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Les réseaux B W 32 et U 32 sont équivalents

Pierre LoyerPatrick Solé — 1994

Journal de théorie des nombres de Bordeaux

On montre que le réseau de Barnes-Wall de rang 32 est équivalent au réseau à double congruence U 32 de Martinet. La preuve utilise la notion de voisinage de Kneser et des résultats de Koch et Venkov sur le défaut du voisinage (“Nachbardefekt”).

Universal codes and unimodular lattices

Robin ChapmanPatrick Solé — 1996

Journal de théorie des nombres de Bordeaux

Binary quadratic residue codes of length p + 1 produce via construction B and density doubling type II lattices like the Leech. Recently, quaternary quadratic residue codes have been shown to produce the same lattices by construction A modulo 4 . We prove in a direct way the equivalence of these two constructions for p 31 . In dimension 32, we obtain an extremal lattice of type II not isometric to the Barnes-Wall lattice B W 32 . The equivalence between construction B modulo 4 plus density doubling and construction...

2 -modular lattices from ternary codes

Robin ChapmanSteven T. DoughertyPhilippe GaboritPatrick Solé — 2002

Journal de théorie des nombres de Bordeaux

The alphabet 𝐅 3 + v 𝐅 3 where v 2 = 1 is viewed here as a quotient of the ring of integers of 𝐐 ( - 2 ) by the ideal (3). Self-dual 𝐅 3 + v 𝐅 3 codes for the hermitian scalar product give 2 -modular lattices by construction A K . There is a Gray map which maps self-dual codes for the Euclidean scalar product into Type III codes with a fixed point free involution in their automorphism group. Gleason type theorems for the symmetrized weight enumerators of Euclidean self-dual codes and the length weight enumerator of hermitian self-dual...

On Robin’s criterion for the Riemann hypothesis

YoungJu ChoieNicolas LichiardopolPieter MoreePatrick Solé — 2007

Journal de Théorie des Nombres de Bordeaux

Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality σ ( n ) : = d | n d < e γ n log log n is satisfied for n 5041 , where γ denotes the Euler(-Mascheroni) constant. We show by elementary methods that if n 37 does not satisfy Robin’s criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that n must be divisible by a fifth power > 1 . As consequence we obtain that RH holds true iff every natural number divisible by a fifth power...

Symmetric flows and broadcasting in hypercubes

Jean-Claude BermondA. BonnecazeT. KodateStéphane PérennesPatrick Solé — 1999

Annales de l'institut Fourier

In this paper, we propose a method which enables to construct almost optimal broadcast schemes on an n -dimensional hypercube in the circuit switched, Δ -port model. In this model, an initiator must inform all the nodes of the network in a sequence of rounds. During a round, vertices communicate along arc-disjoint dipaths. Our construction is based on particular sequences of nested binary codes having the property that each code can inform the next one in a single round. This last property is insured...

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