lect27 - Error Correcting Codes: Combinatorics, Algorithms...

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Unformatted text preview: Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 27: Berlekamp-Welch Algorithm October 31, 2007 Lecturer: Atri Rudra Scribe: Michel Kulhandjian In the last lecture, we discussed unique decoding of RS codes and briefly went through the Berlekamp-Welch algorithm. In todays lecture we will study Berlekamp-Welch algorithm in more detail. Recall that the [ n,k,n- k +1] q Reed-Solomon code regards a message as a polynomial P ( X ) of a degree at most k- 1 , and the encoding of a message m = ( m ,...,m k- 1 ) is ( P ( 1 ) ,...,P ( n )) . Here, m i F q , k n q , and P ( X ) = k- 1 i =0 m i X i . Now let us look at the decoding problem of Reed-Solomon codes. Suppose we are given distinct values 1 ,..., n where i F q with received word y = ( y 1 ,...,y n ) F n q and parameters k and e < n- k +1 2 , where e is an upper bound on the number of errors which occurred during transmission. Our goal is to find a polynomial P ( X ) F q [ X ] of degree at most k- 1 , such that P ( i ) 6 = y i for at most e values of i [ n ] (assuming such a P ( X ) exists). Although this problem is quite non-trivial one, a polynomial time solution can be found for this problem. This solution dates back to 1960 when Peterson [2] came up with a decoding algorithm for the more general BCH code that runs in time O ( n 3 ) . Later Berlekamp and Massey sped up this algorithm so that it runs in O ( n 2 ) . There is an implementation using Fast Fourier Transform that runs in time O ( n log n ) . We will not discuss these faster algo- rithms, but will study another algorithm due to Berlekamp and Welch. More precisely, we will use the Gemmell-Sudan description of the Berlekamp-Welch algorithm[1]....
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lect27 - Error Correcting Codes: Combinatorics, Algorithms...

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