Gen Mat Is Vandermonde
ReedSolomon.genMatIsVandermonde
Plain-language statement
The Vandermonde matrix is the generator matrix for an RS code of length ι and dimension deg.
Exact Lean statement
lemma genMatIsVandermonde [Fintype ι] [Field F] [inst : NeZero m] {α : ι ↪ F} :
fromColGenMat (Vandermonde.nonsquare (ι' := m) α) = ReedSolomon.code α mFormal artifact
Lean source
lemma genMatIsVandermonde [Fintype ι] [Field F] [inst : NeZero m] {α : ι ↪ F} : fromColGenMat (Vandermonde.nonsquare (ι' := m) α) = ReedSolomon.code α m := by classical unfold fromColGenMat ReedSolomon.code ext x; rw [LinearMap.mem_range, Submodule.mem_map] refine ⟨ fun ⟨coeffs, h⟩ ↦ ⟨polynomialOfCoeffs coeffs, h.symm ▸ ?p₁⟩, fun ⟨p, h⟩ ↦ ⟨Fin.liftF' p.coeff, ?p₂⟩ ⟩ · rw [ ←coeff_polynomialOfCoeffs_eq_coeffs (coeffs := coeffs), Vandermonde.mulVecLin_coeff_vandermondens_eq_eval_matrixOfPolynomials (by simp) ] simp [ReedSolomon.evalOnPoints] · exact h.2 ▸ Vandermonde.mulVecLin_coeff_vandermondens_eq_eval_matrixOfPolynomials (natDegree_lt_of_mem_degreeLT h.1)- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/ReedSolomon.lean:138-153
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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.