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As RudiC has already stated, 2^5 is 32 (5^2 is 25). And the output my awk script produces for n "1" bits set is the input sample provided on the first line of output followed by ((2^n) - 2) lines...

If I correctly understand what you're trying to do (and I'm not sure that I do), you might want to try:
#! /bin/bash

input_string="${1:-"bk_00000000011100000000000"}"

echo "$input_string" |...