My Craftsman garage door opener has a roaming/rolling code, and the universal tansmitter works with it.
Homelink's web site lists Genie as a "partner".
HomeLink's Programming FAQ:
I can't program my HomeLink?
You can either call our customer service number (1-800-355-3515) or contact us by email for complete programming assistance. First try the following troubleshooting tips:
** Check the frequency of the device’s hand-held transmitter (usually found on the back). HomeLink is compatible with radio-frequency devices operating between 288-399MHz (some 2001 and newer vehicles may be compatible up to 418MHz. Please contact us for vehicle compatibility).
** Put a brand new, store bought battery in the hand-held transmitter. During the programming process the battery's energy tends to drain (more so with an older battery). A new battery will increase the speed and accuracy in which the hand-held transmitter sends its signal to HomeLink.
** Retrain HomeLink holding the hand-held transmitter at different lengths and angles from the HomeLink button you are trying to program. Try 4-6 inches away at a different angle.
**Cycle the hand-held transmitter during programming: Press and hold the HomeLink button while you press and release the hand-held transmitter every 2 seconds until the indicator light changes from a slow blink to a rapid flash.
**To verify HomeLink has programmed to a non-rolling code ("fixed") device, press and hold the HomeLink button. If the indicator light remains solid, it has picked up the signal and should operate the device. If it remains a slow blink, HomeLink has not picked up the signal from the hand-held remote. Try programming the button again or chose another button to program.
** To verify HomeLink has picked up a rolling code signal: press and hold the programmed HomeLink button. If the indicator light blinks rapidly for two seconds and turns solid, HomeLink has successfully picked up the rolling code signal. Please follow the rolling code instructions outlined in the "programming a garage door opener" section to complete the process.