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Project-declaredLean 4.31.0 · mathlib@fabf563a7c95

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) / 2

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
theorem relativeUniqueDecodingRadius_RS_eq [Fintype ι]  {α : ι ↪ F} [Field F] [DecidableEq F] [NeZero n]  (h : n  Fintype.card ι) :  Code.relativeUniqueDecodingRadius:= ι) (F := F) (C := ReedSolomon.code α n) =    ((1 : 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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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...

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Person-level attribution pending.

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Project-declaredLean 4.31.0

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Plain-language statement

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cryptographyproof systemscoding theory

Source project: ArkLib

Person-level attribution pending.

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Project-declaredLean 4.31.0

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ArkLib.Lattices.Ajtai.gadgetDecompose_lawful

Plain-language statement

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Source project: ArkLib

Person-level attribution pending.

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