Helleseth's conjecture in dimension 32

This page reports a verification of Helleseth's conjecture in dimension 32.
Philippe Langevin, Last modification September 21th 2010.

### Fourier coefficient

Let m be a positive integer, L the field GF(2,m), q the order of L. For a mapping from L into itself, the Fourier coefficient of f at a is defined by

S(f, a) = sum_{x in L} X( f(x) + ax )
where X is the canonical additive character of L. The set of values S(f, a) when a ranges L is called the spectrum of f.
spec* (f) = { S(f, a ) | a in L*}
In the seventies, two conjectures where proposed by Helleseth concerning the spectra of power permutations f(x) = x^d. They are still open !

### Helleseth's conjecture

If d is coprime to q-1 then there exists a non zero element a in L such that

S(x^d, a ) = 0, equivalently 0 lies in spec*(x^d)
note that I checked this claim for m < 26, see this page.

### Four values Helleseth's conjecture

Assuming that m is a power of two. If d is coprime to q-1 then

# spec*(x^d) > 3

### Objectives

The goal of this numerical experiment is to check the secondary conjecture in dimension 32, assumming the main conjecture. In other words, we have to check the non existence of a power permutation f with

spec*( f ) = { 0, A, B}
One will find details on these slides, but, denoting by a the minimal dyadic valuation of A and B, and by b the greater one. The candidates to contradict the conjecture must satisfy several conditions * :
• S(x^d, 1 ) = 0
• A > sqrt(q) or B > sqrt(q)
• AB < 0
• a > m/4
• b >= m/2

### Methodology

We use the fact that the dimension is even to compute efficientely the Fourier coefficient at one of all the power permutations keeping in memory the exponents satisfying S(x^d, 1 ) = 0. The cost of this task is O( q sqrt(q) ). After this, we simply test the list of candidates according to the conditions (*). The total running time was abouth 1 day, no counter example !