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, $f_{\text{inv}}=x^{-1}$ is not $k$th order sum-free for $3\le k\le n-3$. This is the unresolved part of Carlet's conjecture, which gives exact values for which $f_{\text{inv}}$ 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 $f_{\text{inv}}$ is not $k$th order sum-free for $3\le k\le \frac{3}{13}n+0.461$, improving on work previously done by Hou and Zhao.

11 pages