A symbol-based algorithm for decoding bar codes

We investigate the problem of decoding a bar code from a signal measured with a hand-held laser-based scanner. Rather than formulating the inverse problem as one of binary image reconstruction, we instead incorporate the symbology of the bar code into the reconstruction algorithm directly, and search for a sparse representation of the UPC bar code with respect to this known dictionary. Our approach significantly reduces the degrees of freedom in the problem, allowing for accurate reconstruction that is robust to noise and unknown parameters in the scanning device. We propose a greedy reconstruction algorithm and provide robust reconstruction guarantees. Numerical examples illustrate the insensitivity of our symbology-based reconstruction to both imprecise model parameters and noise on the scanned measurements.Comment: 24 pages, 12 figure

Improved linear programming decoding of LDPC codes and bounds on the minimum and fractional distance

We examine LDPC codes decoded using linear programming (LP). Four contributions to the LP framework are presented. First, a new method of tightening the LP relaxation, and thus improving the LP decoder, is proposed. Second, we present an algorithm which calculates a lower bound on the minimum distance of a specific code. This algorithm exhibits complexity which scales quadratically with the block length. Third, we propose a method to obtain a tight lower bound on the fractional distance, also with quadratic complexity, and thus less than previously-existing methods. Finally, we show how the fundamental LP polytope for generalized LDPC codes and nonbinary LDPC codes can be obtained.Comment: 17 pages, 8 figures, Submitted to IEEE Transactions on Information Theor

MIMO Detection for High-Order QAM Based on a Gaussian Tree Approximation

This paper proposes a new detection algorithm for MIMO communication systems employing high order QAM constellations. The factor graph that corresponds to this problem is very loopy; in fact, it is a complete graph. Hence, a straightforward application of the Belief Propagation (BP) algorithm yields very poor results. Our algorithm is based on an optimal tree approximation of the Gaussian density of the unconstrained linear system. The finite-set constraint is then applied to obtain a loop-free discrete distribution. It is shown that even though the approximation is not directly applied to the exact discrete distribution, applying the BP algorithm to the loop-free factor graph outperforms current methods in terms of both performance and complexity. The improved performance of the proposed algorithm is demonstrated on the problem of MIMO detection

Parallel vs. Sequential Belief Propagation Decoding of LDPC Codes over GF(q) and Markov Sources

A sequential updating scheme (SUS) for belief propagation (BP) decoding of LDPC codes over Galois fields, $GF(q)$, and correlated Markov sources is proposed, and compared with the standard parallel updating scheme (PUS). A thorough experimental study of various transmission settings indicates that the convergence rate, in iterations, of the BP algorithm (and subsequently its complexity) for the SUS is about one half of that for the PUS, independent of the finite field size $q$. Moreover, this 1/2 factor appears regardless of the correlations of the source and the channel's noise model, while the error correction performance remains unchanged. These results may imply on the 'universality' of the one half convergence speed-up of SUS decoding

Near-Linear Time Insertion-Deletion Codes and (1+ε\varepsilon)-Approximating Edit Distance via Indexing

We introduce fast-decodable indexing schemes for edit distance which can be used to speed up edit distance computations to near-linear time if one of the strings is indexed by an indexing string $I$. In particular, for every length $n$ and every $\varepsilon >0$, one can in near linear time construct a string $I \in \Sigma'^n$ with $|\Sigma'| = O_{\varepsilon}(1)$, such that, indexing any string $S \in \Sigma^n$, symbol-by-symbol, with $I$ results in a string $S' \in \Sigma''^n$ where $\Sigma'' = \Sigma \times \Sigma'$ for which edit distance computations are easy, i.e., one can compute a $(1+\varepsilon)$-approximation of the edit distance between $S'$ and any other string in $O(n \text{poly}(\log n))$ time. Our indexing schemes can be used to improve the decoding complexity of state-of-the-art error correcting codes for insertions and deletions. In particular, they lead to near-linear time decoding algorithms for the insertion-deletion codes of [Haeupler, Shahrasbi; STOC `17] and faster decoding algorithms for list-decodable insertion-deletion codes of [Haeupler, Shahrasbi, Sudan; ICALP `18]. Interestingly, the latter codes are a crucial ingredient in the construction of fast-decodable indexing schemes