This posting is older than 6 months and can contain outdated information.
The Apple Remote will send a "low battery" code instead of a button 
code, when its battery is low, the first time after waking from sleep. 
The Apple Remote will then send subsequent presses until the remote 
goes to sleep again. If the battery is very low, it may be sending 
"low battery" codes more frequently.
Although the remote is brand new, you said that the situation improved 
after replacing the battery. I certainly believe you that both the 
battery and the remote are new and 6 feet is well within the range of 
the Apple Remote and the receiver, yet, still, the only thing that can 
explain what you are seeing is the Apple Remote sending a low battery 
signal following the pattern described above.
Best regards, 
Felix Schwarz