Just don't forget the .223 is a tiny, tiny round, it draws all its energy from the high muzzle velocity. If you go way down on barrel length you take away its only good feature. While a small rifle is awesome and maneuverable if you end up burning half the powder outside the barrel you get an enormous amount of flash and decreased energy.
A friend of mine has a SIG 556 with, if I remember correctly, an 8'' barrel. It throws so much flame it nearly blinds the shooter and the blast is atrocious. Its a pretty useless little thing.
223 barrel length muzzle velocity chart: (left)
I've always found 14.5 a nice compromise and would not go below 12''