Realization of optimum interleavers

Four realizations of interleavers that reorder a sequence of symbols so that no contiguous sequence of
n_2
symbols in the reordered sequence contains any pair of symbols that were separated by fewer than
n_1
, symbols in the original ordering are introduced. For any
n_1
and
n_2
that satisfy an appropriate relative primeness condition, these interleavers are optimum in the sense that of all possible interleavers providing the indicated symbol separation, one of these four realizations achieves both the minimum possible encoding delay and the minimum possible combined storage capacity for the interleaver and its unscrambler.