lect17 - Error Correcting Codes: Combinatorics, Algorithms...

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Unformatted text preview: Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 17: Proof of a Geometric Lemma October 5, 2007 Lecturer: Atri Rudra Scribe: Sandipan Kundu & Atri Rudra In the last lecture, we proved the Plotkin bound, except for a couple of lemmas which we will prove in this lecture. 1 Geometric Lemma The proof of the Plotkin bound needed the existence of a map from codewords to real vectors with certain properties, which the next lemma guarantees. Lemma 1.1. Let C ⊆ [ q ] n . Then there exists a function f : C-→ R nq such that 1. For every c ∈ C, k f ( c ) k = 1 . 2. For every c 1 6 = c 2 ∈ C, h f ( c 1 ) , f ( c 2 ) i = 1- q q- 1 Δ( c 1 , c 2 ) n . We also used the following lemma, which gives a bound on the number of vectors that can exists such that every pair is at an obtuse angle with each other. Lemma 1.2 (Geometric Lemma) . Let v 1 , v 2 , . . . , v m ∈ R n . 1. If v 1 , v 2 , . . . , v m are all non-zero and h v i , v j i ≤ for all i 6 = j , then m ≤ 2 n . 2. If v 1 , v 2 , . . . , v m are unit vectors and h v i , v j i ≤ - ε < for every i 6 = j , then m ≤ 1 + 1 /ε. Next, we prove the two lemmas above. Proof of Lemma 1.1. We begin by picking a map φ : [ q ] → R q with certain properties. Then we apply φ to all the coordinates of a codeword to define the map...
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lect17 - Error Correcting Codes: Combinatorics, Algorithms...

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