Quadrics and Difference Sets

Abstract

Let $L$ be a linear form on the Galois field $\text{GF}(q^{n+1})$ over $\text{GF}(q)$ ($n\ge 2$). We characterize those integers $s$ coprime to $v=(q^{n+1}-1)/(q-1)$ such that $L(x^s)$ is (or is related to) a quadratic form on $\text{GF}(q^{n+1})$ over $\text{GF}(q)$. This relates to a conjecture of Games concerning quadrics of the form $rD$ in $\text{PG}(n,q)$, where $D$ is a difference set in the cyclic group $Z_v$, acting as a Singer group on the points and hyperplanes of $\text{PG}(n,q)$. It has been shown that Games’ conjecture does not hold except possibly in the case $q=2$: here we establish that it holds exactly when $q=2$. We also suggest a new conjecture. Our result for $q=2$ enables us to prove another conjecture of Games’, concerning $m$-sequences with three-valued periodic cross-correlation function.