Noise-Resilient Group Testing: Limitations and Constructions

We study combinatorial group testing schemes for learning $d$-sparse Boolean vectors using highly unreliable disjunctive measurements. We consider an adversarial noise model that only limits the number of false observations, and show that any noise-resilient scheme in this model can only approximately reconstruct the sparse vector. On the positive side, we take this barrier to our advantage and show that approximate reconstruction (within a satisfactory degree of approximation) allows us to break the information theoretic lower bound of $\tilde{\Omega}(d^2 \log n)$ that is known for exact reconstruction of $d$-sparse vectors of length $n$ via non-adaptive measurements, by a multiplicative factor $\tilde{\Omega}(d)$. Specifically, we give simple randomized constructions of non-adaptive measurement schemes, with $m=O(d \log n)$ measurements, that allow efficient reconstruction of $d$-sparse vectors up to $O(d)$ false positives even in the presence of $\delta m$ false positives and $O(m/d)$ false negatives within the measurement outcomes, for any constant $\delta < 1$. We show that, information theoretically, none of these parameters can be substantially improved without dramatically affecting the others. Furthermore, we obtain several explicit constructions, in particular one matching the randomized trade-off but using $m = O(d^{1+o(1)} \log n)$ measurements. We also obtain explicit constructions that allow fast reconstruction in time \poly(m), which would be sublinear in $n$ for sufficiently sparse vectors. The main tool used in our construction is the list-decoding view of randomness condensers and extractors.Comment: Full version. A preliminary summary of this work appears (under the same title) in proceedings of the 17th International Symposium on Fundamentals of Computation Theory (FCT 2009

Lower Bounds for Sparse Recovery

We consider the following k-sparse recovery problem: design an m x n matrix A, such that for any signal x, given Ax we can efficiently recover x' satisfying ||x-x'||_1 <= C min_{k-sparse} x"} ||x-x"||_1. It is known that there exist matrices A with this property that have only O(k log (n/k)) rows. In this paper we show that this bound is tight. Our bound holds even for the more general /randomized/ version of the problem, where A is a random variable and the recovery algorithm is required to work for any fixed x with constant probability (over A).Comment: 11 pages. Appeared at SODA 201

Pairwise Check Decoding for LDPC Coded Two-Way Relay Block Fading Channels

Partial decoding has the potential to achieve a larger capacity region than full decoding in two-way relay (TWR) channels. Existing partial decoding realizations are however designed for Gaussian channels and with a static physical layer network coding (PLNC). In this paper, we propose a new solution for joint network coding and channel decoding at the relay, called pairwise check decoding (PCD), for low-density parity-check (LDPC) coded TWR system over block fading channels. The main idea is to form a check relationship table (check-relation-tab) for the superimposed LDPC coded packet pair in the multiple access (MA) phase in conjunction with an adaptive PLNC mapping in the broadcast (BC) phase. Using PCD, we then present a partial decoding method, two-stage closest-neighbor clustering with PCD (TS-CNC-PCD), with the aim of minimizing the worst pairwise error probability. Moreover, we propose the minimum correlation optimization (MCO) for selecting the better check-relation-tabs. Simulation results confirm that the proposed TS-CNC-PCD offers a sizable gain over the conventional XOR with belief propagation (BP) in fading channels.Comment: to appear in IEEE Trans. on Communications, 201

An intelligent genetic algorithm for PAPR reduction in a multi-carrier CDMA wireless system

Abstract— A novel intelligent genetic algorithm (GA), called Minimum Distance guided GA (MDGA) is proposed for peak-average-power ratio (PAPR) reduction based on partial transmit sequence (PTS) scheme in a synchronous Multi-Carrier Code Division Multiple Access (MC-CDMA) system. In contrast to traditional GA, our MDGA starts with a balanced ratio of exploration and exploitation which is maintained throughout the process. It introduces a novel replacement strategy which increases significantly the convergence rate and reduce dramatically computational complexity as compared to the conventional GA. The simulation results demonstrate that, if compared to the PAPR reduction schemes using exhaustive search and traditional GA, our scheme achieves 99.52% and 50+% reduction in computational complexity respectively

Generalised Pattern Matching Revisited

In the problem of $\texttt{Generalised Pattern Matching}\ (\texttt{GPM})$ [STOC'94, Muthukrishnan and Palem], we are given a text $T$ of length $n$ over an alphabet $\Sigma_T$, a pattern $P$ of length $m$ over an alphabet $\Sigma_P$, and a matching relationship $\subseteq \Sigma_T \times \Sigma_P$, and must return all substrings of $T$ that match $P$ (reporting) or the number of mismatches between each substring of $T$ of length $m$ and $P$ (counting). In this work, we improve over all previously known algorithms for this problem for various parameters describing the input instance: * $\mathcal{D}\,$ being the maximum number of characters that match a fixed character, * $\mathcal{S}\,$ being the number of pairs of matching characters, * $\mathcal{I}\,$ being the total number of disjoint intervals of characters that match the $m$ characters of the pattern $P$. At the heart of our new deterministic upper bounds for $\mathcal{D}\,$ and $\mathcal{S}\,$ lies a faster construction of superimposed codes, which solves an open problem posed in [FOCS'97, Indyk] and can be of independent interest. To conclude, we demonstrate first lower bounds for $\texttt{GPM}$. We start by showing that any deterministic or Monte Carlo algorithm for $\texttt{GPM}$ must use $\Omega(\mathcal{S})$ time, and then proceed to show higher lower bounds for combinatorial algorithms. These bounds show that our algorithms are almost optimal, unless a radically new approach is developed

Binary Multilevel Convolutional Codes with Unequal Error Protection Capabilities

Binary multilevel convolutional codes (CCs) with unequal error protection (UEP) capabilities are studied. These codes belong to the class of generalized concatenated (GC) codes. Binary CCs are used as outer codes. Binary linear block codes of short length, and selected subcodes in their two-way subcode partition chain, are used as inner codes. Multistage decodings are presented that use Viterbi decoders operating on trellises with similar structure to that of the constituent binary CCs. Simulation results of example binary two-level CC\u27s are also reported