lect26 - Error Correcting Codes Combinatorics Algorithms...

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Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 26: Decoding Concatenated Codes October 29, 2007 Lecturer: Atri Rudra Scribe: Michael Pfetsch & Atri Rudra In the last lecture, we saw Justesen’s strongly explicit asymptotically good code. In today’s lecture, we will begin to answer the natural question of whether we can decode concatenated codes up to half their design distance in polynomial time. 1 Decoder for concatenated codes The concatenation of codes, which was introduced in Lecture 24, is illustrated in Figure 1. Figure 1: Code concatenation and unique decoding of the code C out C in . We begin with a natural decoding algorithm for concatenated codes that “reverses” the encod- ing process. In particular, the code first decodes the inner code and then decodes the outer code. For the time being let us assume that we have a polynomial time unique decoding algorithm for the outer code. This leaves us with the task of coming up with a polynomial time decoding algo- rithm for the inner codes. Our task of coming up such a decoder is made easier by the fact that 1
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the polynomial running time needs to be polynomial in the final block length. This in turn implies that we would be fine if we picked a decoding algorithm that runs in time singly exponential in the inner block length as long as the inner block length is logarithmic in the outer code block length. However, note that the latter is what we have assumed so far and thus, we can use the Maximum Likelihood Decoder (or MLD) for the inner code. More formally, let the input to the decoder be the vector y = ( y 1 , ··· ,y N ) [ q n ] N . The decoding algorithm is a two step process: 1. Step 1 : Compute y 0 = ( y 0 1 , 0 N ) [ q k ] N as follows y 0 i = MLD C in ( y i ) 1 i N.
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This note was uploaded on 01/02/2012 for the course CSE 545 taught by Professor Rudra during the Spring '11 term at SUNY Buffalo.

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lect26 - Error Correcting Codes Combinatorics Algorithms...

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