A function $F:\mathbb{F}_{2}^{n}\to \mathbb{F}_{2}^{m}$ is called $k$th-order sum-free if the sum of its values over any $k$-dimensional affine subspace of $\mathbb{F}_2^n$ is non-zero. Carlet recently introduced this notion and constructed such functions for every $2\le k\le n$. We prove that, for $2\le k\le n-2$ and $m \leq n$, the existence of a (non-degenerate) $\mathbb{F}_{2}^{m}$-valued $k$th-order sum-free function on $\mathbb{F}_{2}^{n}$ is equivalent to the existence of a codimension $m$ linear subcode of the Reed-Muller code $\mathrm{RM}(n-k,n)$ with minimum distance $3\cdot 2^{k-1}$. In particular, this yields a new family of Reed-Muller subcodes that avoid all minimum weight codewords of $\mathrm{RM}(n-k,n)$, and thus have minimum distance $3/2$ times that of $\mathrm{RM}(n-k,n)$. We also derive new necessary conditions for the existence of $k$th-order sum-free functions and present the first nontrivial lower bound on $m$. Finally, we observe that $k$th-order sum-free functions lead to a partition of the Grassmannian of all $k$-dimensional (linear) subspaces of $\mathbb{F}_2^n$ into constant-dimension subspace codes. Under the assumption that functions exist that are $k$th-order sum-free for multiple values of $k$, we obtain an improved partitioning result and a stronger upper bound on the chromatic number of the Grassmann graphs.

翻译：函数$F:\mathbb{F}_{2}^{n}\to \mathbb{F}_{2}^{m}$被称为$k$阶无和函数，若其值在$\mathbb{F}_2^n$的任意$k$维仿射子空间上的和不为零。Carlet最近引入了这一概念，并对所有$2\le k\le n$构造了此类函数。我们证明，当$2\le k\le n-2$且$m \leq n$时，$\mathbb{F}_{2}^{n}$上存在（非退化的）$\mathbb{F}_{2}^{m}$值$k$阶无和函数等价于Reed-Muller码$\mathrm{RM}(n-k,n)$中存在一个余维数为$m$的线性子码，其最小距离为$3\cdot 2^{k-1}$。特别地，这导出了一族新的Reed-Muller子码，它们避开了$\mathrm{RM}(n-k,n)$的所有最小重量码字，因此其最小距离是$\mathrm{RM}(n-k,n)$的$3/2$倍。我们还推导了$k$阶无和函数存在的新必要条件，并给出了$m$的第一个非平凡下界。最后，我们观察到$k$阶无和函数可导出一个将$\mathbb{F}_2^n$的所有$k$维（线性）子空间构成的Grassmannian划分为常维子空间码的划分。在假设存在对多个$k$值均为$k$阶无和函数的条件下，我们得到了一个改进的划分结果以及Grassmann图色数的一个更强上界。