Applying hypersurface bounds to a conjecture by Carlet

Abstract

A function from $\mathbb{F}_{2^n}$ to $\mathbb{F}_{2^n}$ is $k$th order sum-free if the sum of its values over each $k$-dimensional $\mathbb{F}_2$-affine subspace is nonzero. It is conjectured that for $n$ odd and prime, $f_\textrm{inv}=x^{-1}$ is not $k$th order sum-free for $3 \leq k \leq n-3$. This is the unresolved part of Carlet's conjecture, which gives exact values for which $f_\textrm{inv}$ is $k$th order sum-free. We give two results as improvements on an explicit estimate on the number of $q$-rational points of an $\mathbb{F}_q$-definable hypersurface previously proved by Cafure and Matera. We use these results to prove that $f_\textrm{inv}$ is not $k$th order sum-free for $3\leq k \leq \frac{3}{13}n+0.461$, improving on work previously done by Hou and Zhao.