So totally non committal, but I managed to fix my Note 10.1 today I hope it works for you.
I had been trying to upgrade to android 4.4.2 using a custom rooted rom but got stuck in a boot loop with my only option as download mode (no CWM, no reset, nothing just bootloop download only).
As a last resort I downloaded a Note 10.1Android 4.0.4 pit file and flashed it from odin 3.07 (pit file selected, Auto Reboot ticked, Re-partition ticked, F.Reset Time ticked - everything else unticked). Flashed the pit file, let the tablet reset and then it picked up the android update as though all was fine. The device is now working fine and without issue.
The .pit file i used is attached. Cant promise it will work, but I hope you have the same luck I do.
I gave it a shot but I"m getting "No PIT partition found" or something like that when I try. What rom did you flash with the PIT file?