lect19 - Error Correcting Codes Combinatorics Algorithms...

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Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 19: Elias-Bassalygo Bound October 10, 2007 Lecturer: Atri Rudra Scribe: Michael Pfetsch & Atri Rudra In the last lecture, we saw the q-ary version of the Johnson bound on the rate and distance of a code, which we repeat below. 1 Johnson bound Theorem 1.1 (Johnson Bound) . Let C [ q ] n be a code of distance d . If ρ < J q ( d n ) , then C is a ( ρ, qdn ) -list decodable code, where the function J q ( δ ) is defined as J q ( δ ) = 1 - 1 q 1 - 1 - q - 1 . Recall that the best upper bound on R (in terms of δ ) that we have seen so far is a combination of the Plotkin and Hamming bounds (see Figure 1). 2 Elias-Bassalygo bound We begin with the statement of a new upper bound on the rate called the Elias-Bassalygo bound. Theorem 2.1 (Elias-Bassalygo bound) . Every q -ary code of rate R , distance δ , and large enough block length, satisfies the following: R 1 - H q ( J q ( δ )) + o (1) The proof of theorem above uses the following lemma: Lemma 2.2. Given a q -ary code, C [ q ] n and 0 e n , there exists a Hamming ball of radius e with at least | C | V ol q ( 0 ,e ) q n codewords in it. Proof. We will prove the existence of the required Hamming ball by the probabilistic method. Pick a received word y [ q ] n at random. It is easy to check that the expected value of | B q ( y , e ) C | is | C | V ol q ( 0 ,e ) q n . (We have seen this argument earlier when we proved the negative part of the list decoding capacity.) This implies the existence of a y [ q ] n such that | B q ( y , e ) C | ≥ | C | V ol q ( 0 , e ) q n , as desired. 1
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Figure 1: Singleton, Hamming, Plotkin, GV and Elias-Bassalygo bounds on rate versus distance for binary codes. The Elias-Bassalygo bound is shown in red.
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