Minimum Feedback for Collision-Free Scheduling in Massive Random Access

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