lect24 - Error Correcting Codes Combinatorics Algorithms...

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Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 24: Code concatenation October 24, 2007 Lecturer: Atri Rudra Scribe: Yang Wang Recall the following question we have encountered before: Question 0.1. Is there an explicit asymptotically good code? (that is, rate R > 0 and relative distance δ > 0 for small q ). Here, explicit means: (i) polynomial time construction (of some representation of the code), (ii) “super” explicit (like description of RS code). We will answer the question at least in the sense of explicit codes of (i) in this lecture. 1 Code Concatenation Code concatenation was first proposed by Forney[1]. For q 2 , k 1 and Q = q k , consider two codes which we call outer code and inner code: C out : [ Q ] K [ Q ] N , C in : [ q ] k [ q ] n . Note that the alphabet size of C out exactly matches | C in | . Then given m = ( m 1 , . . . , m K ) [ Q ] K , we have the code C out C in : [ q ] kK [ q ] nN defined as C out C in ( m ) = ( C in ( C out ( m ) 1 ) , . . . , C in ( C out ( m ) N )) , where C out ( m ) = C out ( m ) 1 , . . . , m ) N . This construction is also illustrated in Figure 1. We now look at some properties of a concatenated codes.
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