On two-quotient strong starters for $\mathbb{F}_q$

Abstract: Let $G$ be a finite additive abelian group of odd order $n$, and let $G^*=G\setminus{0}$ be the set of non-zero elements. A starter for $G$ is a set $S={{x_i,y_i}:i=1,\ldots,\frac{n-1}{2}}$ such that ${x_1,\ldots,x_\frac{n-1}{2},y_1,\ldots,y_\frac{n-1}{2}}=G^*$ and ${\pm(x_i-y_i):i=1,\ldots,\frac{n-1}{2}}=G^*$. Moreover, if $\left|\left{x_i+y_i:i=1,\ldots,\frac{n-1}{2}\right}\right|=\frac{n-1}{2}$, then $S$ is called a strong starter for $G$. A starter $S$ for $G$ is a $k$ quotient starter if there exists $Q\subseteq G^*$ of cardinality $k$ such that $y_i/x_i\in Q$ or $x_i/y_i\in Q$, for $i=1,\ldots,\frac{n-1}{2}$. In this paper, we give examples of two-quotient strong starters for $\mathbb{F}_q$, where $q=2^kt+1$ is a prime power with $k>1$ a positive integer and $t$ an odd integer greater than 1.