design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

fast hadamard transform based on a simple matrix factorization this correspondence presents a simple factorization of the hadamard matrix which is used to develop a fast algorithm for the hadamard transform the matrix decomposition is of the form of the kronecker products of identity matrices and successively lower order hadamard matrices this decomposition leads very clearly to a sparse matrix factorization of the hadamard matrix

recursive algorithm for sliding walsh hadamard transform in this paper a recursive sliding transform rst algorithm is proposed for the fast implementation of the walsh hadamard transform wht on sliding windows the proposed rst algorithm accelerates the sliding wht process by recursively reducing the order of wht theoretical analysis shows that the proposed rst algorithm requires only 1 33 additions per sample for each wht basis vector regardless of the size or dimension of the basis vector the computational requirement of the proposed rst algorithm is the lowest among existing fast sliding wht algorithms

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

super character theory and comparison arguments for a random walk on the upper triangular matrices abstract consider the random walk on the n n upper triangular matrices with ones on the diagonal and elements over f p where we pick a row at random and either add it or subtract it from the row directly above it the main result of this paper is to prove that the dependency of the mixing time on p is p 2 this is proven by combining super character theory and comparison theory arguments

expander graph arguments for message passing algorithms we show how expander based arguments may be used to prove that message passing algorithms can correct a linear number of erroneous messages the implication of this result is that when the block length is sufficiently large once a message passing algorithm has corrected a sufficiently large fraction of the errors it will eventually correct all errors this result is then combined with known results on the ability of message passing algorithms to reduce the number of errors to an arbitrarily small fraction for relatively high transmission rates the results hold for various message passing algorithms including gallager s hard decision and soft decision with clipping decoding algorithms our results assume low density parity check ldpc codes based on an irregular bipartite graph

comparison of constructions of irregular gallager codes the low density parity check codes whose performance is closest to the shannon limit are gallager codes based on irregular graphs we compare alternative methods for constructing these graphs and present two results first we find a super poisson construction which gives a small improvement in empirical performance over a random construction second whereas gallager codes normally take n sup 2 time to encode we investigate constructions of regular and irregular gallager codes that allow more rapid encoding and have smaller memory requirements in the encoder we find that these fast encoding gallager codes have equally good performance

efficient erasure correcting codes we introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discrete time random process as a result we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability by carefully designing these graphs we can construct for any given rate r and any given real number spl epsiv a family of linear codes of rate r which can be encoded in time proportional to ln 1 spl epsiv times their block length n furthermore a codeword can be recovered with high probability from a portion of its entries of length 1 spl epsiv rn or more the recovery algorithm also runs in time proportional to n ln 1 spl epsiv our algorithms have been implemented and work well in practice various implementation issues are discussed

critical loci for higgs bundles the paper studies the locus in the moduli space of rank 2 higgs bundles over a curve of genus g corresponding to points which are critical for d of the poisson commuting functions defining the integrable system these correspond to the higgs field vanishing on a divisor d of degree d the degree d critical locus is shown to have an induced integrable system related to k d twisted higgs bundles it is embedded in the singular part of the fibration and a description of these singular fibres using hecke curves is given the methods used are also applied to give information about the cohomology classes of components of the nilpotent cone and of the first critical locus it is shown that whereas in the extreme case d 2g 2 the locus is a hyperkahler submanifold this does not hold in general the example of genus 2 is studied concretely and the d 1 integrable system is seen to be described by a pencil of kummer surfaces

linear time encodable and decodable error correcting codes we present a new class of asymptotically good linear error correcting codes these codes can be both encoded and decoded in linear time they can also be encoded by logarithmic depth circuits of linear size and decoded by logarithmic depth circuits of size o nlogn we present both randomized and explicit constructions of these codes

real time pattern matching using projection kernels a novel approach to pattern matching is presented in which time complexity is reduced by two orders of magnitude compared to traditional approaches the suggested approach uses an efficient projection scheme which bounds the distance between a pattern and an image window using very few operations on average the projection framework is combined with a rejection scheme which allows rapid rejection of image windows that are distant from the pattern experiments show that the approach is effective even under very noisy conditions the approach described here can also be used in classification schemes where the projection values serve as input features that are informative and fast to extract

a fast hadamard transform for signals with sublinear sparsity in the transform domain in this paper we design a new iterative low complexity algorithm for computing the walsh hadamard transform wht of an n dimensional signal with a k sparse wht we suppose that n is a power of two and k o n alpha scales sublinearly in n for some alpha in 0 1 assuming a random support model for the nonzero transform domain components our algorithm reconstructs the wht of the signal with a sample complexity o k log 2 n k and a computational complexity o k log 2 k log 2 n k moreover the algorithm succeeds with a high probability approaching 1 for large dimension n our approach is mainly based on the subsampling aliasing property of the wht where by a carefully designed subsampling of the time domain signal a suitable aliasing pattern is induced in the transform domain we treat the resulting aliasing patterns as parity check constraints and represent them by a bipartite graph we analyze the properties of the resulting bipartite graphs and borrow ideas from codes defined over sparse bipartite graphs to formulate the recovery of the nonzero spectral values as a peeling decoding algorithm for a specific sparse graph code transmitted over a binary erasure channel this enables us to use tools from coding theory belief propagation analysis to characterize the asymptotic performance of our algorithm in the very sparse alpha in 0 1 3 and the less sparse alpha in 1 3 1 regime comprehensive simulation results are provided to assess the empirical performance of the proposed algorithm

ldpc codes for 2d arrays binary codes over 2d arrays are very useful in data storage where each array column represents a storage device or unit that may suffer failure in this paper we propose a new framework for probabilistic construction of codes on 2d arrays instead of a pure combinatorial erasure model used in traditional array codes we propose a mixed combinatorial probabilistic model of limiting the number of column failures and assuming a binary erasure channel in each failing column for this model we give code constructions and detailed analysis that allow sustaining a large number of column failures with graceful degradation in the fraction of erasures correctable in failing columns another advantage of the new framework is that it uses low complexity iterative decoding the key component in the analysis of the new codes is to analyze the decoding graphs induced by the failed columns and infer the decoding performance as a function of the code design parameters as well as the array size and failure parameters a particularly interesting class of codes called probabilistically maximum distance separable mds array codes gives fault tolerance that is equivalent to traditional mds array codes the results also include a proof that the 2d codes outperform standard 1d low density parity check codes

euler hadamard dct polynomial matrix abstract the euler hadamard dct polynomial is defined in this paper this polynomial is similar to the euler theorem in that it calculates the unit operation however the euler hadamard dct polynomial is computed by using matrix operations and angle information the computations of euler hadamard dct polynomial can be used to construct higher order of hadamard dct matrices and other real orthogonal matrices specially the inverses of these euler hadamard polynomials are simply from the element inverse and the basic idea is corresponding to the polynomial function x n x n t ni n with hadamard computations h n h n t n i n which is the unit operation of orthogonal matrix otherwise from the geometric view we give a briefly description to the euler hadamard dct polynomial the geometric structure shows that there possibly exist some other orthogonal or element wise inverse matrices or polynomials by using the generalized euler hadamard dct polynomial