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April 2019 Lattice Simplices of Maximal Dimension with a Given Degree
Akihiro Higashitani
Michigan Math. J. 68(1): 193-210 (April 2019). DOI: 10.1307/mmj/1548817531

Abstract

It was proved by Nill that for any lattice simplex of dimension d with degree s that is not a lattice pyramid, we have d+14s1. In this paper, we give a complete characterization of lattice simplices satisfying the equality, that is, the lattice simplices of dimension 4s2 with degree s that are not lattice pyramids. It turns out that such simplices arise from binary simplex codes. As an application of this characterization, we show that such simplices are counterexamples for the conjecture known as the Cayley conjecture. Moreover, by slightly modifying Nill’s inequality we also see the sharper bound d+1f(2s), where f(M)=n=0log2MM/2n for MZ0. We also observe that any lattice simplex attaining this sharper bound always comes from a binary code.

Citation

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Akihiro Higashitani. "Lattice Simplices of Maximal Dimension with a Given Degree." Michigan Math. J. 68 (1) 193 - 210, April 2019. https://doi.org/10.1307/mmj/1548817531

Information

Received: 6 April 2017; Revised: 20 October 2017; Published: April 2019
First available in Project Euclid: 30 January 2019

zbMATH: 07155463
MathSciNet: MR3934609
Digital Object Identifier: 10.1307/mmj/1548817531

Subjects:
Primary: 52B20
Secondary: 14M25, 94B05

Rights: Copyright © 2019 The University of Michigan

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Vol.68 • No. 1 • April 2019
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