Improved Lower Bounds for Permutation Arrays Using Permutation Rational Functions

Abstract

We consider rational functions of the form V(x)/U(x), where both V(x) and U(x) are relatively prime polynomials over the finite field $\mathbb {F}_q$. Polynomials that permute the elements of a field, called permutation polynomials (PPs), have been the subject of research for decades. Let ${\mathcal {P}}^1(\mathbb {F}_q)$ denote $\mathbb {F}_q\cup \{\infty \}$. If the rational function, V(x)/U(x), permutes the elements of ${\mathcal {P}}^1(\mathbb {F}_q)$, it is called a 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 give a formula for $N_{4,4}(q)$. We conjecture that these are 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 $d \in \{5,7,9\}$.

S. Bereg—Research of the first author is supported in part by NSF award CCF-1718994.