lect27 - Error Correcting Codes Combinatorics Algorithms...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 today’s 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 ( α 1 ) ,...,P ( α 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 α 1 ,...,α n where α 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 ( α 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]....
View Full Document

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.

Page1 / 4

lect27 - Error Correcting Codes Combinatorics Algorithms...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online