Minimum Feedback for Collision-Free Scheduling in Massive Random Access

Consider a massive random access scenario in which small set of $k$ active users out large number notation="LaTeX">$n$ potential need to be scheduled notation="LaTeX">$b\ge k$ slots. What is the minimum common feedback needed ensure that scheduling collision-free? Instead naive scheme listing indices order they should transmit, at cost notation="LaTeX">$k\log (n)$ bits, this paper shows for case notation="LaTeX">$b=k$ , rate fixed-length code scales only as notation="LaTeX">$k \log (e)$ plus an additive term notation="LaTeX">$\Theta (\log (n))$ fixed . If variable-length can used, assuming uniform activity among users, average still requires but dependence on reduced notation="LaTeX">$O(1)$ When notation="LaTeX">$b>k$ bits collision-free significantly further reduced. Moreover, similar scaling derived notation="LaTeX">$m$ per slot, when \le mb$ The problem constructing connected perfect hashing family, allows practical codes constructed from algorithms.