A locally recoverable code is a code over a finite alphabet such that the value of any single coordinate of a codeword can be recovered from the values of a small subset of other coordinates. Building on work of Barg, Tamo, and Vlȃduţ, we present several constructions of locally recoverable codes from algebraic curves and surfaces.

We generalize the construction of locally recoverable codes on algebraic curves given by Barg, Tamo and Vlȃduţ [4] to those with arbitrarily many recovery sets by exploiting the structure of fiber products of curves. Employing maximal curves, we create several new families of locally recoverable codes with multiple recovery sets, including codes with two recovery sets from the generalized Giulietti and Korchmáros (GK) curves and the Suzuki curves, and new locally recoverable codes with many recovery sets based on the Hermitian curve, using a fiber product construction of van der Geer and van der Vlugt. In addition, we consider the relationship between local error recovery and global error correction as well as the availability required to locally recover any pattern of a fixed number of erasures.

No abstract

We determine exactly which graph products, also known as Right Angled Artin Groups, embed into Richard Thompson's group V . It was shown by Bleak and Salazar-Diaz that Z 2 * Z was an obstruction. We show that this is the only obstruction. This is shown by proving a graph theory result giving an alternate description of simple graphs without an appropriate induced subgraph.

No abstract