Improved Lower Bounds for Permutation Arrays Using Permutation Rational Functions

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