A multisolution method of phase determination by combined maximization of entropy and likelihood. VI. The use of error-correcting codes as a source of phase permutation and their application to the phase problem in powder, electron and macromolecular crystallography

The use of error-correcting codes as a source of efficient designs of phase permutation schemes is described. Three codes are used, all taken from the Bricogne BUSTER program [Bricogne (1993). Acta Cryst. D49, 37-60]: the Hamming [7, 4, 3], the Nordström-Robinson (16, 256, 6) and the Golay [24, 12, 8] or its punctured [23, 12, 7] form. These are used in a maximum-entropy-likelihood phasing environment to carry out phase permutation of basis-set reflections instead of the usual quadrant permutation or magic integer approaches. The use of codes in this way inevitably introduces some errors in the phase choices, but for most structures this is not significant especially when the gain in sampling efficiency is considered. For example, the Golay [24, 14, 8] allows the permutation of 24 centric phases in such a way that only 4096 phase sets are produced instead of 2²⁴ 16 777 216, and one of these sets has, at most, only four wrong phases. The method is successfully applied to three powder diffraction data sets of increasing complexity, and with increasing degrees of overlap {Mg₃BN₃, Sigma-2 ([Si₆₄O₁₂₈]4C₁₀H₁₇N) and the NU-3 zeolite}, a sparse electron diffraction data set for buckminsterfullerene, C₆₀, and the small protein molecule crambin at 3 Å resolution where 42 reflections are phased with a U-­weighted mean phase error of 58.5°.