Hecke operators with odd determinant and binary frameproof codes beyond the probabilistic bound?

Randriam, Hugues

doi:10.1109/cig.2010.5592905

Cited by 11 publications

(11 citation statements)

References 14 publications

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“…He encouraged me to publish all the details by myself, including heuristics and partial results supporting the underlying conjectures. I did so the year after [Ran10]. Writing everything down had a rather unexpected, happy consequence.…”

Section: A Young Arithmetic Geometer Arrives At Télécommentioning

confidence: 99%

“…His aim was to present a new notion he called "torsion limit". I was very surprised to discover that this new notion was essentially the same that I had introduced in [Ran10], in order to measure how close to a solution of Problem 1 one could get! There was an explanation: actually, Cramer was working with Xing.…”

Section: A Young Arithmetic Geometer Arrives At Télécommentioning

confidence: 99%

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On metric convexity, the discrete Hahn-Banach theorem, separating systems and sets of points forming only acute angles

Randriambololona

2017

IJICOT

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A Gérard pour ses 2 6 et 2 6 +1-èmes anniversaires.Abstract This text has three parts. The first one is largely autobiographical, hence my use of the first person. There I recall how Gérard Cohen influenced important parts of my research.The second is of a more classic mathematical nature. I present a discrete analogue of the Hahn-Banach theorem, which serves as a basis for generalizing the notion of separating systems in the context of metric convexity.The third one aims at building a bridge between two communities of researchers, those interested in separating systems, and those interested in a certain question in combinatorial geometry -sets of points forming only acute angles -who seem not to be aware of each other, while they are working on precisely the same problem! Of course, these three themes are closely intertwined.

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Section: A Young Arithmetic Geometer Arrives At Télécommentioning

confidence: 99%

Section: A Young Arithmetic Geometer Arrives At Télécommentioning

confidence: 99%

On metric convexity, the discrete Hahn-Banach theorem, separating systems and sets of points forming only acute angles

Randriambololona

2017

IJICOT

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“…In fact this condition A(q) > 5 can be relaxed very slightly, as shown in [29]. However, to relax it further to A(q) > 1 would require much deeper results, such as the conjectures proposed in [27] on the existence of curves having many points but few 2-torsion in their class group. This also leads to the following question: do m sym q = m q , or M sym q = M q , or more generally µ sym q (m, l) = µ q (m, l) for all q, m, l?…”

Section: Fixing Some Bounds Of Balletmentioning

confidence: 99%

“…More important, in [38], Xing studied intersecting codes arising from algebraic curves, and he gave a criterion for their existence, that reduces essentially to the second part of system (4). Hence here also the 2-torsion in the class group is an obstruction to get optimal parameters (see [27] for elaborations on this). This problem was essentially solved, or more properly, bypassed by the author in [28] with the method discussed above (although the analog problem for t-torsion, t ≥ 3, is still open).…”

Section: Introductionmentioning

confidence: 99%

Bilinear complexity of algebras and the Chudnovsky-Chudnovsky interpolation method

Randriambololona¹

2011

Preprint

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We give new improvements to the Chudnovsky-Chudnovsky method that provides upper bounds on the bilinear complexity of multiplication in extensions of finite fields through interpolation on algebraic curves. Our approach features three independent key ingredients:• We allow asymmetry in the interpolation procedure. This allows to prove, via the usual cardinality argument, the existence of auxiliary divisors needed for the bounds, up to optimal degree.• We give an alternative proof for the existence of these auxiliary divisors, which is constructive, and works also in the symmetric case, although it requires the curves to have sufficiently many points.• We allow the method to deal not only with extensions of finite fields, but more generally with monogenous algebras over finite fields. This leads to sharper bounds, and is designed also to combine well with base field descent arguments in case the curves do not have sufficiently many points.As a main application of these techniques, we fix errors in, improve, and generalize, previous works of Shparlinski-Tsfasman-Vladut, Ballet, and Cenk-Özbudak. Besides, generalities on interpolation systems, as well as on symmetric and asymmetric bilinear complexity, are also discussed. Contents 1 Tensor rank and bilinear complexity 2 Interpolation systems 3 The extended Chudnovsky-Chudnovsky algorithm 4 Genus 0 or 1 5 Fixing some bounds of Ballet 6 Fixing the Shparlinski-Tsfasman-Vladut asymptotic upper bound

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“…In addition, one could repeat the analysis in alternate situations: one could treat nonprime N , work modulo another prime ℓ, consider T p for another p, and/or work in some higher weight k. While all of these variants are of intrinsic interest, we would like to point out some developments in the computation of modular forms which draw attention to some particular cases. (Separately, the case of N prime, ℓ = 2, p > 2, k = 2 has arisen in the context of error-correcting codes [28]. )…”

Section: Introductionmentioning

confidence: 99%

Mod-2 dihedral Galois representations of prime conductor

Kedlaya

Medvedovsky

2019

Open Book Series

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For all odd primes N up to 500000, we compute the action of the Hecke operator T 2 on the space S 2 (Γ 0 (N ), Q) and determine whether or not the reduction mod 2 (with respect to a suitable basis) has 0 and/or 1 as eigenvalues. We then partially explain the results in terms of class field theory and modular mod-2 Galois representations. As a byproduct, we obtain some nonexistence results on elliptic curves and modular forms with certain mod-2 reductions, extending prior results of Setzer, Hadano, and Kida.

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Hecke operators with odd determinant and binary frameproof codes beyond the probabilistic bound?

Cited by 11 publications

References 14 publications

On metric convexity, the discrete Hahn-Banach theorem, separating systems and sets of points forming only acute angles

On metric convexity, the discrete Hahn-Banach theorem, separating systems and sets of points forming only acute angles

Bilinear complexity of algebras and the Chudnovsky-Chudnovsky interpolation method

Mod-2 dihedral Galois representations of prime conductor

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