Improved Lower Bounds for Permutation Arrays Using Permutation Rational Functions
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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 (PPs), 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}.
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