Are you sure ?. Try

#!/usr/bin/perl
use strict;
use Text::Levenshtein qw(distance);

my $file = $ARGV[0];
open IN, '<',$file or die "$!";
chomp(my @all_IDS=<IN>);
close IN;

for (0..$#all_IDS){
  my $id = shift @all_IDS;
  
  my @ld = distance ($id, @all_IDS);
  print join "\t",$id,@all_IDS,@ld,"\n";
  
  push @all_IDS,$id;
}
__DATA__
A
BB
CCC
DDDD
[download]

Result
A       BB      CCC     DDDD    2       3       4
BB      CCC     DDDD    A       3       4       2
CCC     DDDD    A       BB      4       3       3
DDDD    A       BB      CCC     4       4       4