Back to the OP, in my state, a child can only earn a diploma by going to high school and taking the requisite courses. If the child is homeschooled, they can earn a GED. Yes, they can only acquire the graduation certificate of juvenile delinquents.

The child can avoid the social stigma of having a GED in these ways:

1) Earn an advanced degree. Then nobody cares about the GED, because the degree trumps high school. Of course, not everyone goes on to earn a degree.
2) Earn the GED at a young age, and say so in the resume. Because when an employer sees a GED, they assume "graduated at 25," not 15.
3) Homeschool until the child is ready for high school, then enroll and get a diploma. This is typically the point at which public schools have something to offer gifted kids, because AP.

Just something to think about. For my DD, homeschooling is something we've done before and is still very much an option, but if we go back to it, we're thinking of option 3.