$k$th-order sum-free functions are a natural generalization of APN functions using the concept of (non)vanishing flats. In this paper, we introduce a new combinatorial technique to study the nonvanishing flats of Boolean functions. This approach allows us to determine the number of nonvanishing flats for an infinite family of Boolean functions. We moreover use it to show that any $k$th-order sum-free $(n,n)$-function of algebraic degree $k$ gives rise to an $(n-k)$th-order sum-free $(n,n)$-function of algebraic degree $n-k$. This implies the existence of millions of $(n-2)$th-order sum-free functions.

翻译：$k$阶无和函数是利用（非）零平坦概念对APN函数的自然推广。本文提出一种研究布尔函数非零平坦性的新型组合技术。该方法使我们能够确定一个无限布尔函数族中的非零平坦数量。此外，我们利用该技术证明：任何代数次数为$k$的$k$阶无和$(n,n)$函数都能导出一个代数次数为$n-k$的$(n-k)$阶无和$(n,n)$函数。这一结果意味着存在数百万个$(n-2)$阶无和函数。