lect15 - Error Correcting Codes: Combinatorics, Algorithms...

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Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 15: Gilbert-Varshamov Bound October 2, 2007 Lecturer: Atri Rudra Scribe: Thanh-Nhan Nguyen In the previous lectures, we have only seen upper bounds on the rate of a code (given a fixed relative distance). In this lecture, we study our first (and only) lower bound on rate of a code. 1 Gilbert-Varshamov Bound In today’s lecture we will prove the following result Theorem 1.1. Let q 2 . For every 0 δ < 1 - 1 q , and 0 < ε 1 - H q ( δ ) , there exists a code with rate R 1 - H q ( δ ) - ε , and relative distance δ . The above result was proved for general codes by Gilbert (Section 1.1) and for linear codes by Varshamov (Section 1.2). Hence, the bound is called the Gilbert-Varshamov bound. 1.1 Gilbert Construction Gilbert proved Theorem 1.1 by the following greedy construction (where d = δn ) (i) Start with the empty code: C ← ∅ (ii) If there exists a v [ q ] n such that Δ( v , c ) d
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lect15 - Error Correcting Codes: Combinatorics, Algorithms...

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