Malfunction in the Peterson-Gorenstein- Zierler decoder

Most versions of the Peterson-Gorenstein-Zierler (PGZ) decoding algorithm are not true bounded distance decoding algorithms in the sense that when a received vector is not in the decoding sphere of any codeword, the algorithm does not always declare a decoding failure. For a t-error-correcting BCH code, if the received vector is at distance i, i t, from a codeword in a supercode with BCH distance t+ i+ 1, the decoder will output that codeword from the supercode. If that codeword is not a member of the t-error-correcting code, then decoder malfunction is said to have occurred. We describe the necessary and sufficient conditions for decoder malfunction and show that malfunction can be avoided in the PGZ decoder by checking t equations, where is the number of errors hypothesized by the decoder. A formula for the probability of decoder malfunction is also given, and the significance of decoder malfunction is considered for PGZ decoders and high speed Berlekamp-Massey decoders. Keywords—Decoder malfunction, bounded-distance decoding, BCH coding, Reed-Solomon coding, Peterson-Gorenstein-Zierler algorithm.