On a Conjecture About the Sum-Freedom of the Binary Multiplicative Inverse Function

A recent conjecture by C. Carlet on the sum-freedom of the binary multiplicative inverse function can be stated as follows: For each pair of positive integers $(n,k)$ with $3\le k\le n-3$, there is a $k$-dimensional ${\mathbb{F}}_2$-subspace $E$ of ${\mathbb{F}}_{2^n}$ such that ${\sum }_{0\mathrm{\ne }\in E}1\mathrm{/}u=0$. We confirm this conjecture when $n$ is not a prime.