The Automorphism Group of an Extremal [72,36,16] Code does not contain elements of order 6

The existence of an extremal code of length 72 is a long-standing open problem. Let C be a putative extremal code of length 72 and suppose that C has an automorphism g of order 6. We show that C, as an F_2-module, is the direct sum of two modules, one easily determinable and the other one which has a very restrictive structure. We use this fact to do an exhaustive search and we do not find any code. This proves that the automorphism group of an extremal code of length 72 does not contain elements of order 6.Comment: 15 pages, 0 figures. A revised version of the paper is published on IEEE Transactions on Information Theor

Symmetries of weight enumerators and applications to Reed-Muller codes

Gleason's 1970 theorem on weight enumerators of self-dual codes has played a crucial role for research in coding theory during the last four decades. Plenty of generalizations have been proved but, to our knowledge, they are all based on the symmetries given by MacWilliams' identities. This paper is intended to be a first step towards a more general investigation of symmetries of weight enumerators. We list the possible groups of symmetries, dealing both with the finite and infinite case, we develop a new algorithm to compute the group of symmetries of a given weight enumerator and apply these methods to the family of Reed-Muller codes, giving, in the binary case, an analogue of Gleason's theorem for all parameters.Comment: 14 pages. Improved and extended version of arXiv:1511.00803. To appear in Advances in Mathematics of Communication

Inhomogeneous minima of mixed signature lattices

We establish an explicit upper bound for the Euclidean minimum of a number field which depends, in a precise manner, only on its discriminant and the number of real and complex embeddings. Such bounds were shown to exist by Davenport and Swinnerton-Dyer. In the case of totally real fields, an optimal bound was conjectured by Minkowski and it is proved for fields of small degree. In this note we develop methods of McMullen in the case of mixed signature in order to get explicit bounds for the Euclidean minimum.Comment: To appear in the Journal of Number Theor

Hernández, Alberdi y Nicasio Oroño

Fil: Borello, Rodolfo. Universidad Nacional de Cuyo. Facultad de Filosofía y Letra

Experimental measurement technique for the assessment of the fuel crossover diffusion coefficient in the membrane electrode assembly of a direct methanol fuel cell

Since the cross-over still seems to be the main issue of the direct methanol fuel cells, an experimental evaluation of the diffusive cross-over is performed. Even if the relationship of the rate through the membrane is the sum of the three terms of diffusive, osmotic and drag, the diffusive component is also present at open circuit lowering the Open Circuit Voltage of the single cell up to 50 % with respect to the Nernst potential. The goal of the research is to develop a direct measurement technique of the crossover that can provide the effective values of the parameters that characterize the membrane electrode assembly. The experimental set up consists in the pressure, flow and temperature control and acquisition using Labview. A sensitive analysis for three values of temperatures at 60°C, 65°C and 70°C is performed for first. Then, a small overpressure was generated in the cathode side by a valve located at the cathode outlet. A set of pressure were analysed for 0, 30 and 90 mbar of overpressure at the cathode. The tested fuel cell has a commercial Nafion 117 membrane and carbon paper gas diffusion layers 700 cm2 large. Preliminary results show that the differential concentration term seems to be significantly larger than the osmotic term. The diffusion coefficients are useful for fuel cell modelling and for the calibration of the operating conditions in the sensor less DMFC systems