Sparse group testing codes for low-energy massive random access

Huseyin A. Inan, Peter Kairouz, Ayfer Özgür

Abstract: We present random access protocols for machinetype communication where a massive number of low-energy wireless devices want to occasionally transmit short information packets. We focus on the device discovery problem, with extensions to joint discovery and data transmission as well as data transmission without communicating the device identities. We formulate this problem as a combinatorial group testing problem, where the goal is to exactly identify the set of at most d defective items from a pool of n items. We translate the energy constraint at the wireless physical layer to a constraint on the number of tests each item can participate in, and study the resulting “sparse” combinatorial group testing problem. The celebrated result for the combinatorial group testing problem is that the number of tests t can be made logarithmic in n when d = O (log n). However, state-of-the-art group testing codes require the items to be tested w = Ω((d log n)/(log d + log log n)) times. In our sparse setting, we restrict the number of tests each item can participate in by w <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> . We show that t decreases suddenly from n to (d + 1)√n when w <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> is increased from d to d + 1. If w <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> = ld + 1 for any positive integer l such that ld + 1 ≤ l+1√n we can achieve t = (ld + 1)n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/(i+1)</sup> . We also prove a nearly matching lower bound. These results reveal a favorable trade-off between energy and spectral efficiency for random access.