The non-existence of desarguesian $t$-parallelisms, $t$ an odd prime

Abstract

In this article, it is shown that when $t$ is a prime, partial Desarguesian $%t$-parallelisms in $PG(zt-1,q)$ of $m$ $t$-spreads are equivalent totranslation nets of order $q^{zt}$ and degree $1+q+m(q^{zt}-q)$. Thus, abound is established for the \ number of $t$-spreads in partial Desarguesian $t$-parallelisms, when $t$ is an odd prime. This also shows that therecannot be Desarguesian $t$-parallelisms when $t$ is an odd prime.