Decoder eq some
BerlekampWelch.decoder_eq_some
Project documentation
Correctness theorem for Berlekamp-Welch decoder: If a codeword is close to a polynomial p of degree < k then the decoder succeeds and returns some p. ### Parameters: - e k : ℕ - Error capacity and degree bound - [NeZero n] - Non-zero codeword length - ωs : Fin n → F - Distinct evaluation points (injective mapping) - f : Fin n → F - Recei...
Exact Lean statement
theorem decoder_eq_some {e k : ℕ} [NeZero n] {ωs f : Fin n → F} {p : Polynomial F}
(he : 2 * e < n - k + 1)
(hn : k ≤ n)
(h_inj : Function.Injective ωs)
(h_deg : p.natDegree < k)
(h_dist : Δ₀(f, p.eval ∘ ωs) ≤ e) : decoder e k ωs f = some pFormal artifact
Lean source
theorem decoder_eq_some {e k : ℕ} [NeZero n] {ωs f : Fin n → F} {p : Polynomial F} (he : 2 * e < n - k + 1) (hn : k ≤ n) (h_inj : Function.Injective ωs) (h_deg : p.natDegree < k) (h_dist : Δ₀(f, p.eval ∘ ωs) ≤ e) : decoder e k ωs f = some p := by simp only [decoder] split_ifs with hif · suffices p = 0 from Option.some_inj.2 this.symm refine poly_eq_zero_of_dist_lt h_deg hn h_inj (lt_of_le_of_lt ?p₁ he) transitivity ‖f‖₀ + Δ₀(f, p.eval ∘ ωs) · convert hammingDist_triangle 0 f (p.eval ∘ ωs) using 1 <;> simp · omega · rcases hlinsolve : linsolve (BerlekampWelchMatrix e k ωs f) (Rhs e ωs f) · simp only [reduceCtorEq]; exact linsolve_always_some_berlekamp_welch h_deg h_dist hlinsolve · by_cases hp : p = 0 · have : ‖f‖₀ ≤ e := by aesop omega · have h_cond := linsolve_to_BerlekampWelch_condition hlinsolve have h := Q'_div_E'_eq_p h_deg he hn h_dist h_inj (solutionToQ_ne_zero (not_le.1 hif) (BerlekampWelchCondition_iff_Solution.2 h_cond) h_inj) hp h_cond simp_all- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/BerlekampWelch/BerlekampWelch.lean:95-119
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