Only if they are non-negative integers...

The positive integers can be mapped 1-to-1 on the entire set of integers (both sets have the same cardinality). You can construct a mapping so that the bit index is unique for any integer. For example:

my $index = ( $integer >= 0 )
            ? 2 * $index
            : -2 * $index - 1;
[download]

Then $index is non-negative even for positive integers, and non-negative odd for negative integers.

BTW, there is an interesting idea for Bottle Golf at Aleph-0 on Mathworld.