Distribución de pesos de códigos cíclicos a partir de sumas exponenciales y curvas algebraicas

Tesis (Lic. en Matemática)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, 2020.Este trabajo trata sobre el espectro o distribución de pesos de códigos lineales y cíclicos. Esto es en general una tarea ardua y sólo se conoce el espectro de algunas familias de códigos. Estudiaremos distintas formas de encontrar dichas distribuciones de pesos a través de diferentes caminos. Primero veremos resultados generales para códigos lineales, que en particular dan una respuesta general al caso de los códigos MDS. Luego, nos enfocaremos en códigos cíclicos generales viéndolos como códigos traza (combinando los teoremas de Delsarte y las identidades de MacWilliams). A partir de aquí haremos uso de dos estrategias generales, una que involucra ciertas sumas exponenciales (Gauss, Weil y/o Kloosterman) y otra basada en el conteo de puntos racionales de curvas algebraicas asociadas a los códigos (típicamente de Artin-Schreier). Usaremos estas técnicas para obtener los espectros de familias de códigos muy conocidas como Hamming, BCH y Reed-Muller. Finalmente, aplicaremos estos métodos a dos familias de códigos menos conocidos como los códigos de Melas y de Zetterberg. En los casos binario y ternario, el cálculo de dichos espectros se puede realizar usando curvas elípticas y la traza de operadores de Hecke de ciertas formas modulares asociadas a ellas. El trabajo contiene numerosos ejemplos, muchos de ellos nuevos.This work deals with the spectrum or weight distribution of linear and cyclic codes. This is in general a difficult task and the spectrum is only known for some families of codes. We will study different ways to find these distributions through different ways. We will first see general results for linear codes, which in particular give a general answer to the case of MDS codes. Then, we will focus on general cyclic codes by viewing them as trace codes (combining Delsarte's theorems and MacWilliams identities). From this point on we will use two general strategies, one that involves certain exponential sums (Gauss, Weil or Kloosterman) and another one based on counting the number of rational points of algebraic curves (typically Artin-Schreier) associated with the codes. We will use these techniques to obtain the spectra of well-known families of codes such as Hamming, BCH, and Reed-Muller codes. Finally, we will apply these methods to two lesser known code families, the Melas codes and the Zetterberg codes. In the binary and ternary cases, the computation of the mentioned spectra can be performed by using elliptic curves and the trace of Hecke operators of certain modular forms associated to them. The work contains several examples, many of them new.Fil: Chiapparoli, Paula Mercedes. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina

Quantum stabilizer codes and beyond

The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. This dissertation makes a threefold contribution to the mathematical theory of quantum error-correcting codes. Firstly, it extends the framework of an important class of quantum codes -- nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes. These codes are expected to have efficient error recovery schemes than stabilizer codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work establishes a close link between subsystem codes and classical codes showing that the subsystem codes can be constructed from arbitrary classical codes. Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum codes and considers more realistic channels than the commonly studied depolarizing channel. It gives systematic constructions of asymmetric quantum stabilizer codes that exploit the asymmetry of errors in certain quantum channels.Comment: Ph.D. Dissertation, Texas A&M University, 200