Quantum codes and locally recoverable codes from evaluation codes

Abstract

This PhD thesis addresses two current problems belonging to the fields of classical and quantum information theory. The first is the repair problem in distributed and cloud storage systems in the setting where multiple nodes may fail simultaneously. These codes are called (r,\delta)-locally recoverable codes. To handle it, we consider monomial-Cartesian codes, resulting in new (r,\delta)-locally recoverable codes. The other problem is the construction of good quantum error-correcting codes. Those we provide are quantum stabilizer codes derived from two families of classical codes: generalized (or twisted) monomial-Cartesian codes and evaluation codes at the roots of trace-depending polynomials. Both quantum code constructions have excellent parameters.