Constructive spherical codes on layers of flat tori

A new class of spherical codes is constructed by selecting a finite subset of flat tori from a foliation of the unit sphere S^{2L-1} of R^{2L} and designing a structured codebook on each torus layer. The resulting spherical code can be the image of a lattice restricted to a specific hyperbox in R^L in each layer. Group structure and homogeneity, useful for efficient storage and decoding, are inherited from the underlying lattice codebook. A systematic method for constructing such codes are presented and, as an example, the Leech lattice is used to construct a spherical code in R^{48}. Upper and lower bounds on the performance, the asymptotic packing density and a method for decoding are derived.Comment: 9 pages, 5 figures, submitted to IEEE Transactions on Information Theor

Fast Scrambling of mutual information in Kerr-AdS5_{\textbf{5}}

We compute the disruption of mutual information in a TFD state dual to a Kerr black hole with equal angular momenta in $AdS_5$ due to an equatorial shockwave. The shockwave respects the axi-symmetry of the Kerr geometry with specific angular momenta $\mathcal{L}_{\phi_1}$ & $\mathcal{L}_{\phi_2}$. The sub-systems considered are hemispheres in the $left$ and the $right$ dual CFTs with the equator of the $S^3$ as their boundary. We compute the change in the mutual information by determining the growth of the HRT surface at late times. We find that at late times leading upto the scrambling time the minimum value of the instantaneous Lyapunov index $\lambda_L^{(min)}$ is bounded by $\kappa=\frac{2\pi T_H}{(1-\mu \,\mathcal{L}_+)}$ and is found to be greater than $2\pi T_H$ in certain regimes with $T_H$ and $\mu$ denoting the black hole's temperature and the horizon angular velocity respectively while $\mathcal{L}_+=\mathcal{L}_{\phi_1}+\mathcal{L}_{\phi_2}$. We also find that for non-extremal geometries the null perturbation obeys $\mathcal{L}_+<\mu^{-1}$ for it to reach the outer horizon from the $AdS$ boundary. The scrambling time at very late times is given by $\kappa\tau_*\approx\log \mathcal{S}$ where $\mathcal{S}$ is the Kerr entropy. We also find that the onset of scrambling is delayed due to a term proportional to $\log(1-\mu\,\mathcal{L}_+)^{-1}$ which is not extensive and does not scale with the entropy of Kerr black hole.Comment: 18-pages, 4-figure