  We consider rational functions of the form $V(x)/U(x)$, where both $V(x)$ and
$U(x)$ are polynomials over the finite field $\mathbb{F}_q$. Polynomials that
permute the elements of a field, called {\it permutation polynomials ($PPs$)},
have been the subject of research for decades. Let ${\mathcal
P}^1(\mathbb{F}_q)$ denote $\mathbb{Z}_q \cup \{\infty\}$. If the rational
function, $V(x)/U(x)$, permutes the elements of ${\mathcal P}^1(\mathbb{F}_q)$,
it is called a {\em permutation rational function (PRf)}. Let $N_d(q)$ denote
the number of PPs of degree $d$ over $\mathbb{F}_q$, and let $N_{v,u}(q)$
denote the number of PRfs with a numerator of degree $v$ and a denominator of
degree $u$. It follows that $N_{d,0}(q) = N_d(q)$, so PRFs are a generalization
of PPs. The number of monic degree 3 PRfs is known [11]. We develop efficient
computational techniques for $N_{v,u}(q)$, and use them to show $N_{4,3}(q) =
(q+1)q^2(q-1)^2/3$, for all prime powers $q \le 307$, $N_{5,4}(q) >
(q+1)q^3(q-1)^2/2$, for all prime powers $q \le 97$, and $N_{4,4}(p) =
(p+1)p^2(p-1)^3/3$, for all primes $p \le 47$. We conjecture that these
formulas are, in fact, true for all prime powers $q$. Let $M(n,D)$ denote the
maximum number of permutations on $n$ symbols with pairwise Hamming distance
$D$. Computing improved lower bounds for $M(n,D)$ is the subject of much
current research with applications in error correcting codes. Using PRfs, we
obtain significantly improved lower bounds on $M(q,q-d)$ and $M(q+1,q-d)$, for
$d \in \{5,7,9\}$.
