Robust locally testable codes and products of codes

We continue the investigation of locally testable codes, i.e., error‐correcting codes for which membership of a given word in the code can be tested probabilistically by examining it in very few locations. We give two general results on local testability: First, motivated by the recently proposed notion of robust probabilistically checkable proofs, we introduce the notion of robust local testability of codes. We relate this notion to a product of codes introduced by Tanner and show a very simple composition lemma for this notion. Next, we show that codes built by tensor products can be tested robustly and somewhat locally by applying a variant of a test and proof technique introduced by Raz and Safra in the context of testing low‐degree multivariate polynomials (which are a special case of tensor codes). Combining these two results gives us a generic construction of codes of inverse polynomial rate that are testable with poly‐logarithmically many queries. We note that these locally testable tensor codes can be obtained from any linear error correcting code with good distance. Previous results on local testability, albeit much stronger quantitatively, rely heavily on algebraic properties of the underlying codes. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Catalytic and Asymmetric Fluorolactonisations of Carboxylic Acids through Anion Phase Transfer

Catalytic fluorolactonisations of aromatic carboxylic acids have been developed. The reactions proceed under mild conditions using the commercially available reagent Selectfluor. A weak phase transfer of the reagent mediated by Na₂CO₃ allows the reaction to be conducted in non‐polar solvents. Furthermore, by the use of a catalytic amount of (DHQ)₂PHAL (hydroquinine 1,4‐phthalazinediyl diether), the first asymmetric fluorolactonisation has been achieved. The corresponding isobenzofuran core can be found in many biologically active molecules.     

Generalized spheroidal spacetimes in 5-D Einstein–Maxwell–Gauss–Bonnet gravity

The field equations for static EGBM gravity are obtained and transformed to an equivalent form through a coordinate redefinition. A form for one of the metric potentials that generalizes the spheroidal ansatz of Vaidya–Tikekar superdense stars and additionally prescribing the electric field intensity yields viable solutions. Some special cases of the general solution are considered and analogous classes in the Einstein framework are studied. In particular the Finch–Skea ansatz is examined in detail and found to satisfy the elementary physical requirements. These include positivity of pressure and density, the existence of a pressure free hypersurface marking the boundary, continuity with the exterior metric, a subluminal sound speed as well as the energy conditions. Moreover, the solution possesses no coordinate singularities. It is found that the impact of the Gauss–Bonnet term is to correct undesirable features in the pressure profile and sound speed index when compared to the equivalent Einstein gravity model. Furthermore graphical analyses suggest that higher densities are achievable for the same radial values when compared to the 5-dimensional Einstein case. The case of a constant gravitational potential, isothermal distribution as well as an incompressible fluid are studied. All exact solutions derived exhibit an equation of state explicitly.