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e5a97c6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 | import OAI.Combinatorics.MatrixRemoval.CanonicalCopyBound
import OAI.Combinatorics.MatrixRemoval.OrderedRepairDistance
import OAI.Combinatorics.MatrixRemoval.SequenceAssembly
import OAI.Combinatorics.MatrixRemoval.Consequence
namespace OAI
noncomputable section
namespace Problem348
theorem counterexample_sequence :
∀ h : ℕ, 1 ≤ h →
let d : ℕ := 386 * h + 2
let n : ℕ := d * 2 ^ h
∃ A : BinaryMatrix n,
fixedDistance A ≥ 1 / ((d : ℝ) ^ 2) ∧
(copyCount fixedH A : ℝ) / ((n : ℝ) ^ 132) ≤
(1 / ((d : ℝ) ^ 2)) * (1 / ((2 : ℝ) ^ h)) := by
apply SequenceAssembly.counterexample_sequence_of_integer_bounds
intro h hh
refine ⟨Construction.hostMatrix (OrderedHost.rowIndex h) (OrderedHost.columnIndex h), ?_,
OrderedHost.canonical_copyCount_le h⟩
intro B hfree
exact OrderedHost.hammingDistance_ge hh B hfree
theorem no_polynomial_removal_bound :
∀ c C : ℝ, 0 < c → 0 < C →
∃ n : ℕ, 1 ≤ n ∧
∃ ε : ℝ, 0 < ε ∧ ε < 1 ∧
∃ A : BinaryMatrix n,
fixedDistance A ≥ ε ∧
(copyCount fixedH A : ℝ) <
c * Real.rpow ε C * ((n : ℝ) ^ 132) := by
exact no_polynomial_removal_bound_of_sequence counterexample_sequence
end Problem348
end
end OAI
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