3 Elias-bassalygo Theorem
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چکیده
Problem Set 1 is due next Wednesday. 2 Overview Today we will do the following. • Prove the Elias-Bassalygo-Johnson Bound introduced in the previous lecture. Recall that the proof consists of the two lemmas described roughly below – List decodable codes have poor rate. – Good error correcting codes are nicely decodable. • Introduce some algebraic codes along with some helpful idea from algebra. The following is the Elias Bassalygo Theorem. Note that this version of the theorem is for the case q = 2. Although a similar statement exists for all q, the proof is significantly more involved. Theorem 1. Any code of relative distance δ has rate R ≤ 1 − H(τ) where τ ≤ 1 2 (1 − √ 1 − 2δ). Recall that H is the entropy function defined a few lectures ago. The bound on τ comes in part from the solution for τ in the equation 2τ (1 − τ) = δ. The proof of this theorem consists of the following two lemmas. Lemma 2. A (τ, poly(n))-list decodable code has rate R ≤ 1 − H(τ). Before continuing on to the next lemma we will define list decodable codes.
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تاریخ انتشار 2013