Explicit measurements with almost optimal thresholds for compressed sensing

We consider the deterministic construction of a measurement matrix and a recovery method for signals that are block sparse. A signal that has dimension N = nd, which consists of n blocks of size d, is called (s, d)-block sparse if only s blocks out of n are nonzero. We construct an explicit linear mapping Φ that maps the (s, d)-block sparse signal to a measurement vector of dimension M, where s•d <N(1-(1-M/N)^(d/(d+1))-o(1). We show that if the (s, d)- block sparse signal is chosen uniformly at random then the signal can almost surely be reconstructed from the measurement vector in O(N^3) computations

List decoding of the biorthogonal wavelet code with predetermined code distance on a field of odd characteristic

Представлено теоретическое обоснование возможности списочного декодирования для биортогональных вейвлет-кодов W[n,n/2,d] с заданным кодовым расстоянием над полями нечётной характеристики. Для входного сообщения длины n задача списочного декодирования заключается в нахождении всех кодовых слов, расстояние Хэмминга до которых не превосходит заданного значения. Для кода W[n,n/2,d] эта задача сводится к задаче списочного декодирования для кода Рида — Соломона RS[n, n — d + 1] посредством преобразования входящего сообщения и последующего применения к его результатам улучшенной версии алгоритма Гурусвами — Судана. Результаты декодирования для кода W[n,n/2,d] находятся путём решения системы линейных уравнений относительно коэффициентов информационного многочлена, полученной из преобразования Фурье кодового слова вейвлет-кода, для каждого найденного информационного слова кода RS[n, n—d+1], являющегося в решаемой системе столбцом свободных членов. Приведены примеры результатов списочного декодирования для кода W[26,13,12], на которых длина результирующего списка равна 2

Reed-Solomon list decoding from a system-theoretic perspective

In this paper, the Sudan-Guruswami approach to list decoding of Reed-Solomon (RS) codes is cast in a system-theoretic framework. With the data, a set of trajectories or time series is associated which is then modeled as a so-called behavior. In this way, a connection is made with the behavioral approach to system theory. It is shown how a polynomial representation of the modeling behavior gives rise to the bivariate interpolating polynomials of the Sudan-Guruswami approach. The concept of "weighted row reduced" is introduced and used to achieve minimality. Two decoding methods are derived and a parametrization of all bivariate interpolating polynomials is given

Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts

We compute minimal bases of solutions for a general interpolation problem, which encompasses Hermite-Pad\'e approximation and constrained multivariate interpolation, and has applications in coding theory and security. This problem asks to find univariate polynomial relations between $m$ vectors of size $\sigma$; these relations should have small degree with respect to an input degree shift. For an arbitrary shift, we propose an algorithm for the computation of an interpolation basis in shifted Popov normal form with a cost of $\mathcal{O}\tilde{~}(m^{\omega-1} \sigma)$ field operations, where $\omega$ is the exponent of matrix multiplication and the notation $\mathcal{O}\tilde{~}(\cdot)$ indicates that logarithmic terms are omitted. Earlier works, in the case of Hermite-Pad\'e approximation and in the general interpolation case, compute non-normalized bases. Since for arbitrary shifts such bases may have size $\Theta(m^2 \sigma)$, the cost bound $\mathcal{O}\tilde{~}(m^{\omega-1} \sigma)$ was feasible only with restrictive assumptions on the shift that ensure small output sizes. The question of handling arbitrary shifts with the same complexity bound was left open. To obtain the target cost for any shift, we strengthen the properties of the output bases, and of those obtained during the course of the algorithm: all the bases are computed in shifted Popov form, whose size is always $\mathcal{O}(m \sigma)$. Then, we design a divide-and-conquer scheme. We recursively reduce the initial interpolation problem to sub-problems with more convenient shifts by first computing information on the degrees of the intermediate bases.Comment: 8 pages, sig-alternate class, 4 figures (problems and algorithms

Iterative Algebraic Soft-Decision List Decoding of Reed-Solomon Codes

In this paper, we present an iterative soft-decision decoding algorithm for Reed-Solomon codes offering both complexity and performance advantages over previously known decoding algorithms. Our algorithm is a list decoding algorithm which combines two powerful soft decision decoding techniques which were previously regarded in the literature as competitive, namely, the Koetter-Vardy algebraic soft-decision decoding algorithm and belief-propagation based on adaptive parity check matrices, recently proposed by Jiang and Narayanan. Building on the Jiang-Narayanan algorithm, we present a belief-propagation based algorithm with a significant reduction in computational complexity. We introduce the concept of using a belief-propagation based decoder to enhance the soft-input information prior to decoding with an algebraic soft-decision decoder. Our algorithm can also be viewed as an interpolation multiplicity assignment scheme for algebraic soft-decision decoding of Reed-Solomon codes.Comment: Submitted to IEEE for publication in Jan 200

Multi-Trial Guruswami–Sudan Decoding for Generalised Reed–Solomon Codes

An iterated refinement procedure for the Guruswami--Sudan list decoding algorithm for Generalised Reed--Solomon codes based on Alekhnovich's module minimisation is proposed. The method is parametrisable and allows variants of the usual list decoding approach. In particular, finding the list of \emph{closest} codewords within an intermediate radius can be performed with improved average-case complexity while retaining the worst-case complexity.Comment: WCC 2013 International Workshop on Coding and Cryptography (2013