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2-flats of course
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These are called Sidon sets. Generally, these are sets such that all pairwise sums are disjoint, for characteristic 2 this coincides with sets not containing 42-flats. There are quite a few papers in Boolean functions literature about them, including some bounds and known values.

These are called Sidon sets. Generally, these are sets such that all pairwise sums are disjoint, for characteristic 2 this coincides with sets not containing 4-flats. There are quite a few papers in Boolean functions literature about them, including some bounds and known values.

These are called Sidon sets. Generally, these are sets such that all pairwise sums are disjoint, for characteristic 2 this coincides with sets not containing 2-flats. There are quite a few papers in Boolean functions literature about them, including some bounds and known values.

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These are called Sidon sets. Generally, these are sets such that all pairwise sums are disjoint, for characteristic 2 this coincides with sets not containing 4-flats. There are quite a few papers in Boolean functions literature about them, including some bounds and known values.