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

TL;DR

This paper constructs examples of two-quotient strong starters in finite fields of characteristic two, expanding the understanding of combinatorial structures useful in design theory and finite geometry.

Contribution

It introduces new constructions of two-quotient strong starters specifically for finite fields of the form _q with q=2^k t+1, where k>1 and t is odd.

Findings

Examples of two-quotient strong starters for _q are provided.

The constructions apply to prime powers with specific form q=2^k t+1.

Results extend known classes of strong starters in finite fields.

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.