I would think at that distance, the #3 would be the minimum you'd want. I ran about 75 feet for my 100A subpanel and used #4. I ran all #4 cause the price was right ($0).
Check with you local building dept. or Electrical inspector and do what they say and you'll be good to go.
#4 copper is only good for 85 amps BUT can be breakered at 90 since that is the next breaker size up
as long as the connected load is no more than 85 amps. It is undersized for a 100 amp panel.
Table 310.16
#2 THHN wire is good for 130 amps. That is what I ran for my shop with a #8 ground at 140 ft. #3 THHN is good for 110 amps. 2/0 THHN is good for 195 amps. 2/0 requires a #4 ground.
You can't use the 90c chart for wire ampacities in this situation, you must use the 75c chart. The ampacities you posted are all incorrect.
110.14(C)(1)
but you have to go to #2 for aluminum at 100 amps.
#2 aluminum can be breakered at 90 amps max in this situation, not 100.
Table 310.16
Here's the minimum wire sizes straight out of the code book. Also at 150' I wouldn't really consider voltage drop to be an issue. It really only becomes borderline when you're actually pulling close to 100 amps and even then it's still very close to the recommended 3%.
Copper THWN
Two #3 phases(hot) conductors
One #3 grounded(neutral) conductor
One #8 grounding conductor
Recommended: 1 1/4" pvc minimum
Aluminum USE
Two #1 phases(hot) conductors
One #1 grounded(neutral) conductor
One #6 grounding conductor
Recommended: 2" pvc minimum
A few things to keep in mind:
1)You can
usually get away with dropping the neutral 1-2 sizes with no problems from the inspector, best to ask first though.
2)The USE aluminum can be direct buried and sleeved in pipe where it exits the ground. I would definitely recommend installing it in pipe all the way though, it's good insurance. Just to put things in perspective: If it saves one service call from an electrician to fix damaged wires, it'll pay for itself.
3)Burial depth is 18" to the top of the conduit/direct burial wire. 24" if it passes under a driveway, parking lot, etc.
This is all I could think of, hopefully this will help you.