Decoding the 6-error-correcting Z4-linear Calderbank-McGuire code

Calderbank and McGuire discovered 2 remarkable Z₄ -linear codes (see Des. Codes Cryptography, vol.10, no.2, Feb. 1997 and IEEE Trans. Inform. Theory, vol.42, no.1, p.217-26, Jan.1996). The binary Gray images of these codes have respective parameters (64,2 ³⁷,12) and (64,2³²,14) and thus have 2 (resp. 4) times as many codewords as the best known linear codes of the same length and minimum distance. A decoding algorithm for the 5-error-correcting code is given by C. Rong et al. (see IEEE Trans. Inform. Theory, vol.45, no.5, p.1423-34, July 1999). The approach there (following the ideas of the pioneers of Z₄-codes) is to split the study into several cases according to the Lee type of the error vector. Then the Galois ring algebra is used to decide, whether the syndromes are compatible with an error vector of the prescribed type. Unfortunately, it seems to be very difficult to apply this method to the case of the six-error-correcting code. A different approach to that of C. Rong et al. is required. Using the ideas presented here it is easy to also develop a list decoding algorithm for the 5-error-correcting code. The author discusses this possibility