Enumerating Permutation Polynomials over finite fields by degree

We prove an asymptotic formula for the number of permutation for which the associated permutation polynomial has degree smaller than $q-2$.Comment: LaTeX2e amsart 5 page

Enumerating permutation polynomials over finite fields by degree II

AbstractThis note is a continuation of a paper by the same authors that appeared in 2002 in the same journal. First we extend the method of the previous paper proving an asymptotic formula for the number of permutations for which the associated permutation polynomial has d coefficients in specified fixed positions equal to 0. This also applies to the function Nq,d that counts the number of permutations for which the associated permutation polynomial has degree <q-d-1. Next we adopt a more precise approach to show that the asymptotic formula Nq,d∼q!/qd holds for d⩽αq and α=0.03983

Estimates on the Size of Symbol Weight Codes

The study of codes for powerlines communication has garnered much interest over the past decade. Various types of codes such as permutation codes, frequency permutation arrays, and constant composition codes have been proposed over the years. In this work we study a type of code called the bounded symbol weight codes which was first introduced by Versfeld et al. in 2005, and a related family of codes that we term constant symbol weight codes. We provide new upper and lower bounds on the size of bounded symbol weight and constant symbol weight codes. We also give direct and recursive constructions of codes for certain parameters.Comment: 14 pages, 4 figure

On permutation polynomials over finite fields /

A permutation polynomial (PP) over a nite eld Fq is a polynomial in Fq[x] which induces a bijective map from Fq to itself. PPs are of great theoretical interest and are also needed for applications. This thesis starts with some basic facts about PPs. Recent results about one of the most important open problems in this topic: counting PPs of a given degree, are presented. Well known classes of PPs are the linear polynomials, the monomials xk, with gcd(k, q 1) = 1, the linearized polynomials, and the Dickson polynomials. It turns out that nding new classes of PPs is not easy. We also focus on this problem and give a survey of some recent constructions

On the cycle structure of permutation polynomials

L. Carlitz observed in 1953 that for any a € F*q, the transposition (0 a) can be represented by the polynomial Pa(x) = -a[2](((x - a)[q-2] + a-[1])[q-2] - a)[q-2] which shows that the group of permutation polynomials over Fq is generated by the linear polynomials ax + b, a, b € Fq, a≠0, and x[q-2]. Therefore any permutation polynomial over Fq can be represented as Pn = (...((a[0]x + a[1])[q-2] +a[2]) [q-2] ... + a[n])[q-2] + a[n+1], for some n ≥ 0. In this thesis we study the cycle structure of permutation polynomials Pn, and we count the permutations Pn, n ≤ 3, with a full cycle. We present some constructions of permutations of the form Pn with a full cycle for arbitrary n. These constructions are based on the so called binary symplectic matrices. The use of generalized Fibonacci sequences over Fq enables us to investigate a particular subgroup of Sq, the group of permutations on Fq. In the last chapter we present results on this special group of permutations