 |
Center for Nonlinear Studies |
How can one arrange $d+k$ many vectors in $\mathbb R^d$ so that they are as close to orthogonal as possible? Such arrangements are known as projective codes (or antipodal spherical codes) and are a natural generalization of balanced error-correcting codes. In this talk, we will consider the case when $k$ is constant and $d\to\infty$, i.e.\ when there is a "small excess" of vectors. In this realm, we show that there is an intimate connection to the existence of systems of ${k+1\choose 2}$ equiangular lines in $\mathbb R^k$ and use this to obtain tight bounds for $k\in\{1,2,3,7,23\}$ and outperform the Welchbound otherwise. To expose this relationship, we show how to "dualize" the problem and instead discuss bounding the first moment of isotropicprobability masses (a.k.a.\ probabilistic tight frames) on $\mathbb R^k$, which may be of independent interest. While we will focus mainly on the real case in this talk, all of theseresults translate naturally to the complex case (and even to the quaternions), wherein the answer relates to Zauner's conjecture on the existence of systems of $k^2$ equiangular lines in $\mathbb C^k$, also known as SIC-POVM in physics literature.
Host: Nathan Lemons (T-5)