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lect28(1) - Error Correcting Codes Combinatorics Algorithms...

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Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 28: Generalized Minimum Distance Decoding November 5, 2007 Lecturer: Atri Rudra Scribe: Sandipan Kundu & Atri Rudra 1 Decoding From Errors and Erasures So far, we have seen the following result concerning decoding of RS codes: Theorem 1.1. An [ n, k ] q RS code can be corrected from e errors (or s erasures) as long as e < n - k +1 2 (or s < n - k + 1 ) in O ( n 3 ) time. Next, we show that we can get the best of the errors and erasures worlds simultaneously: Theorem 1.2. An [ n, k ] q RS code can be corrected from e errors and s erasures in O ( n 3 ) time as long as 2 e + s < n - k + 1 . (1) Proof. Given a received word y ( F n q ∪{ ? } ) n with s erasures and e errors, let y be the sub-vector with no erasures. This implies y F n - s q , which is a valid received word for an [ n - s, k ] q RS code. Now run the Berlekamp-Welch algorithm on y . It can correct y as long as e < ( n - s ) - k + 1 2 . This condition is implied by (1). Thus, we have proved one can correct e errors under (1). Now we have to prove that one can correct the s erasures under (1). Let z be the output after correcting e errors. Now we extend z to z ( F ∪ { ? } ) n in the natural way. Finally, run the erasure decoding algorithm on z . This works as long as s < ( n - k + 1) , which in turn is true by (1). The time complexity of the algorithm above is O ( n 3 ) as both the Berlekamp-Welch algorithm and the erasure decoding algorithm can be implemented in cubic time. Next, we will use the errors and erasure decoding algorithm above to design decoding algo- rithms for certain concatenated codes that can be decoded up to half their design distance. 2 Generalized Minimum Distance Decoding In the last lecture, we studied the natural decoding algorithm for concatenated codes. In particular, we performed MLD on the inner code and then fed the resulting vector to a unique decoding 1
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algorithm for the outer code. As was mentioned last time, a drawback of this algorithm is that it does not take into account the information that MLD can offer. E.g., the situations where a given inner code received word has a Hamming distance of one vs. (almost) half the inner code distance
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