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

on expander codes sipser and spielman see ibid vol 42 p 1717 22 nov 1996 have introduced a constructive family of asymptotically good linear error correcting codes expander codes together with a simple parallel algorithm that will always remove a constant fraction of errors we introduce a variation on their decoding algorithm that with no extra cost in complexity provably corrects up to 12 times more errors

map estimation via agreement on trees message passing and linear programming we develop and analyze methods for computing provably optimal maximum a posteriori probability map configurations for a subclass of markov random fields defined on graphs with cycles by decomposing the original distribution into a convex combination of tree structured distributions we obtain an upper bound on the optimal value of the original problem i e the log probability of the map assignment in terms of the combined optimal values of the tree problems we prove that this upper bound is tight if and only if all the tree distributions share an optimal configuration in common an important implication is that any such shared configuration must also be a map configuration for the original distribution next we develop two approaches to attempting to obtain tight upper bounds a a tree relaxed linear program lp which is derived from the lagrangian dual of the upper bounds and b a tree reweighted max product message passing algorithm that is related to but distinct from the max product algorithm in this way we establish a connection between a certain lp relaxation of the mode finding problem and a reweighted form of the max product min sum message passing algorithm

analysis and design of tuned turbo codes it has been widely observed that there exists a fundamental tradeoff between the minimum hamming distance properties and the iterative decoding convergence behavior of turbo like codes while capacity achieving code ensembles typically are asymptotically bad in the sense that their minimum distance does not grow linearly with block length and they therefore exhibit an error floor at moderate to high signal to noise ratios asymptotically good codes usually converge further away from channel capacity in this paper we introduce the concept of tuned turbo codes a family of asymptotically good hybrid concatenated code ensembles where asymptotic minimum distance growth rates convergence thresholds and code rates can be tradedoff using two tuning parameters and by decreasing the asymptotic minimum distance growth rate is reduced in exchange for improved iterative decoding convergence behavior while increasing raises the asymptotic minimum distance growth rate at the expense of worse convergence behavior and thus the code performance can be tuned to fit the desired application by decreasing a similar tuning behavior can be achieved for higher rate code ensembles

fast decoding of expander codes expander codes are tanner codes defined on sparse graphs that have good expansion properties sipser and spielman 1996 showed that there is a linear time decoding algorithm for expander codes when the vertex expansion is at least 3 4 and the number of errors corrected is a constant fraction of the code length later feldman et al 2007 gave a decoding algorithm that allows the expansion to be 2 3 1 3c where c is the left degree of the underlying bipartite graph at the expense of polynomial time decoding complexity recently viderman 2013 further improved the expansion parameter to 2 3 1 6c and the decoding algorithm runs in linear time these results are for expander codes whose inner codes are parity check codes by using stronger inner codes chilappagari et al 2010 showed that there is a linear time decoding algorithm for every vertex expansion greater than 1 2 in this paper it is shown that for every vertex expansion there is a linear time decoding algorithm for expander codes using inner codes with minimum distance depending on the vertex expansion and that the number of errors corrected is a constant fraction of the code length

confluence expansions of the generalized hypergeometric function by confluencing a subset of upper and lower parameters in the generalized hypergeometric function f p q a 1 a p c 1 c q z with the variable z one obtains a lower order hypergeometric

signal space characterization of iterative decoding by tracing the flow of computations in the iterative decoders for low density parity check codes we formulate a signal space view for a finite number of iterations in a finite length code on a gaussian channel maximum a posteriori map codeword decoding or maximum likelihood decoding decodes to the codeword signal that is closest to the channel output in euclidean distance in contrast we show that iterative decoding decodes to the pseudosignal that has highest correlation with the channel output the set of pseudosignals corresponds to pseudocodewords only a vanishingly small number of which correspond to codewords we show that some pseudocodewords cause decoding errors but that there are also pseudocodewords that frequently correct the deleterious effects of other pseudocodewords

the admm penalized decoder for ldpc codes linear programming lp decoding for low density parity check codes was introduced by feldman et al and has been shown to have theoretical guarantees in several regimes furthermore it has been reported in the literature via simulation and via instanton analysis that lp decoding displays better error rate performance at high signal to noise ratios snr than does belief propagation bp decoding however at low snrs lp decoding is observed to have worse performance than bp in this paper we seek to improve lp decoding at low snrs while maintaining lp decoding s high snr performance our main contribution is a new class of decoders obtained by applying the alternating direction method of multipliers admm algorithm to a set of non convex optimization problems these non convex problems are constructed by adding a penalty term to the objective of lp decoding the goal of the penalty is to make pseudocodewords which are non integer vertices of the lp relaxation more costly we name this class of decoders admm penalized decoders for low and moderate snrs we simulate admm penalized decoding with ell 1 and ell 2 penalties we find that these decoders can outperform both bp and lp decoding for high snrs where it is difficult to obtain data via simulation we use an instanton analysis and find that asymptotically admm penalized decoding performs better than bp but not as well as lp unfortunately since admm penalized decoding is not a convex program we have not been successful in developing theoretical guarantees however the non convex program can be approximated using a sequence of linear programs an approach that yields a reweighted lp decoder we show that a two round reweighted lp decoder has an improved theoretical recovery threshold when compared with lp decoding in addition we find via simulation that reweighted lp decoding significantly attains lower error rates than lp decoding at low snrs

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

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

impact of redundant checks on the lp decoding thresholds of ldpc codes feldman et al 11 asked whether the performance of the linear programming lp decoder can be improved by adding redundant parity checks to tighten the lp relaxation we prove in this paper that for low density parity check codes even if we include all redundant parity checks asymptotically there is no gain in the lp decoder threshold on the binary symmetric channel under certain conditions on the base tanner graph first we show that if the tanner graph has bounded check degree and satisfies a condition which we call asymptotic strength then including high degree redundant parity checks in the lp does not significantly improve the threshold of the lp decoder in the following sense for each constant delta 0 there is a constant k 0 such that the threshold of the lp decoder containing all redundant checks of degree at most k improves by at most delta upon adding to the lp all redundant checks of degree larger than k we conclude that if the graph satisfies an additional condition which we call rigidity then including all redundant checks does not improve the threshold of the base lp we call the graph asymptotically strong if the lp decoder corrects a constant fraction of errors even if the log likelihood ratios of the correct variables are arbitrarily small by building on a construction due feldman et al 9 and its recent improvement by viderman 24 we show that asymptotic strength follows from sufficiently large variable to check expansion we also give a geometric interpretation of asymptotic strength in terms pseudocodewords we call the graph rigid if the minimum weight of a sum of check nodes involving a cycle tends to infinity as the block length tends to infinity under the assumptions that the graph girth is logarithmic and the minimum check degree is at least 3 rigidity is equivalent to the nondegeneracy property that adding at least logarithmically many checks does not give a constant weight check we argue that nondegeneracy is a typical property of random check regular tanner graphs

admm lp decoding of non binary ldpc codes in mathbb f _ 2 m in this paper we develop efficient decoders for non binary low density parity check codes using the alternating direction method of multipliers admm we apply admm to two decoding problems the first problem is linear programming lp decoding in order to develop an efficient algorithm we focus on non binary codes in fields of characteristic two this allows us to transform each constraint in mathbb f 2 m to a set of constraints in mathbb f 2 that has a factor graph representation applying admm to the lp decoding problem results in two types of non trivial sub routines the first type requires us to solve an unconstrained quadratic program we solve this problem efficiently by leveraging new results obtained from studying the above factor graphs the second type requires euclidean projections onto polytopes that are studied in the literature such projections can be solved efficiently using off the shelf techniques which scale linearly in the dimension of the vector to project admm lp decoding scales linearly with block length linearly with check degree and quadratically with field size the second problem we consider is a penalized lp decoding problem this problem is obtained by incorporating a penalty term into the lp decoding objective the purpose of the penalty term is to make non integer solutions pseudocodewords more expensive and hence to improve decoding performance the admm algorithm for the penalized lp problem requires euclidean projection onto a polytope formed by embedding the constraints specified by the non binary single parity check code which can be solved by applying the admm technique to the resulting quadratic program empirically this decoder achieves a much reduced error rate than lp decoding at low signal to noise ratios

message passing algorithms and improved lp decoding linear programming lp decoding for low density parity check codes and related domains such as compressed sensing has received increased attention over recent years because of its practical performance coming close to that of iterative decoding algorithms and its amenability to finite blocklength analysis several works starting with the work of feldman showed how to analyze lp decoding using properties of expander graphs this line of analysis works for only low error rates about a couple of orders of magnitude lower than the empirically observed performance it is possible to do better for the case of random noise as shown by daskalakis and koetter and vontobel building on work of koetter and vontobel we obtain a novel understanding of lp decoding which allows us to establish a 0 05 fraction of correctable errors for rate codes this comes very close to the performance of iterative decoders and is significantly higher than the best previously noted correctable bit error rate for lp decoding our analysis exploits an explicit connection between lp decoding and message passing algorithms and unlike other techniques directly works with the primal linear program an interesting byproduct of our method is a notion of a locally optimal solution that we show to always be globally optimal i e it is the nearest codeword such a solution can in fact be found in near linear time by a reweighted version of the min sum algorithm obviating the need for lp our analysis implies in particular that this reweighted version of the min sum decoder corrects up to a 0 05 fraction of errors

using linear programming to decode binary linear codes a new method is given for performing approximate maximum likelihood ml decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel the decoding algorithm is based on a linear programming lp relaxation that is defined by a factor graph or parity check representation of the code the resulting lp decoder generalizes our previous work on turbo like codes a precise combinatorial characterization of when the lp decoder succeeds is provided based on pseudocodewords associated with the factor graph our definition of a pseudocodeword unifies other such notions known for iterative algorithms including stopping sets irreducible closed walks trellis cycles deviation sets and graph covers the fractional distance d sub frac of a code is introduced which is a lower bound on the classical distance it is shown that the efficient lp decoder will correct up to spl lceil d sub frac 2 spl rceil 1 errors and that there are codes with d sub frac spl omega n sup 1 spl epsi an efficient algorithm to compute the fractional distance is presented experimental evidence shows a similar performance on low density parity check ldpc codes between lp decoding and the min sum and sum product algorithms methods for tightening the lp relaxation to improve performance are also provided

computing linear transformations with unreliable components we consider the problem of computing a binary linear transformation when all circuit components are unreliable two models of unreliable components are considered probabilistic errors and permanent errors we introduce the encoded technique that ensures that the error probability of the computation of the linear transformation is kept bounded below a small constant independent of the size of the linear transformation even when all logic gates in the computation are noisy by deriving a lower bound we show that in some cases the computational complexity of the encoded technique achieves the optimal scaling in error probability further we examine the gain in energy efficiency from the use of a voltage scaling scheme where gate energy is reduced by lowering the supply voltage we use a gate energy reliability model to show that tuning gate energy appropriately at different stages of the computation dynamic voltage scaling in conjunction with encoded can lead to orders of magnitude energy savings over the classical uncoded approach finally we also examine the problem of computing a linear transformation when noiseless decoders can be used providing upper and lower bounds to the problem

guessing facets polytope structure and improved lp decoder we investigate the structure of the polytope underlying the linear programming lp decoder introduced by feldman karger and wainwright we first show that for expander codes every fractional pseudocodeword always has at least a constant fraction of nonintegral bits we then prove that for expander codes the active set of any fractional pseudocodeword is smaller by a constant fraction than that of any codeword we further exploit these geometrical properties to devise an improved decoding algorithm with the same order of complexity as lp decoding that provably performs better the method is very simple it first applies ordinary lp decoding and when it fails it proceeds by guessing facets of the polytope and then resolving the linear program on these facets while the lp decoder succeeds only if the ml codeword has the highest likelihood over all pseudocodewords we prove that the proposed algorithm when applied to suitable expander codes succeeds unless there exists a certain number of pseudocodewords all adjacent to the ml codeword on the lp decoding polytope and with higher likelihood than the ml codeword we then describe an extended algorithm still with polynomial complexity that succeeds as long as there are at most polynomially many pseudocodewords above the ml codeword

on trapping sets and guaranteed error correction capability of ldpc codes and gldpc codes the relation between the girth and the guaranteed error correction capability of left regular low density parity check ldpc codes when decoded using the bit flipping serial and parallel algorithms is investigated a lower bound on the size of variable node sets which expand by a factor of at least 3 4 is found based on the moore bound this bound combined with the well known expander based arguments leads to a lower bound on the guaranteed error correction capability the decoding failures of the bit flipping algorithms are characterized using the notions of trapping sets and fixed sets the relation between fixed sets and a class of graphs known as cage graphs is studied upper bounds on the guaranteed error correction capability are then established based on the order of cage graphs the results are extended to left regular and right uniform generalized ldpc codes it is shown that this class of generalized ldpc codes can correct a linear number of worst case errors in the code length under the parallel bit flipping algorithm when the underlying tanner graph is a good expander a lower bound on the size of variable node sets which have the required expansion is established

borel subalgebras and categories of highest weight modules for toroidal lie algebras abstract in this article we begin an investigation of the conjugacy classes of borel subalgebras together with verma modules induced from standard borel subalgebras of a toroidal lie algebra t in two variables we define for each highest weight a category o of representations of t that contain these verma modules and show that when a certain central element acts nontrivially this category is equivalent to the category o m for an extension of a suitable affine kac moody algebra m t from this equivalence we obtain a bgg type resolution and bgg duality theorem in the setting of o

a sub graph expansion contraction method for error floor computation in this paper we present a computationally efficient method for estimating error floors of low density parity check ldpc codes over the binary symmetric channel bsc without any prior knowledge of its trapping sets tss given the tanner graph g of a code and the decoding algorithm mathcal d the method starts from a list of short cycles in g and expands each cycle by including its sufficiently large neighborhood in g variable nodes of the expanded sub graphs mathcal g text exp are then corrupted exhaustively by all possible error patterns and decoded by mathcal d operating on mathcal g text exp union of support of the error patterns for which mathcal d fails on each mathcal g text exp defines a subset of variable nodes that is a ts the knowledge of the minimal error patterns and their strengths in each tss is used to compute an estimation of the frame error rate this estimation represents the contribution of error events localized on tss and therefore serves as an accurate estimation of the error floor performance of mathcal d at low bsc cross over probabilities we also discuss trade offs between accuracy and computational complexity our analysis shows that in some cases the proposed method provides a million fold improvement in computational complexity over standard monte carlo simulation

iterative approximate linear programming decoding of ldpc codes with linear complexity the problem of low complexity linear programming lp decoding of low density parity check ldpc codes is considered an iterative algorithm similar to min sum and belief propagation for efficient approximate solution of this problem was proposed by vontobel and koetter in this paper the convergence rate and computational complexity of this algorithm are studied using a scheduling scheme that we propose in particular we are interested in obtaining a feasible vector in the lp decoding problem that is close to optimal in the following sense the distance normalized by the block length between the minimum and the objective function value of this approximate solution can be made arbitrarily small it is shown that such a feasible vector can be obtained with a computational complexity which scales linearly with the block length combined with previous results that have shown that the lp decoder can correct some fixed fraction of errors we conclude that this error correction can be achieved with linear computational complexity this is achieved by first applying the iterative lp decoder that decodes the correct transmitted codeword up to an arbitrarily small fraction of erroneous bits and then correcting the remaining errors using some standard method these conclusions are also extended to generalized ldpc codes

the minimum distance of turbo like codes worst case upper bounds are derived on the minimum distance of parallel concatenated turbo codes serially concatenated convolutional codes repeat accumulate codes repeat convolute codes and generalizations of these codes obtained by allowing nonlinear and large memory constituent codes it is shown that parallel concatenated turbo codes and repeat convolute codes with sub linear memory are asymptotically bad it is also shown that depth two serially concatenated codes with constant memory outer codes and sublinear memory inner codes are asymptotically bad most of these upper bounds hold even when the convolutional encoders are replaced by general finite state automata encoders in contrast it is proven that depth three serially concatenated codes obtained by concatenating a repetition code with two accumulator codes through random permutations can be asymptotically good

finite length analysis of low density parity check codes on the binary erasure channel in this paper we are concerned with the finite length analysis of low density parity check ldpc codes when used over the binary erasure channel bec the main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of ldpc codes when decoded iteratively we also give expressions for upper bounds on the average bit and block erasure probability for regular ldpc ensembles and the standard random ensemble under maximum likelihood ml decoding finally we present what we consider to be the most important open problems in this area

an analysis of the block error probability performance of iterative decoding asymptotic iterative decoding performance is analyzed for several classes of iteratively decodable codes when the block length of the codes n and the number of iterations i go to infinity three classes of codes are considered these are gallager s regular low density parity check ldpc codes tanner s generalized ldpc gldpc codes and the turbo codes due to berrou et al it is proved that there exist codes in these classes and iterative decoding algorithms for these codes for which not only the bit error probability p sub b but also the block frame error probability p sub b goes to zero as n and i go to infinity

lp decoding of regular ldpc codes in memoryless channels we study error bounds for linear programming decoding of regular low density parity check ldpc codes for memoryless binary input output symmetric channels we prove bounds on the word error probability that are inverse doubly exponential in the girth of the factor graph for memoryless binary input awgn channel we prove lower bounds on the threshold for regular ldpc codes whose factor graphs have logarithmic girth under lp decoding specifically we prove a lower bound of 0 735 upper bound of eb n0 2 67 db on the threshold of 3 6 regular ldpc codes whose factor graphs have logarithmic girth our proof is an extension of a recent paper of arora daskalakis and steurer stoc 2009 who presented a novel probabilistic analysis of lp decoding over a binary symmetric channel their analysis is based on the primal lp representation and has an explicit connection to message passing algorithms we extend this analysis to any mbios channel

improved bounds on the word error probability of ra 2 codes with linear programming based decoding this paper deals with the linear programming based decoding algorithm of feldman and karger for repeat accumulate turbo like codes we present a new structural characterization that captures the event that decoding fails based on this structural characterization we develop polynomial algorithms that given an ra 2 code compute upper and lower bounds on the word error probability p sub w for the binary symmetric and the additive white gaussian noise awgn channels our experiments with an implementation of these algorithms for bounding p sub w demonstrate in many interesting cases an improvement in the upper bound on the word error probability by a factor of over 1000 compared to the bounds by feldman et al the experiments also indicate that the improvement in upper bound increases as the codeword length increases and the channel noise decreases the computed lower bounds on the word error probability in our experiments are roughly ten times smaller than the upper bound

a separation algorithm for improved lp decoding of linear block codes maximum likelihood ml decoding is the optimal decoding algorithm for arbitrary linear block codes and can be written as an integer programming ip problem feldman relaxed this ip problem and presented linear programming lp based decoding in this paper we propose a new separation algorithm to improve the error correcting performance of lp decoding for binary linear block codes we use an ip formulation with indicator variables that help in detecting the violated parity checks we derive gomory cuts from the ip and use them in our separation algorithm an efficient method of finding cuts induced by redundant parity checks rpc is also proposed under certain circumstances we can guarantee that these rpc cuts are valid and cut off the fractional optimal solutions of lp decoding it is demonstrated on three ldpc codes and two bch codes that our separation algorithm performs significantly better than lp decoding and belief propagation bp decoding

error correction capability of column weight three ldpc codes in this paper the error correction capability of column weight three low density parity check ldpc codes when decoded using the gallager a algorithm is investigated it is proved that a necessary condition for a code to correct all error patterns with up to k ges 5 errors is to avoid cycles of length up to 2k in its tanner graph as a consequence of this result it is shown that given any alpha 0 exist n such that forall n n no code in the ensemble of column weight three codes can correct all alphan or fewer errors the results are extended to the bit flipping algorithms

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

lp decoding corrects a constant fraction of errors we show that for low density parity check ldpc codes whose tanner graphs have sufficient expansion the linear programming lp decoder of feldman karger and wainwright can correct a constant fraction of errors a random graph will have sufficient expansion with high probability and recent work shows that such graphs can be constructed efficiently a key element of our method is the use of a dual witness a zero valued dual solution to the decoding linear program whose existence proves decoding success we show that as long as no more than a certain constant fraction of the bits are flipped by the channel we can find a dual witness this new method can be used for proving bounds on the performance of any lp decoder even in a probabilistic setting our result implies that the word error rate of the lp decoder decreases exponentially in the code length under the binary symmetric channel bsc this is the first such error bound for ldpc codes using an analysis based on pseudocodewords recent work by koetter and vontobel shows that lp decoding and min sum decoding of ldpc codes are closely related by the graph cover structure of their pseudocodewords in their terminology our result implies that that there exist families of ldpc codes where the minimum bsc pseudoweight grows linearly in the block length

high dimensional variable selection with sparse random projections measurement sparsity and statistical efficiency we consider the problem of high dimensional variable selection given n noisy observations of a k sparse vector rp estimate the subset of non zero entries of a significant body of work has studied behavior of l1 relaxations when applied to random measurement matrices that are dense e g gaussian bernoulli in this paper we analyze sparsified measurement ensembles and consider the trade off between measurement sparsity as measured by the fraction of non zero entries and the statistical efficiency as measured by the minimal number of observations n required for correct variable selection with probability converging to one our main result is to prove that it is possible to let the fraction on non zero entries 0 at some rate yielding measurement matrices with a vanishing fraction of non zeros per row while retaining the same statistical efficiency as dense ensembles a variety of simulation results confirm the sharpness of our theoretical predictions

iterative linear programming decoding of nonbinary ldpc codes with linear complexity the problem of low complexity linear programming lp decoding of nonbinary low density parity check ldpc codes is considered and an iterative lp decoding algorithm is presented results that were previously derived for binary ldpc codes are extended to the nonbinary case both simple and generalized nonbinary ldpc codes are considered it is shown how the algorithm can be implemented efficiently using a finite field fast fourier transform then the convergence rate of the algorithm is analyzed the complexity of the algorithm scales linearly in the block length and it can approximate up to an arbitrarily small relative error the objective function of the exact lp solution when applied to a typical code from an appropriate nonbinary ldpc code ensemble the algorithm can correct a constant fraction of errors in linear in the block length computational complexity computer experiments with the new iterative lp decoding algorithm show that in the error floor region it can have better performance compared to belief propagation decoding with similar computational requirements

linear programming decoding of spatially coupled codes for a given family of spatially coupled codes we prove that the linear programming lp threshold on the binary symmetric channel bsc of the tail biting graph cover ensemble is the same as the lp threshold on the bsc of the derived spatially coupled ensemble this result is in contrast with the fact that spatial coupling significantly increases the belief propagation threshold to prove this we establish some properties related to the dual witness for lp decoding more precisely we prove that the existence of a dual witness which was previously known to be sufficient for lp decoding success is also necessary and is equivalent to the existence of certain acyclic hyperflows we also derive a sublinear in the block length upper bound on the weight of any edge in such hyperflows both for regular low density parity check lpdc codes and spatially coupled codes and we prove that the bound is asymptotically tight for regular ldpc codes moreover we show how to trade crossover probability for lp excess on all the variable nodes for any binary linear code