Codes with Locality for Two Erasures

In this paper, we study codes with locality that can recover from two erasures via a sequence of two local, parity-check computations. By a local parity-check computation, we mean recovery via a single parity-check equation associated to small Hamming weight. Earlier approaches considered recovery in parallel; the sequential approach allows us to potentially construct codes with improved minimum distance. These codes, which we refer to as locally 2-reconstructible codes, are a natural generalization along one direction, of codes with all-symbol locality introduced by Gopalan \textit{et al}, in which recovery from a single erasure is considered. By studying the Generalized Hamming Weights of the dual code, we derive upper bounds on the minimum distance of locally 2-reconstructible codes and provide constructions for a family of codes based on Tur\'an graphs, that are optimal with respect to this bound. The minimum distance bound derived here is universal in the sense that no code which permits all-symbol local recovery from $2$ erasures can have larger minimum distance regardless of approach adopted. Our approach also leads to a new bound on the minimum distance of codes with all-symbol locality for the single-erasure case.Comment: 14 pages, 3 figures, Updated for improved readabilit

Approximation of Entropy Numbers

The purpose of this article is to develop a technique to estimate certain bounds for entropy numbers of diagonal operator on spaces of p-summable sequences for finite p greater than 1. The approximation method we develop in this direction works for a very general class of operators between Banach spaces, in particular reflexive spaces. As a consequence of this technique we also obtain that the entropy number of a bounded linear operator T between two separable Hilbert spaces is equal to the entropy number of the adjoint of T. This gives a complete answer to the question posed by B. Carl [4] in the setting of separable Hilbert spaces.Comment: 10 page

Synthesis and superconductivity of new BiS2 based superconductor PrO0.5F0.5BiS2

We report synthesis and superconductivity at 3.7K in PrO0.5F0.5BiS2. The newly discovered material belongs to the layered sulfide based REO0.5F0.5BiS2 compounds having ZrCuSiAs type structure. The bulk polycrystalline compound is synthesized by vacuum encapsulation technique at 7800C in single step. Detailed structural analysis has shown that the as synthesized PrO0.5F0.5BiS2 is crystallized in tetragonal P4/nmm space group with lattice parameters a = 4.015(5) {\AA}, c = 13.362(4) {\AA}. Bulk superconductivity is observed in PrO0.5F0.5BiS2 below 4K from magnetic and transport measurements. Electrical transport measurements showed superconducting transition temperature (Tc) onset at 3.7K and Tc ({\rho}=0) at 3.1K. Hump at Tc related to superconducting transition is not observed in heat capacity measurement and rather a Schottky-type anomaly is observed at below ~6K. The compound is slightly semiconducting in normal state. Isothermal magnetization (MH) exhibited typical type II behavior with lower critical field (Hc1) of around 8Oe.Comment: Short note 10 pages text+figs. First report on PrO.5F.5BiS2 Su

Gravity and Large Extra Dimensions

The idea that quantum gravity can be realized at the TeV scale is extremely attractive to theorists and experimentalists alike. This proposal leads to extra spacial dimensions large compared to the electroweak scale. Here we give a very systematic view of the foundations of the theories with large extra dimensions and their physical consequences.Comment: 26 pages, 3 diagram

Superconductivity at 5K in NdO0.5F0.5BiS2

We report appearance of superconductivity at 5K in NdO0.5F0.5BiS2 and supplement the discovery [1] of the same in layered sulfide based ZrCuSiAs type compounds. The bulk polycrystalline compound is synthesized by conventional solid state route via vacuum encapsulation technique. Detailed structural analysis showed that the studied compound is crystallized in tetragonal P4/nmm space group with lattice parameters a = 3.9911(3) {\AA}, c = 13.3830(2) {\AA}. Bulk superconductivity is established in NdO0.5F0.5BiS2 at 5K by both transport and magnetic measurements. Electrical transport measurements showed superconducting Tc onset at 5.2K and Tc ({\rho}=0) at 4.7K. Under applied magnetic field both Tc onset and Tc ({\rho} =0) decrease to lower temperatures and an upper critical field [Hc2(0)] of above 23kOe is estimated. Both AC and DC magnetic susceptibility measurements showed bulk superconductivity below 5K. Isothermal magnetization (MH) exhibited typical type II behavior with lower critical field (Hc1) of around 15Oe. Isothermal magnetization (MH) exhibited typical type-II behavior with lower critical field (Hc1) of around 15Oe. Specific heat [Cp(T)] is investigated in the temperature range of 1.9-50K in zero external magnetic field. A Schottky-type anomaly is observed at low temperature below 7K. This low temperature Schottky can be attributed to the change in the entropy of the system.Comment: 10 pages text + Figs (New Version):comments/suggestion welcome ([email protected]