If the 8820 on this page
is your remote, then there isn't much you can do. Even though this remote is made by UEI, it does not support keymoves. If it did, you could just enter in EFC 15822 and assign the is EFC to a suitable button.
If you really want to use this remote, you could buy/buid a JP2 interface and program it via computer. Or you could buy a RCA RCRP05B (<$20) or get Cox to allow you to use a 7820B. Those can be manually programmed to send the Home IR signal.