## The Michigan Mathematical Journal

### Lattice Simplices of Maximal Dimension with a Given Degree

Akihiro Higashitani

#### 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 $d+1\leq 4s-1$. In this paper, we give a complete characterization of lattice simplices satisfying the equality, that is, the lattice simplices of dimension $4s-2$ 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 $d+1\leq f(2s)$, where $f(M)=\sum_{n=0}^{\lfloor \log_{2}M\rfloor }\lfloor M/2^{n}\rfloor$ for $M\in \mathbb{Z}_{\geq 0}$. We also observe that any lattice simplex attaining this sharper bound always comes from a binary code.

#### Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1548817531

Digital Object Identifier
doi:10.1307/mmj/1548817531

Mathematical Reviews number (MathSciNet)
MR3934609

#### Citation

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. https://projecteuclid.org/euclid.mmj/1548817531

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