Checking Helleseth's conjecture in dimension less or equal to 32


This page reports the numerical experiments (January 2012) related to numerical verification of Helleseth's conjecture in dimension less or equal to 32.


Helleseth'conjecture

Let m be a positive integer, X a nontrivial additive character of GF(2,m). The Helleseth's conjecture claims that for all integer s coprime with (2^m-1), there exists a non zero element a in GF(2,m) such that

sum_{x in GF(2,m)} X( x^s + ax ) = 0.
execpt in the case where s = 1, 2, 4 etc...

Numerical facts

Helleseth conjecture is true for all m less than 26, and it is true for all even m less or equal to 32. Note that the first point was done here. In this page, we explain how to get the second point.

Methodology




Philippe Langevin, Last modification January 2012.