Relative Unique Decoding Radius RS eq
ReedSolomon.relativeUniqueDecodingRadius_RS_eq
Plain-language statement
Relative unique decoding radius for RS code with arbitrary finite index type ι.
Exact Lean statement
theorem relativeUniqueDecodingRadius_RS_eq [Fintype ι]
{α : ι ↪ F} [Field F] [DecidableEq F] [NeZero n]
(h : n ≤ Fintype.card ι) :
Code.relativeUniqueDecodingRadius (ι := ι) (F := F) (C := ReedSolomon.code α n) =
((1 : ℝ≥0) - n / Fintype.card ι) / 2Formal artifact
Lean source
theorem relativeUniqueDecodingRadius_RS_eq [Fintype ι] {α : ι ↪ F} [Field F] [DecidableEq F] [NeZero n] (h : n ≤ Fintype.card ι) : Code.relativeUniqueDecodingRadius (ι := ι) (F := F) (C := ReedSolomon.code α n) = ((1 : ℝ≥0) - n / Fintype.card ι) / 2 := by have h_card_ne_zero: Fintype.card ι ≠ 0 := by by_contra h_card_eq_zero have h_n_eq_0 : n = 0 := by omega have h_n_ne_0 : n ≠ 0 := by exact Ne.symm (NeZero.ne' n) exact h_n_ne_0 h_n_eq_0 rw [Code.relativeUniqueDecodingRadius, ReedSolomon.dist_eq_of_le h] simp only [Nat.cast_add, Nat.cast_tsub, Nat.cast_one, add_tsub_cancel_right] conv_lhs => rw [NNReal.sub_div, NNReal.sub_div, div_div, mul_comm, ←div_div] rw [div_self (Nat.cast_ne_zero.mpr h_card_ne_zero)] conv_rhs => rw [NNReal.sub_div, div_div, mul_comm, ←div_div]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/ReedSolomon.lean:523-538
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Source project: ArkLib
Person-level attribution pending.
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ArkLib.Lattices.Ajtai.gadgetDecompose_coeff
Plain-language statement
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Source project: ArkLib
Person-level attribution pending.
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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.