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


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.


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