Lecture 8 : More Results on List Decoding
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چکیده
Proof. Let C be a binary code with minimum relative distance δ. Suppose x ∈ {0, 1} is arbitrary and c1, c2, . . . , cL ∈ C are codewords such that for all i, ∆ (x, ci) < ρn. Since each ci is a codeword, we have that ∆ (ci, cj) ≥ δn for all i 6= j. Let Φ : {0, 1} → Rn be the map sending 0 to 1 and 1 to −1. Define u = Φ (x), and, for all i, define vi = Φ (ci). Let ū = 1 √ n u denote the normalization of u; similarly, for all i, let v̄i = 1 √ n vi.
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