its not the alternator for two main reasons:
1. it has nothing to do with starting the car
2. cars can run for a few minutes off the battery before dying if the alternator goes bad
and a bonus third reason, just for this situation:
3. that you jumped it, drove it, parked it, and restarted it right away shows that its charging--just that the battery can't hold the charge for very long
since the battery is definitely still under warranty, start there. the starter is a wayyyy distant second place because the car is asking for the radio code, meaning that theres inadequate backup voltage (from the battery). if theres a literal grinding noise, like ERRRRRRRRRRRRRR when you try to start it, sure, could be the starter. there will be no mistaking a grinding starter for anything else. if you hear relays making a really fast clicking noise and the gauge cluster has lights and the radio turns on, its the battery.