ED Mathématiques et Informatique
Noncommutative cyclic algebras in coding theory with applications to duality and decoding
by Fabrice DRAIN (IMB - Institut de Mathématiques de Bordeaux)
The defense will take place at 10h00 - Salle de conférences Institut de mathématiques de Bordeaux, 351 cours de la libération F-33405 Talence
in front of the jury composed of
- Xavier CARUSO - Directeur de recherche - Université de Bordeaux - Directeur de these
- Elena BERARDINI - Chaire de professeur junior - Université de Bordeaux - CoDirecteur de these
- Delphine BOUCHER - Maîtresse de conférences - Université de Rennes - Examinateur
- Yves AUBRY - Maître de conférences - Université de Toulon - Examinateur
- Jean-Marc COUVEIGNES - Professeur des universités - Université de Bordeaux - Examinateur
- Alin BOSTAN - Directeur de recherche - INRIA SACLAY - Examinateur
- Emmanuel HALLOUIN - Maître de conférences - Université de Toulouse 3 Paul-Sabatier - Rapporteur
- Christian MAIRE - Professeur des universités - Université de Besançon - Rapporteur
In 2023, E. Berardini and X. Caruso introduced sum-rank metric codes called linearized algebraic geometry (LAG), obtained by evaluating Riemann--Roch spaces on division algebras over function fields. These codes generalize both Algebraic Geometry codes and Linearized Reed Solomon codes and combine their advantages. They are almost maximum sum-rank distance with a defect relative to the Singleton bound equal to the genus of the order determined by a gauge of the underlying division algebra and their asymptotic length is not limited by the size of the alphabet. Our main objective in this thesis is to design and prove a polynomial decoding algorithm for these LAG codes. More generally, we are interested in noncommutative cyclic algebras in coding theory. Our approach is based on skew polynomials. In our first contribution, we count and enumerate self-dual skew cyclic codes over a finite field using a cubic algorithm for which we provide a SageMath implementation. In our second contribution, we propose a polynomial decoding algorithm for LAG codes under the hypothesis of non-ramification at the evaluation places. To design it, we establish a Riemann--Roch theorem on our division algebra and show that the duals of LAG codes are themselves LAG codes on the adjoint division algebra. Moreover, we provide a SageMath implementation of this decoding algorithm. We then show how our Riemann--Roch Theorem can be deduced from a more general theorem by Tamagawa by means of computing the discriminant of our gauge order. Then, we apply this same Tamagawa theory to construct more general LAG codes defined on abelian algebras (ALAG), ie quotients of multivariate Ore polynomial rings with coefficients in an abelian extension of function fields. Finally, we provide a SageMath implementation of an ALAG decoding algorithm whose formal proof is yet to be completed.
ED Sciences Physiques et de l'Ingénieur
Experimental characterization of particle sources generated from laser-plasma interactions in the picosecond regime on the petawatt PETAL facility
by Lucas RIBOTTE (Centre Lasers Intenses et Applications)
The defense will take place at 10h00 - Amphithéâtre 3 351 cours de la Libération, Bâtiment A9, 33400 Talence
in front of the jury composed of
- Emmanuel D'HUMIERES - Professeur des universités - Université de Bordeaux - Directeur de these
- Sandrine DOBOSZ DUFRENOY - Directrice de recherche - CEA Saclay - Rapporteur
- Vincent BAGNOUD - Professeur - GSI Helmholtzzentrum für Schwerionenforschung GmbH - Rapporteur
- Florin NEGOITA - Research Scientist II - ELI-NP - Examinateur
- Félicie ALBERT - Research Scientist - Lawrence Livermore National Laboratory - Examinateur
- Julien FUCHS - Professeur - Technion Israel Institute of Technology, Faculty of Physics - Examinateur
- Xavier DAVOINE - Ingénieur de recherche - CEA DAM DIF - Examinateur
Laser-plasma interactions are an innovative way to create exotic particle beams that have both applied and fundamental prospects. Picosecond time duration and kilojoules energy laser pulses are formidable tools to induce the acceleration of compact, bright and energetic particle beams in conditions that only a few installations in the world can reach. This work focused on the qualification of the particle source terms from several laser-plasma interaction experiments conducted with the ps-kJ class laser PETAL. The experimental results demonstrate the production of beams of electrons, protons, neutrons and photons in four experimental campaigns at the LMJ-PETAL facility of the CEA-DAM. This thesis combines a throurough experimental work on the facility instrumentation, with Monte-Carlo (Geant4) and particle-in-cell (CALDER) simulations aiming at numerically reproducing the particle sources measured in each experiment. The first part of this work is about the diagnostics employed during the experimental campaigns: sensitive screens (imaging plates and radiochromic films), magnetic spectrometers, sensitive screens stacks (proton radiography and bremsstrahlung cannons), neutron time-of-flight ultra-fast organic scintillator detectors, nuclear activation spectrometry. Special attention is dedicated to the extraction methods for each diagnostic to get the best possible measurement from raw results. The instrumentation being the same throughout all the experimental campaigns, this first part gives the tools to get the results described in the following sections. The second part of the thesis addresses experiments with a micrometric solid target to accelerate protons through the TNSA (Target Normal Sheath Acceleration) process. These proton beams are then used to generate an intense neutron source with a millimetric LiF and Pb converter. The third and last part of this work focuses on laser shots performed on supersonic Helium gas jet targets at pressures around 60 bars. The laser-plasma acceleration process at hand here is self-modulated laser wakefield acceleration (SM-LWFA), able to generate relativistic electron beams with record-breaking charges of the order of the µC. The properties of these beams are studied in depth in a dedicated chapter. The next and final chapter then deals with the generation of secondary sources of particles from SM-LWFA electrons. Neutrons sources were experimentally produced, while electron-positron pair and photon production are numerically studied.