Réseau de Leech

Le réseau de Leech est un réseau remarquable dans l'espace euclidien de dimension 24. Il est relié au code de Golay. Historique Ernst Witt le découvre en 1940 mais ne publie pas cette découverte qui sera finalement attribuée à John Leech en 1965. Caractéristiques Le réseau de Leech est caractérisé comme étant le seul pair en dimension 24 qui ne contient pas de racines, c'est-à-dire de vecteur v tel que (v,v)=2. Il a été construit par John Leech. Le groupe des automorphismes du réseau de Leech est le groupe de Conway Co0. Il y a exactement 24 . Ils ont été classifiés par en 1973. Un gros trou est un point dans ℝ24 à distance maximale de tous les points du réseau de Leech. Conway, Parker et Sloane ont montré qu'il y a exactement 23 types de gros trous, avec une correspondance précise entre les gros trous et les réseaux de Niemeier. Une des méthodes pour construire ce réseau consiste à faire intervenir un code (24,12,8) particulier (c'est-à-dire un codage p

On the number of lattice points in a small sphere and a recursive lattice decoding algorithm

Annika Meyer

Let L be a lattice in . This paper provides two methods to obtain upper bounds on the number of points of L contained in a small sphere centered anywhere in . The first method is based on the observation that if the sphere is sufficiently small then the lattice points contained in the sphere give rise to a spherical code with a certain minimum angle. The second method involves Gaussian measures on L in the sense of Banaszczyk (Math Ann 296:625-635, 1993). Examples where the obtained bounds are optimal include some root lattices in small dimensions and the Leech lattice. We also present a natural decoding algorithm for lattices constructed from lattices of smaller dimension, and apply our results on the number of lattice points in a small sphere to conclude on the performance of this algorithm.

Springer US2013

Pin-by-Pin Treatment of LWR Cores with Cross Section Equivalence Procedures and Higher-Order Transport Methods

Petra Mala

This thesis presents a systematic study on the merits and limitations on using pin-by-pin resolution and transport theory based approaches for nuclear core design calculations. Starting from the lattice codes and an optimal cross section generation scheme, it compares different methods, transport approximations, and spatial discretizations used in pin-by-pin homogenized codes. It is in the interest of nuclear power plant operators to employ more heterogeneous core loadings in order to improve the fuel utilization and decrease the amount of spent fuel. This necessarily increases the requirements on the accuracy of the computation tools used for the core design and safety analysis. One possibility is employing 3D core solvers with higher spatial resolution, e.g. pin-cell wise. The comparison of several lattice codes indicates that already the proper generation of diffusion coefficients and higher-order scattering moments for pin-cell geometry is not straightforward. Out of three available lattice codes, two generated unphysical diffusion coefficient when using the inscatter approximation, while the last code was not able to provide the higher scattering moments. Several few-assembly test cases with either high neutron leakage effects or MOX/uranium interfaces showed that the quality of the results of diffusion and SP3 solvers depends critically on the choice of the diffusion coefficient: in the outscatter approximation, it can cause major deviations, while the inscatter seems overall more adequate. Regarding the transport solvers, results obtained with the SN solver DORT showed very good performance for MOX/uranium interfaces, but major deviations occurred for problems with large power gradients. On the other hand, the MOC solver nTRACER showed in all cases small average error, but 2 - 3 % error around the interface of different assemblies. A study on spatial discretization indicated that the finite difference method applied on pin-cells does not properly capture the big flux changes between MOX and uranium fuel, while the nodal expansion method is more accurate but too slow. It was suggested to use the finite difference method with finer mesh in the outer assembly pin-cells, which increases the required computation time by only 50 % and decreases the pin power errors below 1 % with respect to lattice code results. Due to some problems which were observed with the available diffusion/SP3 solvers, a new SP3 solver was implemented in the DORT-TD platform. Several core tests showed that the SP3 pin-by-pin solver can significantly outperform the state-of-the-art nodal solver SIMULATE-5, in particular for reactor cores with inserted control rods. Finally, the pin-by-pin solvers were coupled to a depletion solver. For that, an accurate and fast interpolation routine had to be implemented. The obtained results of full-cycle depletion with complex core loading showed very good performance in comparison to a heterogeneous transport-based fine-group calculations.

EPFL2018

Pin-by-Pin Treatment of LWR Cores with Cross Section Equivalence Procedures and Higher-Order Transport Methods

Petra Mala

EPFL2018

On the number of lattice points in a small sphere and a recursive lattice decoding algorithm

Universal optimality of the E8 and Leech lattices and interpolation formulas

Pin-by-Pin Treatment of LWR Cores with Cross Section Equivalence Procedures and Higher-Order Transport Methods

Universal optimality of the E8 and Leech lattices and interpolation formulas

Pin-by-Pin Treatment of LWR Cores with Cross Section Equivalence Procedures and Higher-Order Transport Methods

On the number of lattice points in a small sphere and a recursive lattice decoding algorithm