# What does 2-flat mean when discussing APN permutations?

I am currently reading a research paper that uses the term 2-flat. However, I don't understand what this term means.

For example, the paper mentions in its definition of APN that for all distinct $$a$$, $$b$$, $$c$$ and $$d$$ in $$V$$,

$$a + b + c + d = 0 \implies f(a) + f(b) + f(c) + f(d) \neq 0$$

i.e. $$f$$ does not sum to 0 on any 2-flat.

It would be great if anyone could explain what 2-flat means here.

An APN permutation in dimension 6

## 1 Answer

In the context of Boolean functions, "flat" is usually used as a synonym for "affine subspace of $$\mathbb{F}_2^n$$". More generally, an $$n$$-flat in a vector space $$V$$ (which may be considered as an affine space) is an $$n$$-dimensional affine subspace of $$V$$.

Note that a 2-flat of a vector space $$V$$ over $$\mathbb{F}_2$$ is then any set of the form

$$A = \{a, a + b, a + c, a + b + c\}$$

with $$a, b, c \in V$$ distinct. Remark that $$\sum_{x \in A} x = 0$$ and conversely, any four distinct elements in $$V$$ that sum to zero form a 2-flat.