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, multi-access channel where $K$ active users (out of a total of $N$ users) seek to 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 $T\geq (C_1 +\xi_1)\log N$, where $C_1$ and $\xi_1$ are positive constants (independent of $N$), and $\xi_1$ can be arbitrary small, the partition problem can be solved with error probability $P_e^{(N)} \to 0$, for large $N$. Under the same scheme, we also bound $T$ from the other direction, establishing that, for any $T \leq (C_2 - \xi_2) \log N$, the error probability $P_e^{(N)} \to 1$ for large $N$; again $C_2$ and $\xi_2$ are constants and $\xi_2$ can be arbitrarily small. These bounds on the number of steps are lower than the tight achievable lower-bound in terms of $T \geq (C_g +\xi)\log N$ 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.Comment: This paper was submitted in June 2014 to IEEE Transactions on Information Theory, and is under review no

Partition Information and its Transmission over Boolean Multi-Access Channels

In this paper, we propose a novel partition reservation system to study the partition information and its transmission over a noise-free Boolean multi-access channel. The objective of transmission is not message restoration, but to partition active users into distinct groups so that they can, subsequently, transmit their messages without collision. We first calculate (by mutual information) the amount of information needed for the partitioning without channel effects, and then propose two different coding schemes to obtain achievable transmission rates over the channel. The first one is the brute force method, where the codebook design is based on centralized source coding; the second method uses random coding where the codebook is generated randomly and optimal Bayesian decoding is employed to reconstruct the partition. Both methods shed light on the internal structure of the partition problem. A novel hypergraph formulation is proposed for the random coding scheme, which intuitively describes the information in terms of a strong coloring of a hypergraph induced by a sequence of channel operations and interactions between active users. An extended Fibonacci structure is found for a simple, but non-trivial, case with two active users. A comparison between these methods and group testing is conducted to demonstrate the uniqueness of our problem.Comment: Submitted to IEEE Transactions on Information Theory, major revisio

Group testing:an information theory perspective

The group testing problem concerns discovering a small number of defective items within a large population by performing tests on pools of items. A test is positive if the pool contains at least one defective, and negative if it contains no defectives. This is a sparse inference problem with a combinatorial flavour, with applications in medical testing, biology, telecommunications, information technology, data science, and more. In this monograph, we survey recent developments in the group testing problem from an information-theoretic perspective. We cover several related developments: efficient algorithms with practical storage and computation requirements, achievability bounds for optimal decoding methods, and algorithm-independent converse bounds. We assess the theoretical guarantees not only in terms of scaling laws, but also in terms of the constant factors, leading to the notion of the {\em rate} of group testing, indicating the amount of information learned per test. Considering both noiseless and noisy settings, we identify several regimes where existing algorithms are provably optimal or near-optimal, as well as regimes where there remains greater potential for improvement. In addition, we survey results concerning a number of variations on the standard group testing problem, including partial recovery criteria, adaptive algorithms with a limited number of stages, constrained test designs, and sublinear-time algorithms.Comment: Survey paper, 140 pages, 19 figures. To be published in Foundations and Trends in Communications and Information Theor

Statistical Mechanics Analysis of LDPC Coding in MIMO Gaussian Channels

Using analytical methods of statistical mechanics, we analyse the typical behaviour of a multiple-input multiple-output (MIMO) Gaussian channel with binary inputs under LDPC network coding and joint decoding. The saddle point equations for the replica symmetric solution are found in particular realizations of this channel, including a small and large number of transmitters and receivers. In particular, we examine the cases of a single transmitter, a single receiver and the symmetric and asymmetric interference channels. Both dynamical and thermodynamical transitions from the ferromagnetic solution of perfect decoding to a non-ferromagnetic solution are identified for the cases considered, marking the practical and theoretical limits of the system under the current coding scheme. Numerical results are provided, showing the typical level of improvement/deterioration achieved with respect to the single transmitter/receiver result, for the various cases.Comment: 25 pages, 7 figure

Non-Coherent Active Device Identification for Massive Random Access

Massive Machine-Type Communications (mMTC) is a key service category in the current generation of wireless networks featuring an extremely high density of energy and resource-limited devices with sparse and sporadic activity patterns. In order to enable random access in such mMTC networks, base station needs to identify the active devices while operating within stringent access delay constraints. In this paper, an energy efficient active device identification protocol is proposed in which active devices transmit On-Off Keying (OOK) modulated preambles jointly and base station employs non-coherent energy detection avoiding channel estimation overheads. The minimum number of channel-uses required by the active user identification protocol is characterized in the asymptotic regime of total number of devices $\ell$ when the number of active devices $k$ scales as $k=\Theta(1)$ along with an achievability scheme relying on the equivalence of activity detection to a group testing problem. Several practical schemes based on Belief Propagation (BP) and Combinatorial Orthogonal Matching Pursuit (COMP) are also proposed. Simulation results show that BP strategies outperform COMP significantly and can operate close to the theoretical achievability bounds. In a partial-recovery setting where few misdetections are allowed, BP continues to perform well