Universal Feedback Gain for Modulo-Sum Computation Over the Erasure MAC

The problem of computing the modulo-sum of independent messages over a finite-field erasure multiple access channel is investigated. The focus is on the role of delayed state feedback for function computation over state dependent multiple access channels. For the two-user case, a set of new outer bounds on the non-feedback computation capacity region are developed, which strictly improve the state of the art by Khisti
et al.
, 2013. As a result, a previously unsettled question is answered in the affirmative: delayed state feedback strictly increases computation capacity for the two-user erasure multiple access channel universally. The proof leverages the subset entropy inequalities by Madiman and Tetali, 2010, Jiang
et al.
, 2014, and submodularity of conditional entropies. For the achievability part of the non-feedback case, an achievable computation rate region is derived by generalizing the proposed schemes in Khisti
et al.
, 2013. Beyond the two-user case, the
$K$
-user case is also investigated, with emphasis on the large system regime, where
$K$
is the number of users. For the non-feedback case, we propose a grouping scheme which has higher computation rate than that of the conventional “compute-and-forward” (
CF
) scheme. Furthermore, with a growing number of users, the proposed grouping scheme strictly outperforms the conventional “decode-and-forward (
DF
)” scheme when the erasure probability is smaller than
$1-e^{\frac {1}{e}}\approx 0.3078$
. This is in contrast to the two-user case where the currently best known achievability (Khisti
et al.
, 2013) coincides with the better one between
DF
and
CF
. For the case with delayed state feedback, a new hybrid-ARQ-type scheme is proposed, and in the large system regime, it achieves a computation rate scaling like
$\Omega \left({\frac {1}{\log (K)}}\right)$
, much higher than the scaling
$\Theta \left({\frac {1}{K}}\right)$
achieved by the grouping scheme without feedback.