You have a couple of problems.

foreach my $name (readdir DIR) {
        next if /^\./;
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You want next if $name =~ /^\./; instead. The match operator works on $_ by default.

sub process_file {
    my $file = @_;
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That doesn't do what you expect; it assigns the number of elements in @_ to $file. You want either my ($file) = @_; or my $file = shift; or maybe even my $file = $_[0]; (though I don't recommend that last.)

open (FILE, $file) or die "Could not open file: $!";
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Unless your CWD is $dir, you'll need to prepend it to your filename... otherwise it will try to open $file in the current directory.

Oh, you might very well want split ' ', $line instead. That's a special case. Use /\s+/ only if you want to generate a leading null field on lines that begin with space.

Addendum: There are very good (security) reasons to use the three argument form of open() or to, at least, explicitly specify the mode.

-sauoq
"My two cents aren't worth a dime.";