You should not bed a heavy bull-barrelled Ruger 10/22 carbine like you normally bed a heavy bull-barrelled rifle (by bedding the receiver and either free-floating or bedding the heavy bull-barrel). The Ruger 10/22 aluminum-alloy receiver, when bedded, is not strong enough to support a floated, heavy, bull-barrel. Therefore, you should bed the heavy bull-barrel and float the Ruger 10/22 receiver/action.
Most aftermarket 10/22 sporter stocks have the barrel channel already routed out and ready to accept the epoxy, AND the void that houses the action is made a little larger than the 10/22 action. These two features enable you to bed the bull-barrel and free-float the action, as 10/22 experts recommend. But you must fill any gap at the rear of the receiver to transfer some of the bolt stress to the stock, or you may crack the receiver.
Get some Brownells AcraGlass (it comes in a liquid or gel; the gel is easier to work with) and follow the directions closely. Remember, you are bedding the heavy bull-barrel and not the action, so disregard all the instructions regarding preparation of the stock around the action. There should be no epoxy anywhere near the action.
Fill ALL voids and holes with modeling clay to prevent the AcraGlass gel from flowing into the receiver section. If you get the wet AcraGlass onto a part of the metal that does not have any release agent on it (or in the case of the 10/22, the AcraGlass works itself up under the receiver-barrel lug connection), then you will have an impossible time getting it all apart. Use modeling clay to make a dam in the barrel channel (in front of the lug cutout in the stock) to keep the epoxy out of the V-block recess area where the barrel is secured to the action. There is a chance of developing a mechanical lock there between the stock and the barrelled-action.
Use masking tape on the edges of the stock. Do not line the floor of the V-block recess in the stock with masking tape, because the tape would prevent the V-block from resting in its normal position. In that case, the barrel would not rest as far down as if the tape were absent, creating "play" in the barrelled-action when the action screw was fully tightened.
Use an Exacto knife to cut bedding channels in the stock. Gouge out the stock where you want the AcraGlass to be, giving the AcraGlass something to "bite" into.
For a release agent, use several coats of Johnson floor wax on all other parts. Spread the wax on, buff it to a high shine, and let it dry for 10 minutes. (As an alternative, you can use a very thin coat of non-butter-flavored Pam cooking spray, or a heavy coat of CLP on the barrel and receiver.)
After suitable application of the release agent to the barrel, receiver, and other parts, mix the Brownell's AcraGlass bedding compound (the resin and the hardener). Pour the AcraGlass into the stock and bed the barrel channel for its full length. Keep the AcraGlass away from the V-block recess area in the stock. As an alternative, you could also just build a "bump pad" for the heavy bull-barrel toward the front of the stock, but consistent pressure on the barrel is difficult to achieve.
Bed the rear portion of the stock where the gap is, but do not bed the action. It is very important to transfer some of the stress to the stock, or you could crack the receiver. Either make a "bar" across the rear of the stock by adding a "contact point" at the rear of the receiver in line with the bolt, or simply fill the gap completely (it would probably look better that way anyway).
Put the barrelled-action back into the stock and let the resin cure. Do not screw the action down fully. Tighten the screw to 1/2 turn from its maximum, and after a few hours, loosen the screw a bit and re-tighten to its original position. This leaves a bit more room for the AcraGlass to harden. The screw position will be at the maximum after the AcraGlass hardens, and on the chance the action is bottoming out elsewhere (like at the rear), this prevents a permanent stress on the action.
Before the AcraGlass has completely dried and hardened, use a plastic knife along the masking tape to trim the resin from the edges of the stock.
After allowing suitable time for the resin to cure (at least 24 hours), remove the barreled-action from the stock. Make sure the hammer is cocked and push-button safety is halfway in between safe and off; otherwise, when the barreled-action releases from the bedding material suddenly, the safety will mar the stock.
Removing the barreled-action from the stock after the epoxy has cured can be tricky. If possible, use a padded vise and try using a rubber mallet on the action screw and on the barrel itself.
Once you have removed the barreled-action from the stock, wipe the release agent from the barrel and other parts. Use 1,1,1 Trichloroethane degreaser for best results. Screw the barreled-action back together with the stock, and you're ready to go!