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.
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.
As Aleph said, an $n$-flat in $\mathbb{F}_2^n$ (ocassionally denoted as $\mathrm{AG}(n,2)$), is an $n$-dimensional affine subspace.
It is worth noting that an affine combination in $\mathbb{F}_2^n$ is a sum $\sum_{i=1}^m \alpha_i x_i$ where $\alpha_i \in \mathbb{F}_2$ such that $\sum_{i=1}^m \alpha_i = 1$. In short, this means that affine combinations in $\mathbb{F}_2^n$ are the sums of an odd number of vectors. The affine span of some set of vectors is then the set of all affine combinations of vectors in the given set. Finally, an $n$-flat is defined to be the affine span of $n+1$ affinely independent vectors.
