A Geometrical Realisation of Quasi-Cyclic Codes

We study and enumerate cyclic codes which include generalised Reed-Solomon codes as function field codes. This geometrical approach allows to construct longer codes and to get more information on the parameters defining the codes. We provide a closed formula in terms of Stirling numbers for the number of irreducible polynomials and we relate it with other formulas existing in the literature. Further, we study quasi-cyclic codes as orbit codes in the Grassmannian parameterizing constant dimension codes. In addition, we review Horn’s algorithm and apply it to construct classical codes by their defining ideals

Applications of Derandomization Theory in Coding

Randomized techniques play a fundamental role in theoretical computer science and discrete mathematics, in particular for the design of efficient algorithms and construction of combinatorial objects. The basic goal in derandomization theory is to eliminate or reduce the need for randomness in such randomized constructions. In this thesis, we explore some applications of the fundamental notions in derandomization theory to problems outside the core of theoretical computer science, and in particular, certain problems related to coding theory. First, we consider the wiretap channel problem which involves a communication system in which an intruder can eavesdrop a limited portion of the transmissions, and construct efficient and information-theoretically optimal communication protocols for this model. Then we consider the combinatorial group testing problem. In this classical problem, one aims to determine a set of defective items within a large population by asking a number of queries, where each query reveals whether a defective item is present within a specified group of items. We use randomness condensers to explicitly construct optimal, or nearly optimal, group testing schemes for a setting where the query outcomes can be highly unreliable, as well as the threshold model where a query returns positive if the number of defectives pass a certain threshold. Finally, we design ensembles of error-correcting codes that achieve the information-theoretic capacity of a large class of communication channels, and then use the obtained ensembles for construction of explicit capacity achieving codes. [This is a shortened version of the actual abstract in the thesis.]Comment: EPFL Phd Thesi

Resource management techniques for sustainable networks with energy harvesting nodes

Premi extraordinari doctorat UPC curs 2015-2016, àmbit Enginyeria de les TICThis dissertation proposes novel techniques for assigning resources of wireless networks by considering that the coverage radii are small, implying that some power consumption sinks not considered so far shouldnow be introduced, and by considering that the devices are battery-powered terminals provided with energy harvesting capabilities. In this framework, two different configurations in terms of harvesting capabilities are considered. First, we assume that the energy source is external and not controllable, e.g. solar energy. In this context, the proposed design should adapt to the energy that is currently being harvested. We also study the effect of having a finite backhaul connection that links the wireless access network with the core network. On the other hand, we propose a design in which the transmitter feeds actively the receivers with energy by transmitting signals that receivers use for recharging their batteries. In this case, the power transfer design should be carried out jointly with the power control strategy for users that receive information as both procedures, transfer of information and transfer of power, are implemented at the transmitter and make use of a common resource, i.e., power. Apart from techniques for assigning the radio resources, this dissertation develops a procedure for switching on and off base stations. Concerning this, it is important to notice that the traffic profile is not constant throughout the day. This is precisely the feature that can be exploited to define a strategy based on a dynamic selection of the base stations to be switched off when the traffic load is low, without affecting the quality experienced by the users. Thanks to this procedure, we are able to deploy smaller energy harvesting sources and smaller batteries and, thus, to reduce the cost of the network deployment. Finally, we derive some procedures to optimize high level decisions of the network operation in which variables from several layers of the protocol stack are involved. In this context, admission control procedures for deciding which user should be connected to which base station are studied, taking into account information of the average channel information, the current battery levels, etc. A multi-tier multi-cell scenario is assumed in which base stations belonging to different tiers have different capabilities, e.g., transmission power, battery size, end energy harvesting source size. A set of strategies that require different computational complexity are derived for scenarios with different user mobility requirements.Aquesta tesis doctoral proposa tècniques per assignar els recursos disponibles a les xarxes wireless considerant que els radis de cobertura són petits, el que implica que altres fonts de consum d’energia no considerades fins al moment s’hagin d’introduir dins els dissenys, i considerant que els dispositius estan alimentats amb bateries finites i que tenen a la seva disposició fonts de energy harvesting. En aquest context, es consideren dues configuracions diferents en funció de les capacitats de l’energia harvesting. En primer lloc, s’assumirà que la font d’energia és externa i incontrolable com, per exemple, l’energia solar. Els dissenys proposats han d’adaptar-se a l’energia que s’està recol·lectant en un precís moment. En segon lloc, es proposa un disseny en el qual el transmissor és capaç d’enviar energia als receptors mitjançant senyals de radiofreqüència dissenyats per aquest fi, energia que és utilitzada per recarregar les bateries. A part de tècniques d’assignació de recursos radio, en aquesta tesis doctoral es desenvolupa un procediment dinàmic per apagar i encendre estacions base. És important notar que el perfil de tràfic no és constant al llarg del dia. Aquest és precisament el patró que es pot explotar per definir una estratègia dinàmica per poder decidir quines estaciones base han de ser apagades, tot això sense afectar la qualitat experimentada pels usuaris. Gràcies a aquest procediment, es possible desplegar fonts d'energy harvesting més petites i bateries més petites. Finalment, aquesta tesis doctoral presenta procediments per optimitzar decisions de nivell més alt que afecten directament al funcionament global de la xarxa d’accés. Per prendre aquestes decisions, es fa ús de diverses variables que pertanyen a diferents capes de la pila de protocols. En aquest context, aquesta tesis aborda el disseny de tècniques de control d’admissió d’usuaris a estacions base en entorns amb múltiples estacions base, basant-se amb la informació estadística dels canals, i el nivell actual de les bateries, entre altres. L'escenari considerat està format per múltiples estacions base, on cada estació base pertany a una família amb diferents capacitats, per exemple, potència de transmissió o mida de la bateria. Es deriven un conjunt de tècniques amb diferents costos computacionals que són d'utilitat per a poder aplicar a escenaris amb diferents mobilitats d’usuaris.Award-winningPostprint (published version