The Michigan Mathematical Journal

Lattice Simplices of Maximal Dimension with a Given Degree

Akihiro Higashitani

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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.

Article information

Michigan Math. J., Volume 68, Issue 1 (2019), 193-210.

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

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Mathematical Reviews number (MathSciNet)

Primary: 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]
Secondary: 14M25: Toric varieties, Newton polyhedra [See also 52B20] 94B05: Linear codes, general


Higashitani, Akihiro. Lattice Simplices of Maximal Dimension with a Given Degree. Michigan Math. J. 68 (2019), no. 1, 193--210. doi:10.1307/mmj/1548817531.

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