E le dist over 3
ProximityToRS.e_le_dist_over_3
Plain-language statement
Lemma 4.4, [AHIV22] (mutual-exclusion corollary). Either all points on the affine line are e-close to the Reed–Solomon code, or at most ‖RS‖₀ points are. The assumptions v ≠ 0 and ‖RS‖₀ < |F| are necessary for mutual exclusion: if v = 0, the affine line degenerates to a singleton and the two branches can hold simultaneously.
Exact Lean statement
lemma e_le_dist_over_3
{deg : ℕ}
{α : ι ↪ F} {e : ℕ} {u v : ι → F}
(he : (e : ℚ≥0) < ‖(RScodeSet α deg)‖₀ / 3)
(hv : v ≠ 0)
(hFd : ‖(RScodeSet α deg)‖₀ < Fintype.card F) :
Xor
(∀ x ∈ Affine.affineLineAtOrigin (F := F) u v, Δ₀(x, ReedSolomon.code α deg) ≤ e)
(numberOfClosePts u v deg α e ≤ ‖(RScodeSet α deg)‖₀)Formal artifact
Lean source
lemma e_le_dist_over_3 {deg : ℕ} {α : ι ↪ F} {e : ℕ} {u v : ι → F} (he : (e : ℚ≥0) < ‖(RScodeSet α deg)‖₀ / 3) (hv : v ≠ 0) (hFd : ‖(RScodeSet α deg)‖₀ < Fintype.card F) : Xor (∀ x ∈ Affine.affineLineAtOrigin (F := F) u v, Δ₀(x, ReedSolomon.code α deg) ≤ e) (numberOfClosePts u v deg α e ≤ ‖(RScodeSet α deg)‖₀) := by classical have hline : (∀ x ∈ Affine.affineLineAtOrigin (F := F) u v, Δ₀(x, ReedSolomon.code α deg) ≤ e) ∨ numberOfClosePts u v deg α e ≤ ‖(RScodeSet α deg)‖₀ := e_le_dist_over_3_strong (F := F) (ι := ι) (α := α) (e := e) (u := u) (v := v) he rcases hline with h_all | h_few · exact Or.inl ⟨h_all, all_close_not_few_close_pts (F := F) (ι := ι) (hv := hv) (hFd := hFd) h_all⟩ · exact Or.inr ⟨h_few, fun h_all ↦ all_close_not_few_close_pts (F := F) (ι := ι) (hv := hv) (hFd := hFd) h_all h_few⟩- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/ProximityGap/AHIV22.lean:1070-1091
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Project documentation
This lemma proves the final algebraic step in the DG25 Theorem 3.1 proof. It shows that if R > e + 1, then e * (R / (R - 1)) < e + 1. The intuition is that the fraction R / (R - 1) is always greater than 1, but as R gets larger, it gets closer to 1. The hypothesis R > e + 1 provides a strong enough bound to ensure the product e * (fraction) do...
Source project: ArkLib
Person-level attribution pending.
Gadget Decompose coeff
ArkLib.Lattices.Ajtai.gadgetDecompose_coeff
Plain-language statement
The k-th coefficient (k < deg φ) of a gadget-decomposition block is exactly the corresponding digit of the corresponding input coefficient.
Source project: ArkLib
Person-level attribution pending.
Gadget Decompose lawful
ArkLib.Lattices.Ajtai.gadgetDecompose_lawful
Plain-language statement
The base-b gadget decomposition is a lawful gadget decomposition.
Source project: ArkLib
Person-level attribution pending.