Consider building a hash of all 256 possible strings.

#!/usr/bin/perl
use strict;

my %convert=();
my %char = (
  '00'=>'a', '01'=>'b',
  '10'=>'c', '11'=>'d',
);
for my $n (0..255){
  my $bin = sprintf "%08b",$n;
  (my $str = $bin) =~ s/(..)/$char{$1}/eg;
  $convert{$str} = int $bin;
}

for ( qw( aaaa abcd dcba abab cdcd abba caab dddd) ){
  print $_.' '.$convert{$_}."\n";
}
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