International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 90

Lattice centring is an instance of the duality between periodization and decimation: the extra translational periodicity of ρ induces a decimation of described by the `reflection conditions' on h. As was pointed out in Section 1.3.4.2.2.3, nonprimitive lattices are introduced in order to retain the same matrix representation for a given geometric symmetry operation in all the arithmetic classes in which it occurs. From the computational point of view, therefore, the main advantage in using centred lattices is that it maximizes decomposability (Section 1.3.4.2.2.4); reindexing to a primitive lattice would for instance often destroy the diagonal character of the matrix representing a dyad.
In the usual procedure involving three successive onedimensional transforms, the loss of efficiency caused by the duplication of densities or the systematic vanishing of certain classes of structure factors may be avoided by using a multiplexing/demultiplexing technique (Ten Eyck, 1973):
The threedimensional factorization technique of Section 1.3.4.3.4.1 is particularly well suited to the treatment of centred lattices: if the decimation matrix of N contains as a factor a matrix which `integerizes' all the nonprimitive lattice vectors, then centring is reflected by the systematic vanishing of certain classes of vectors of decimated data or results, which can simply be omitted from the calculation. An alternative possibly is to reindex on a primitive lattice and use different representative matrices for the symmetry operations: the loss of decomposability is of little consequence in this threedimensional scheme, although it substantially complicates the definition of the cocycles and .
References
Ten Eyck, L. F. (1973). Crystallographic fast Fourier transforms. Acta Cryst. A29, 183–191.