no. 3
Counterexamples to the Woods Conjecture in dimensions d≥24
Chen, Hao1 ; Xu, Liqing1
1 The College of Information Science and Technology/Cyber Security, Jinan University Guangzhou, Guangdong Province 510632, China
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 d≤9. En 2016, Regev, Shapira et Weiss ont trouvé des contre-exemples à la conjecture de Woods pour d≥30. Dans cet article, nous donnons des contre-exemples à la conjecture de Woods pour d≥24. La question reste donc ouverte pour les dimensions 10≤d≤23.

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 d≤9. In 2016 Regev, Shapira and Weiss gave counterexamples for the Woods conjecture for d≥30. In this paper we give counterexamples to the Woods conjecture in dimensions d≥24. Then the unknown dimensions of the Woods conjecture are 14 dimensions 10≤d≤23.

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DOI : 10.5802/jtnb.1105
Classification : 11H31, 11H99
Mots clés : Lattice, Woods conjecture, Minkowski conjecture
Chen, Hao 1 ; Xu, Liqing 1

1 The College of Information Science and Technology/Cyber Security, Jinan University Guangzhou, Guangdong Province 510632, China 
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Chen, Hao; Xu, Liqing. 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. http://archive.numdam.org/articles/10.5802/jtnb.1105/

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