Lecture 12 : Near - Capacity Polynomial - Time Codes for BSC p 11 / 08 / 2006

Derandomized GMD can generally be summarized as follows. Each inner block zi is encoded as yi, with some weight wi representing the likelihood that the block was decoded incorrectly. For sorted wi1 < wi2 < . . . win , we erase all blocks past some threshold Θ, leaving only wi1 , wi2, . . . , wij to be decoded. We try this for all distinguishing values of Θ, that is, for all possible values of j. Picking the j that minimizes the distance between that codeword and the received codeword gets the most likely codeword; this is guaranteed to be correct for fewer than dD 2 errors. Note that this requires running the decoding algorithm d/2 times, in order to try at most every possible weight threshold, (0 ≤ wi ≤ d/2). Thus the overall running time is O(d · T imeRS−Decoding).