Asymptotic Error Free Partitioning Over Noisy Boolean Multiaccess Channels

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In this paper, we consider the problem of partitioning active users in a manner that facilitates multi-access without collision. The setting is of a noisy, synchronous, Boolean, and multi-access channel, where K active users (out of a total of N users) seek channel access. A solution to the partition problem places each of the N users in one of K groups (or blocks), such that no two active nodes are in the same block. We consider a simple, but non-trivial and illustrative, case of K=2 active users and study the number of steps T used to solve the partition problem. By random coding and a suboptimal decoding scheme, we show that for any T ≥ (C1 + ξ1)log N, where C1 and ξ1 are positive constants (independent of N), and where ξ1 can be arbitrary small, the partition problem can be solved with error probability Pe(N) → 0, for large N. Under the same scheme, we also bound T from the other direction, establishing that, for any T ≤ (C2 - ξ2)log N, the error probability Pe(N) → 1 for large N; again, C2 and ξ2 are constants, and ξ2 can be arbitrarily small. These bounds on the number of steps are lower than the tight achievable lower bound in terms of T ≥ (Cg + ξ)log N for group testing (in which all active users are identified, rather than just partitioned). Thus, partitioning may prove to be a more efficient approach for multi-access than group testing.

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IEEE Transactions on Information Theory

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