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Almost optimum $\ell$-covering of $\mathbb{Z}_n$
- Publication Year :
- 2022
-
Abstract
- A subset $B$ of the ring $\mathbb{Z}_n$ is referred to as a $\ell$-covering set if $\{ ab \pmod n | 0\leq a \leq \ell, b\in B\} = \mathbb{Z}_n$. We show that there exists a $\ell$-covering set of $\mathbb{Z}_n$ of size $O(\frac{n}{\ell}\log n)$ for all $n$ and $\ell$, and how to construct such a set. We also provide examples where any $\ell$-covering set must have a size of $\Omega(\frac{n}{\ell}\frac{\log n}{\log \log n})$. The proof employs a refined bound for the relative totient function obtained through sieve theory and the existence of a large divisor with a linear divisor sum. The result can be used to simplify a modular subset sum algorithm.
- Subjects :
- Computer Science - Discrete Mathematics
Mathematics - Combinatorics
05B40
G.2.1
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2207.05017
- Document Type :
- Working Paper