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Exact uniform bilinear character mean

Lax342547.BilinearMean · concepts/Lax342547/BilinearMean.lean · lax-342547

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    Natural Language Statement

    Lemma

    The mean binary bilinear character under independent uniform coordinates is exactly two to minus the actual matrix rank, by orthogonality and the kernel cardinality.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    Each proof establishes this claim relative to its assumptions.

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    1import Lax342547.BilinearWalsh
    2
    3/-!
    4---
    5title: Exact uniform bilinear character mean
    6type: lemma
    7---
    8The mean binary bilinear character under independent uniform coordinates is exactly two to minus the actual matrix rank, by orthogonality and the kernel cardinality.
    9-/
    10
    11namespace Lax342547.BilinearMean
    12
    13open Lax342547.MomentSpace Lax342547.Walsh
    14open scoped BigOperators
    15
    16axiom bilinear_character_average {I J : Type} [Fintype I] [Fintype J]
    17 [DecidableEq I] [DecidableEq J] (A : Matrix I J Binary) :
    18 (∑ x : I → Binary, ∑ y : J → Binary, phase x (A.mulVec y))/
    19 ((2 : ℝ)^Fintype.card I*2^Fintype.card J) = 1/(2 : ℝ)^A.rank
    20
    21end Lax342547.BilinearMean
    22
    Show Proof
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