Nonsingular splittings over finite fields

Abstract

We say that $M$ and $S$ form a \textsl{splitting} of $G$ if every nonzero element $g$ of $G$ has a unique representation of the form $g=ms$ with $m\in M$ and $s\in S$, while $0$ has no such representation. The splitting is called {\it nonsingular} if $\gcd (|G|,a)=1$ for any $a\in M$. In this paper, we focus our study on nonsingular splittings of cyclic groups. We introduce a new notation --direct KM logarithm and we prove that if there is a prime $q$ such that $M$ splits ${\mathbb{Z}}_q$, then there are infinitely many primes $p$ such that $M$ splits ${\mathbb{Z}}_p$.

References in corpus (1)

• A General Upper Bound on the Size of Constant-Weight Conflict-Avoiding Codes