonnegativity constraint rivals the state-of-the-art closed-form bound recently proved by Hanson and Petridis. We conjecture that this linear programming bound improves on the Hanson-Petridis bound infinitely often, and we derive the dual program to facilitate proving this conjecture., Comment: Wavelets and Sparsity XVIII

We apply the method of moments to prove a recent conjecture of Haikin, Zamir and Gavish (2017) concerning the distribution of the singular values of random subensembles of Paley equiangular tight frames. Our analysis applies more generally to real equiangular tight frames of redundancy 2, and we suspect similar ideas will eventually produce more general results for arbitrary choices of redundancy.

The line packing problem is concerned with the optimal packing of points in real or complex projective space so that the minimum distance between points is maximized. Until recently, all bounds on optimal line packings were known to be derivable from Delsarte's linear program. Last year, Bukh and Cox introduced a new bound for the line packing problem using completely different techniques. In this paper, we use ideas from the Bukh--Cox proof to find a new proof of the Welch bound, and then we use ideas from Delsarte's linear program to find a new proof of the Bukh--Cox bound. Hopefully, these unifying principles will lead to further refinements., Comment: 4 pages, conference: SampTA 2019

The primary focus is on Constant Amplitude Zero Autocorrelation sequences, or CAZAC sequences for short. We provide both an exposition and some new results relating to CAZAC sequences. We then apply them to Gabor frames and prove a condition for checking when Gabor systems generated by a CAZAC sequence are tight frames. We also construct multiple examples using this condition. Finally, we explore natural generalizations of CAZAC sequences to the torus and the real line and provide examples highlighting the differences between the generalizations. In the last section, we highlight some work done in image processing. In particular, we present results in single-pixel camera imaging and the demosaicing problem in image processing.