Abstract:
Locally recoverable codes arised to treat the repair problem for large scale distributed and cloud storage systems. This problem consists of recovering the information of a failing node from the others. A locally recoverable code with locality $r$, $C$, is an error-correcting code such that any position in $C$ can be recovered from at most $r$ other positions of $C$. An improvement are $(r,\delta)$-locally recoverable codes, which are designed for simultaneous multiple device failures. They admit a Singleton-like bound, and optimal $(r,\delta)$-locally recoverable codes are those achieving that bound. We give several families of optimal $(r,\delta)$-locally recoverable codes. Most of the constructions of these codes are new, and they are designed for repairing one position by accesing at most $r$ positions but tolerating other $\delta-1$ erasures. These codes belong to a family of error-correcting codes which enlarges that of $J$-affine variety codes which also allow us to obtain $(r,\delta)$-locally recoverable codes. These results were obtained jointly with C. Galindo and F. Hernando.