Is MCA projected Generator of is MCA
LinearTransformations.isMCA_projectedGenerator_of_isMCA
Plain-language statement
If the MCA condition IsMCA holds for a projected generator, then IsMCA holds for the original generator G with the zero-extension defined above.
Exact Lean statement
lemma isMCA_projectedGenerator_of_isMCA (LC : LinearCode ι F) [Nonempty S] (G : Generator S ℓ F)
(κ : Set ℓ) [Fintype κ] (U : κ → (ι → F)) (γ : I) (x : S) :
IsMCA (projectedGenerator G κ) LC x U γ → IsMCA G LC x (zeroExtend κ U) γFormal artifact
Lean source
lemma isMCA_projectedGenerator_of_isMCA (LC : LinearCode ι F) [Nonempty S] (G : Generator S ℓ F) (κ : Set ℓ) [Fintype κ] (U : κ → (ι → F)) (γ : I) (x : S) : IsMCA (projectedGenerator G κ) LC x U γ → IsMCA G LC x (zeroExtend κ U) γ := by have vecMul_projectedGenerator : Matrix.vecMul (projectedGenerator G κ x) U = Matrix.vecMul (G x) (zeroExtend κ U) := by ext i simp only [Matrix.vecMul, dotProduct] rw [← Finset.sum_subset (Finset.subset_univ (Set.toFinset κ))] · refine Finset.sum_bij (fun j _ => j) ?_ ?_ ?_ ?_ <;> simp [projectedGenerator, zeroExtend] · intro x _ hx; simp [zeroExtend]; aesop have zeroExtend_val (j : κ) : zeroExtend κ U j.val = U j := by simp [zeroExtend, j.property] rintro ⟨T, hT₁, hT₂, j, hT₃⟩ exact ⟨T, hT₁, by convert hT₂ using 1; exact funext fun _ => by simp [vecMul_projectedGenerator], ⟨j, by rw [zeroExtend_val] ; assumption⟩⟩- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/ProximityGap/MCAGenerator.lean:91-107
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Person-level attribution pending.
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Plain-language statement
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