Printing "Connection to the socket was successful" doesn't make it so. You're getting the error because $socket is not defined (like the error message says), and it's undefined because your new failed. Use the following construct to get a more meaningful error message:

my $socket = IO::Socket::INET->new(...)
   or die("Unable to create socket: $!\n");
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If you're trying to create a server socket, you need to specify a Listen argument. Furthermore, you should probably omit LocalHost.

If you're trying to create a client socket, you need to specify a PeerAddr and the PeerPort argument, and you should remove LocalHost and LocalPort.