Counterexamples to the Woods Conjecture in dimensions d24
Journal de Théorie des Nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 723-726.

Soit N d le maximum des rayons de recouvrement des réseaux d-dimensionnels unimodulaires possédants d vecteurs minimaux indépendants. En 1972, A. C. Woods a conjecturé que N d d 2. En 2005, C. T. McMullen a démontré que la conjecture de Woods implique la célèbre conjecture de Minkowski. La conjecture de Woods est prouvée pour d9. En 2016, Regev, Shapira et Weiss ont trouvé des contre-exemples à la conjecture de Woods pour d30. Dans cet article, nous donnons des contre-exemples à la conjecture de Woods pour d24. La question reste donc ouverte pour les dimensions 10d23.

Let N d be the greatest value of covering radius over all well-rounded unimodular d dimensional lattices. In 1972 A. C. Woods conjectured that N d d 2. C. T. McMullen proved that the Woods conjecture implies the celebrated Minkowski’s conjecture in 2005. The Woods conjecture has been proved for d9. In 2016 Regev, Shapira and Weiss gave counterexamples for the Woods conjecture for d30. In this paper we give counterexamples to the Woods conjecture in dimensions d24. Then the unknown dimensions of the Woods conjecture are 14 dimensions 10d23.

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DOI : https://doi.org/10.5802/jtnb.1105
Classification : 11H31,  11H99
Mots clés : Lattice, Woods conjecture, Minkowski conjecture
@article{JTNB_2019__31_3_723_0,
     author = {Hao Chen and Liqing Xu},
     title = {Counterexamples to the {Woods} {Conjecture} in dimensions $d \ge 24$},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {723--726},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {3},
     year = {2019},
     doi = {10.5802/jtnb.1105},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1105/}
}
Hao Chen; Liqing Xu. Counterexamples to the Woods Conjecture in dimensions $d \ge 24$. Journal de Théorie des Nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 723-726. doi : 10.5802/jtnb.1105. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1105/

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