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lect30 - Error Correcting Codes Combinatorics Algorithms...

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Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 30: Achieving the BSC p capacity (II) Tuesday, November 6, 2007 Lecturer: Atri Rudra Scribe: Nathan Russell & Atri Rudra In the last lecture, we started with the description of our BSC p capacity achieving code C * , which is a concatenated code C out C in , where C out and C in satisfying the following properties: (i) C out : The outer code with block length N and rate 1 - ε 2 over F 2 k , with k = O (log N ) . Further, the outer code has a unique decoding algorithm D out that can correct at most γ fraction of worst-case errors in time T out ( N ) . (ii) C in : The inner code has dimension k , dimension n and a rate of 1 - H ( p ) - ε/ 2 . Further, there is a decoding algorithm D in that runs in T in ( k ) time and has decoding error probability no more than γ 2 over BSC p . In today’s lecture, we will analyze the properties of C * and also see how to get our hands on C out and C in with the desired properties. For the rest of the lecture, we will assume that p is an absolute constant. Note that this implies that k = Θ( n ) and thus, we will use k and n interchangeably in our asymptotic bounds. Finally, we will use N = nN to denote the block length of C * . 1 Decoding Error Probability We begin this section by analyzing the natural decoding algorithm that we saw in the last lecture: Step 1 : Let y 0 i = D in ( y i ) , 1 i N . Step 2 : Run D out on y 0 = ( y 0 1 , . . . , y 0 N ) . By the properties of D in , for any fixed i , there is an error at y 0 i with probability γ 2 . Each such error is independent, since errors in BSC p itself are independent by definition. Because of this, and by linearity of expectation, the expected number of errors in y 0 is γN 2 . Taken together, those two facts allow us to conclude that, by the Chernoff bound, the probability that the total number of errors will be more than γN is at most e - γN 6 . Since the decoder D out fails only when there are more than γN errors, this is also the decoding error probability. Expressed in asymptotic terms, the error probability is 2 - Ω( γ N n ) .
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