Asymptotic Error Free Partitioning Over Noisy Boolean Multiaccess Channels

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 P_e^(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 P_e^(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 ≥ (C_g + 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.