CanaDAM 2013 Memorial University of Newfoundland, June 10 - 13, 2013 www.cms.math.ca//2013

Finite Fields in Combinatorics II
Chair: Petr Lisonek (Simon Fraser University)
Org: Petr Lisonek (Simon Fraser University) and David Thomson (Carleton University)
[PDF]

MARK GIESBRECHT, University of Waterloo
Decomposition of additive polynomials and matrix similarity classes  [PDF]

We explore problems of efficient computations with additive (linearized) polynomials over finite fields, including decomposition/factorization and classifying the number of distinct composition patterns. We connect this to the similarity class of the Frobenius operator of the polynomials.

JING HE, Carleton University
A new class of almost perfect sequences and a new family of Zero Correlation Zone sequences  [PDF]

Using maximal length sequences and multiplicative characters, we construct a class of sequences with almost perfect autocorrelation. Then we interleave two sequences in this class to construct a zero correlation zone (ZCZ) sequence family with large size.

XIANG-DONG HOU, University of South Florida
A Class of Permutation Binomials over Finite Fields  [PDF] [SLIDES]

Let $q$ a prime power and $f=ax+x^{2q-1}$, where $a\in\Bbb F_q^*$. It was recently conjectured that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if and only if one of the following holds: (i) $a=1$, $q\equiv 1\pmod 4$; (ii) $a=-3$, $q\equiv \pm1\pmod{12}$; (iii) $a=3$, $q\equiv -1\pmod 6$. We will confirm this conjecture. We will also describe the context from which this conjecture arose.

DANIEL KATZ, California State University, Northridge
Weil Sums of Binomials with Three-Valued Spectra  [PDF] [SLIDES]

Weil sums of binomials arise naturally in number theory, and have direct applications in cryptography, digital sequence design, and coding theory. Consider the Weil sum $W_{q,d}(a)=\sum_{x\in{\mathbb F}_q}\psi_q(x^d+ax)$, with $\psi_q$ the canonical additive character of finite field ${\mathbb F}_q$, $\gcd(d,q-1)=1$, $d$ not a power of $p$ modulo $q-1$, and $a\in{\mathbb F}_q^*$. Fix $q$ and $d$ and consider the spectrum of values obtained as $a$ runs through ${\mathbb F}_q^*$. At least three values must appear, and we discuss recent results about the case where precisely three appear, including our recent proof of the characteristic $3$ case of a 1976 conjecture of Helleseth.

DAVID THOMSON, Carleton University
On a conjecture of Golomb and Moreno  [PDF]

A polynomial $f$ over a finite field with $f(0)=0$ and $f(xd)-f(x)$ being a permutation for all $d\neq1$ is a \emph{Costas polynomial}. Costas polynomials are semi-multiplicative analogues of \emph{planar functions}. The Golomb-Moreno conjecture states that a Costas polynomial over a prime field is a monomial.

In this talk, we draw connections between Costas polynomials and related combinatorial objects. We also give a partial proof of the Golomb-Moreno conjecture: we show that $3/4$ of the terms of a Costas polynomial must equal $0$. We also give an equivalent conjecture in terms of the number of \emph{moved} elements of the field under $f$.