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Complexitylib.Algebraic.LowerBound.Cutwidth.Aggregate.Geometry.Multioutput.RestrictionRank.Defs

Output components that stay nonaffine on large flats #

A prime-field-valued function is affine on an affine flat when it preserves the ternary sum x + y + z of any three points of the flat; its values off the flat are unconstrained. These functions form a submodule. A Boolean map is nonaffine on flats of dimension D when no nonzero linear combination of its output coordinates is affine on any flat with at least 2^D points. Equivalently, D - 1 bounds the dimension of every flat on which a nonzero component is affine.

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def Algebraic.Aggregate.Geometry.flatAffineFunctions {n : ℕ} (S : AffineFlat n) :
Submodule (ZMod 2) ((Fin n → Bool) → ZMod 2)

Prime-field-valued functions that preserve ternary addition on one flat. Their values outside the flat are arbitrary.

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    def Algebraic.Aggregate.Geometry.NonaffineOnFlats {n m : ℕ} (F : (Fin n → Bool) → Fin m → Bool) (D : ℕ) :
    Prop

    No nonzero linear combination of output bits is affine on a flat with at least 2^D points.

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