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 Ham ming [7, 4, 3], the Nordstrom-Robinson (16, 256, 6) and the Golay [24, 12, 8] or its punctured [23, 12, 7] form. These are used in a maximum-entropy-l ikelihood phasing environment to carry out phase permutation of basis-set r eflections instead of the usual quadrant permutation or magic integer appro aches. The use of codes in this way inevitably introduces some errors in th e 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 2(24) = 16 777 216, and one o f these sets has, at most, only four wrong phases. The method is successful ly applied to three powder diffraction data sets of increasing complexity, and with increasing degrees of overlap {Mg3BN3, Sigma-2 ([Si64O128]. 4C(10) H(17)N) and the NU-3 zeolite}, a sparse electron diffraction data set for b uckminsterfullerene, C-60, and the small protein molecule crambin at 3 Angs trom resolution where 42 reflections are phased with a U-weighted mean phas e error of 58.5 degrees.