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07-02-2020 | Original Paper | Issue 5/2021

Applicable Algebra in Engineering, Communication and Computing 5/2021

Discrete Gaussian measures and new bounds of the smoothing parameter for lattices

Journal:
Applicable Algebra in Engineering, Communication and Computing > Issue 5/2021
Authors:
Zhongxiang Zheng, Chunhuan Zhao, Guangwu Xu
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Abstract

In this paper, we start with a discussion of discrete Gaussian measures on lattices. Several results of Banaszczyk are analyzed, a simple form of uncertainty principle for discrete Gaussian measure is formulated. In the second part of the paper we prove two new bounds for the smoothing parameter of lattices. Under the natural assumption that \(\varepsilon\) is suitably small, we obtain two estimations of the smoothing parameter:
1.
$$\begin{aligned} \displaystyle \eta _{\varepsilon }({{\mathbb {Z}}}) \le \sqrt{\frac{\ln \big (\frac{\varepsilon }{44}+\frac{2}{\varepsilon }\big )}{\pi }}. \end{aligned}$$
This is a practically useful case. For this case, our upper bound is very close to the exact value of \(\eta _{\varepsilon }({{\mathbb {Z}}})\) in that \(\sqrt{\frac{\ln \big (\frac{\varepsilon }{44}+\frac{2}{\varepsilon }\big )}{\pi }}-\eta _{\varepsilon }({{\mathbb {Z}}})\le \frac{\varepsilon ^2}{552}\).
 
2.
For a lattice \({{{\mathcal {L}}}}\subset {{\mathbb {R}}}^n\) of dimension n,
$$\begin{aligned} \displaystyle \eta _{\varepsilon }({{{\mathcal {L}}}}) \le \sqrt{\frac{\ln \big (n-1+\frac{2n}{\varepsilon }\big )}{\pi }}\tilde{bl}({{{\mathcal {L}}}}). \end{aligned}$$
 

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