I have a toilet drain in a straight line from the toilet to the basement (length about six feet). The stack is close to the toilet, and the sink drains into the stack. I would like to relocated the stack about 15 inches, to move it into a wall. This would require a short horizontal run for the stack from the toilet drain to the new location of the vertical stack. The sink would still drain into the stack.

Can there be a short run of horizontal pipe as part of a toilet vent/stack system?

Thanks

So will the sink still be draining into this stack, if so no horizontal run but a slope of 1/4"/f must be maintained.

I'm a littoral guy so when you say
I have a toilet drain in a straight line from the toilet to the basement (length about six feet). you're telling me that your toilet doesn't connect to the stack but drops directly into the basement? And that the basement's only 6 feet under the toilet?
If you're asking if your toilet can move next to the stack and still be vented the answer is yes. Is your first statement correct or did I misunderstand it? Cheers, Tom