On the Hardness of Decoding the Gale-Berlekamp Code

The Gale-Berlekamp (in short, GB) code is the dual code of the binary product code in which the horizontal and vertical constituent codes are both the parity code. It is shown that the problem of deciding whether there is a codeword of the GB code within a prescribed distance from a given received word, is NP-complete. The problem remains hard (in a well-defined sense) even if the decoder is allowed unlimited preprocessing that depends only on the code length. While the intractability of maximum-likelihood decoding for specific codes has already been shown by Bruck and Naor and Lobstein, the result herein seems to be the first that shows hardness for familiar (or "natural") codes. In contrast, it is also shown that, with respect to any memoryless binary symmetric channel with crossover probability less than 1/2, maximum-likelihood decoding can be implemented in linear time for all error events except for a portion that occurs with vanishing probability.