How about finding the smallest index that contains a value equal or greater than your target

sub binary_search {
    my ($str, $a) = @_;
    my $l = 0;
    my $h = @$a;
    while ($l < $h) {
        my $p = int (($l + $h) / 2);
        if ($a->[$p] lt $str) {
            $l = $p + 1;
        }
        else {
            $h = $p;
        }
    }
    $l
}
[download]

update: minor changes to the sub.