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Random information on the term “BRR”:
Balanced repeated replication is a statistical technique for estimating the sampling variability of a statistic obtained by stratified sampling.
Consider first an idealized situation, where each stratum of our sample contains only two units. Then each half-sample will contain exactly one of these, so that the half-samples share the stratification of the full sample. If there are s strata, we would ideally take all 2s ways of choosing the half-stratum; but if s is large, this may be infeasible.
If fewer half-samples must be taken, they are selected so as to be “balanced” (hence the name of the technique). Let H be a Hadamard matrix of size s, and choose one row per half-sample. (It doesn’t matter which rows; the important fact is that all the rows of H are orthogonal.) Now, for each half-sample, choose which unit to take from each stratum according to the sign of the corresponding entry in H: that is, for half-sample h, we choose the first unit from stratum k if Hhk = −1 and the second unit if Hhk = +1. The orthogonality of rows of H ensures that our choices are uncorrelated between half-samples.