.224 is a correct size bullet for reloading .223/5.56
77 grains is almost as heavy as it gets in .223; usually used for long distance competition shooting. Some of these bullets may not feed properly in standard AR mags, and some can only be used with specially throated match barrels. Is this the use you have in mind?
Some times it can also be tricky figuring out how to seat these very long bullets, and may require making special measurements of your gun's freebore.
Pardon me if I am butting my nose in unwarranted, but I was just wondering because this is an unusual choice for someone who is new to reloading.