Resumen de Quantum codes and locally recoverable codes from evaluation codes
This PhD thesis addresses two current problems that belong to the fields of classical and quantum information theory. These are the repair problem in distributed and cloud storage systems and the construction of good quantum error-correcting codes. We mainly use tools coming from algebra and algebraic geometry.
The work is divided in four parts. The first part contains some preliminaries on classical (Chapter 1) and quantum (Chapter 2) error-correcting codes. Suitable classical error-correcting codes, particularly evaluation codes, are the main instrument to give a solution to the problems we address.
The second part is devoted to construct codes dealing the repair problem in the setting where simultaneous multiple device failures may happen. These codes are called (r,\delta)-locally recoverable codes. Chapter 3 shows interesting advances by considering monomial-Cartesian codes as (r,\delta)-locally recoverable codes. These codes come with a natural bound for their minimum distance and we determine those giving rise to (r,\delta)-optimal locally recoverable codes for that minimum distance, which are in fact (r,\delta)-optimal. We prove that a large subfamily of monomial-Cartesian codes admits subfield-subcodes with the same parameters of certain (r,\delta)-optimal monomial-Cartesian codes but over smaller supporting fields. This fact allows us to determine infinitely many new (r,\delta)-optimal locally recoverable codes.
Our constructions of new and good quantum-error correcting codes are given in the third part of this PhD thesis. Chapter 4 shows how to construct new stabilizer quantum error-correcting codes from generalized (or twisted) monomial-Cartesian codes. Our construction uses an explicitly defined twist vector, and we present formulae for the minimum distance and dimension. Generalized monomial-Cartesian codes arise from polynomials in m variables. When m=1 our codes are quantum maximum distance separable, and when m=2 and our lower bound for the minimum distance is 3, the obtained codes are at least Hermitian almost maximum distance separable. Continuing with the case m=2 we prove that, for an infinite family of parameters, our codes beat the quantum Gilbert-Varshamov bound. Our construction gives rise to many codes whose parameters improve those appearing in the literature.
Quantum error-correcting codes with good parameters can also be constructed by evaluating polynomials at the roots of the trace polynomial. In Chapter 5, we propose to evaluate polynomials at the roots of what we call trace-depending polynomials. They are given by a nonzero constant plus the trace of a polynomial. We show that this procedure gives rise to stabilizer quantum error-correcting codes with a new wide range of lengths and with excellent parameters. Namely, we are able to provide new binary records according to Markus Grassl tables and non-binary codes improving the ones available in the literature.
Some ideas on future research can be found in the brief fourth part which finishes this memoir.
