Linsolve always some berlekamp welch
BerlekampWelch.linsolve_always_some_berlekamp_welch
Plain-language statement
If only up to e errors happened linsolve cannot fail to find a solution.
Exact Lean statement
lemma linsolve_always_some_berlekamp_welch
[NeZero n]
(hp_deg : p.natDegree < k)
(h_ham : (Δ₀(f, p.eval ∘ ωs) : ℕ) ≤ e) :
linsolve (BerlekampWelchMatrix e k ωs f) (Rhs e ωs f) ≠ noneFormal artifact
Lean source
lemma linsolve_always_some_berlekamp_welch [NeZero n] (hp_deg : p.natDegree < k) (h_ham : (Δ₀(f, p.eval ∘ ωs) : ℕ) ≤ e) : linsolve (BerlekampWelchMatrix e k ωs f) (Rhs e ωs f) ≠ none := fun contr ↦ by refine linsolve_none contr ⟨E_and_Q_to_a_solution e (E ωs f p e) (Q ωs f p e), ?p₁⟩ rw [←IsBerlekampWelchSolution_def] simp [ BerlekampWelchCondition_iff_Solution, solutionToQ_from_Q hp_deg h_ham, solutionToE_from_E hp_deg h_ham, E_and_Q_BerlekampWelch_condition hp_deg h_ham]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/BerlekampWelch/Existence.lean:148-159
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
Affine gaps lifted to interleaved codes
affine_gaps_lifted_to_interleaved_codes
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.