Nobody has spotted the obvious, that the code for reading the file is painfully slow.

A significant improvement would be:

while (read($fh, my $char, 1) && $eof) {
    
    if ($char =~ m/[ACGT]/ ) {
        $i++;

        if ($i >= $start_genome && $i < $end_genome) {
            push (@genome, $char);        
        } elsif ($i > $end_genome) {
            $eof = 0;
                # do the searching here, instead of outside the loop!
        }
        
   }
}
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