Lecture 28 : Generalized Minimum Distance Decoding November 5 , 2007

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∪ {?}) 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) 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 algorithms for certain concatenated codes that can be decoded up to half their design distance.