New lower bounds for permutation arrays using contraction

Abstract

We consider rational functions of the form V (x)/U (x), where both V (x) and U (x) are polynomials over the finite field F q . Polynomials that permute the elements of a field, called permutation polynomials (P P s), have been the subject of research for decades. Let P 1 (F q ) denote Z q ∪ {∞}. If the rational function, V (x)/U (x), permutes the elements of P 1 (F q ), it is called a permutation rational function (PRF). Let N d (q) denote the number of PPs of degree d over 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 ≤ 307, N 5,4 (q) > (q + 1)q 3 (q − 1) 2 /2, for all prime powers q ≤ 97, and N 4,4 (p) = (p + 1)p 2 (p − 1) 3 /3, for all primes p ≤ 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 ∈ {5, 7, 9}.

Equivalence Relations for Computing Permutation Polynomials

“…To illustrate G-cycles on coefficients, consider GF (2 4 ). The elements of GF (2 4 ) are partitioned into 6 disjoint equivalence classes (i.e., G-cycles on coefficients) by the G-map, namely the equivalence classes [0], [1], [2], [4], [6], and [8]. To see this, observe that where mod 15 arithmetic is used in the exponents.…”