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

an efficient algorithm to find all small size stopping sets of low density parity check matrices in this work we introduce an efficient algorithm to find all stopping sets of size less than some threshold of a fixed low density parity check ldpc matrix the solution is inspired by the algorithm proposed by rosnes and ytrehus in 2005 to find an exhaustive list of all small size turbo stopping sets in a turbo code the efficiency of the proposed algorithm is demonstrated by several numerical examples for instance we have applied the algorithm to the well known 3 5 regular 155 64 tanner code and found all stopping sets of size at most 18 in about 1 min on a standard desktop computer also we have verified that the minimum stopping set size of the 4896 2474 ramanujan margulis code is indeed 24 and that the corresponding multiplicity is exactly 204 furthermore we have applied the algorithm to the ieee 802 16e ldpc codes and determined the minimum stopping set size and the corresponding multiplicity exactly for these codes finally as an application we present a greedy algorithm to find a small number of redundant parity checks to add to the original parity check matrix in order to remove all stopping sets in the corresponding tanner graph of size less than the minimum distance an extensive case study of the 155 64 tanner code illustrates the usefulness of the algorithm and we present a 110 times 155 redundant parity check matrix for this code with no stopping sets of size less than the minimum distance simulation results of iterative decoding on the binary erasure channel show performance improvements for low to medium erasure probabilities when this redundant parity check matrix is used for decoding

algebraic construction of sparse matrices with large girth in this correspondence we present a method for constructing sparse matrices that have a compact description and whose associated bipartite graphs have large girth based on an arbitrary seed matrix of nonnegative integers a new matrix is constructed which replaces each entry of the seed matrix with a sum of permutation matrices algebraic conditions that lead to short cycles in the associated bipartite graph are analyzed and methods to achieve large girth in two special cases are presented in one all the permutation matrices are circulants in the other they are all affine permutation matrices when used to define a low density parity check ldpc code the compact description should lead to efficient implementation and the large girth to good error correction performance the method is adaptable to a variety of rates and a variety of row and column degrees

the trapping redundancy of linear block codes we generalize the notion of the stopping redundancy in order to study the smallest size of a trapping set in tanner graphs of linear block codes in this context we introduce the notion of the trapping redundancy of a code which quantifies the relationship between the number of redundant rows in any parity check matrix of a given code and the size of its smallest trapping set trapping sets with certain parameter sizes are known to cause error floors in the performance curves of iterative belief propagation bp decoders and it is therefore important to identify decoding matrices that avoid such sets bounds on the trapping redundancy are obtained using probabilistic and constructive methods and the analysis covers both general and elementary trapping sets numerical values for these bounds are computed for the 2640 1320 margulis code and the class of projective geometry codes and compared with some new code specific trapping set size estimates

a class of quasi cyclic ldpc codes over gf 2 m low density parity check ldpc codes over gf 2m are an extension of binary ldpc codes performances of gf 2m ldpc codes have been shown to be higher than binary ldpc codes but the complexity of the encoders decoders increases hence there is a substantial lack of implementations for ldpc codes over gf 2m codes this paper presents a class of quasi cyclic ldpc codes over gf 2m these codes can alleviate the encoding decoding complexity without excessive loss of performances it is shown how from a performance point of view such codes are better than m times bigger binary codes and as good as 2m longer binary codes

low density parity check codes based on finite geometries a rediscovery and new results this paper presents a geometric approach to the construction of low density parity check ldpc codes four classes of ldpc codes are constructed based on the lines and points of euclidean and projective geometries over finite fields codes of these four classes have good minimum distances and their tanner 1981 graphs have girth 6 finite geometry ldpc codes can be decoded in various ways ranging from low to high decoding complexity and from reasonably good to very good performance they perform very well with iterative decoding furthermore they can be put in either cyclic or quasi cyclic form consequently their encoding can be achieved in linear time and implemented with simple feedback shift registers this advantage is not shared by other ldpc codes in general and is important in practice finite geometry ldpc codes can be extended and shortened in various ways to obtain other good ldpc codes several techniques of extension and shortening are presented long extended finite geometry ldpc codes have been constructed and they achieve a performance only a few tenths of a decibel away from the shannon theoretical limit with iterative decoding

on k t subnormal covering codes the concept of a k t subnormal covering code is defined it is discussed how an amalgamated direct sumlike construction can be used to combine such codes the existence of optimal q n m 1 codes c is discussed such that by puncturing the first coordinate of c one obtains a code with q 1 subnorm 2

reversible low density parity check codes low density parity check ldpc codes may be decoded using a circuit implementation of the sum product algorithm which maps the factor graph of the code by reusing the decoder for encoding both tasks can be performed using the same circuit thus reducing area and verification requirements motivated by this iterative encoding techniques based upon the graphical representation of the code are proposed code design constraints which ensure encoder convergence are presented and then used to design iteratively encodable codes while also preventing 4 cycle creation we show how the jacobi method for iterative matrix inversion can be applied to finite field matrices viewed as message passing and employed as the core of an iterative encoder we present an algebraic construction of 4 cycle free iteratively encodable codes using circulant matrices analysis of these codes identifies a weakness in their structure due to a repetitive pattern in the factor graph the graph supports pseudo codewords of low pseudo weight in order to remove the repetitive pattern in the graph we propose a recursive technique for generating iteratively encodable codes the new codes offer flexibility in the choice of code length and rate and performance that compares well to randomly generated quasi cyclic and extended euclidean geometry codes

high rate quantum low density parity check codes assisted by reliable qubits quantum error correction is an important building block for reliable quantum information processing a challenging hurdle in the theory of quantum error correction is that it is significantly more difficult to design error correcting codes with desirable properties for quantum information processing than for traditional digital communications and computation a typical obstacle to constructing a variety of strong quantum error correcting codes is the complicated restrictions imposed on the structure of a code recently promising solutions to this problem have been proposed in quantum information science where in principle any binary linear code can be turned into a quantum error correcting code by assuming a small number of reliable quantum bits this paper studies how best to take advantage of these latest ideas to construct desirable quantum error correcting codes of very high information rate our methods exploit structured high rate low density parity check codes available in the classical domain and provide quantum analogues that inherit their characteristic low decoding complexity and high error correction performance even at moderate code lengths our approach to designing high rate quantum error correcting codes also allows for making direct use of other major syndrome decoding methods for linear codes making it possible to deal with a situation where promising quantum analogues of low density parity check codes are difficult to find

a new construction of structured binary regular ldpc codes based on steiner systems with parameter t 2 this paper presents a novel method for constructing structured regular low density parity check ldpc codes based on a special type of combinatorial designs known as steiner systems this code design approach can be considered as a generalization of the well known method which uses the point block incidence matrix of a steiner 2 design for the code construction though the given method can be applied on any steiner system s t k v in this paper we focus only on steiner systems with t 3 furthermore we show that not only a steiner system x b itself but also its residual design with respect to an arbitrary point x x can be employed for code construction we also present a technique for constructing binary and non binary qc ldpc codes from steiner systems the tanner graph of the constructed codes is free of 4 cycles and hence the codes have girth at least six simulation results show that the so constructed codes perform well over the awgn channel with iterative message passing decoding

construction of low density parity check codes based on balanced incomplete block designs this correspondence presents a method for constructing structured regular low density parity check ldpc codes based on a special type of combinatoric designs known as balance incomplete block designs codes constructed by this method have girths at least 6 and they perform well with iterative decoding furthermore several classes of these codes are quasi cyclic and hence their encoding can be implemented with simple feedback shift registers

construction of nonbinary cyclic quasi cyclic and regular ldpc codes a finite geometry approach this paper presents five methods for constructing nonbinary ldpc codes based on finite geometries these methods result in five classes of nonbinary ldpc codes one class of cyclic ldpc codes three classes of quasi cyclic ldpc codes and one class of structured regular ldpc codes experimental results show that constructed codes in these classes decoded with iterative decoding based on belief propagation perform very well over the awgn channel and they achieve significant coding gains over reed solomon codes of the same lengths and rates with either algebraic hard decision decoding or kotter vardy algebraic soft decision decoding at the expense of a larger decoding computational complexity

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

iterative decoding of one step majority logic deductible codes based on belief propagation previously the belief propagation bp algorithm has received a lot of attention in the coding community mostly due to its near optimum decoding for low density parity check ldpc codes and its connection to turbo decoding in this paper we investigate the performance achieved by the bp algorithm for decoding one step majority logic decodable osmld codes the bp algorithm is expressed in terms of likelihood ratios rather than probabilities as conventionally presented the proposed algorithm fits better the decoding of osmld codes with respect to its numerical stability due to the fact that the weights of their check sums are often much higher than that of the corresponding ldpc codes although it has been believed that osmld codes are far inferior to ldpc codes we show that for medium code lengths say between 200 1000 bits the bp decoding of osmld codes can significantly outperform bp decoding of their equivalent ldpc codes the reasons for this behavior are elaborated

exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method the authors present a systematic and formal approach toward finding solitary wave solutions of nonlinear evolution and wave equations from the real exponential solutions of the underlying linear equations the physical concept is one of the mixing of these elementary solutions through the nonlinearities in the system the emphasis is however on the mathematical aspects i e the formal procedure necessary to find single solitary wave solutions by means of examples it is shown how various cases of pulse type and kink type solutions are to be obtained by this method an exhaustive list of equations so treated is presented in tabular form together with the particular intermediate relations necessary for deriving their solutions the extension of the technique to construct n soliton solutions and indicate connections with other existing methods is outlined

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

further results on k t subnormal covering codes the concept of k t subnormal covering codes is discussed generalizing some of the earlier results in a similar way k t normal covering codes are defined using the results including some new constructions upper bounds for covering codes are improved it is shown how simulated annealing can be used to find acceptable partitions for codes

modified bounds for covering codes the covering radius of binary codes is studied bounds on k n r the minimum cardinality of any binary code of length n and covering radius r are found modifications of the van wee lower bounds are proved for k n r the minimal number of codewords in any binary code of length n and covering radius r the first of the two van wee bounds is based on studying the hamming spheres of radius 1 centered at the points which have distance r to the code c the points covered by more than one codeword are divided into several classes and better estimates for some of these classes are obtained using a suitable averaging process the lower bound for k n r when r or 2 is improved the second van wee bound studies spheres of radius 2 centered at the points which have distance r 1 or r to the code c these points are divided essentially into two classes the points that are covered by only one codeword of c and the points that are covered by more than one codeword

high rate quasi cyclic low density parity check codes derived from finite affine planes this paper shows that several attractive classes of quasi cyclic qc low density parity check ldpc codes can be obtained from affine planes over finite fields one class of these consists of duals of one generator qc codes presented here for codes contained in this class are the exact minimum distance and a lower bound on the multiplicity of the minimum weight codewords further it is shown that the minimum hamming distance of a code in this class is equal to its minimum additive white gaussian noise awgn pseudoweight also discussed is a class consisting of codes from circulant permutation matrices and an explicit formula for the rank of the parity check matrix is presented for these codes additionally it is shown that each of these codes can be identified with a code constructed from a constacyclic maximum distance separable code of dimension 2 the construction is similar to the derivation of reed solomon rs based ldpc codes presented by chen and djurdjevic experimental results show that a number of high rate qc ldpc codes with excellent error performance are contained in these classes

on minimum wer performance of a class of ldpc codes for the bsc e under extended bounded distance decoding we study the word error rate wer performance and the wer minimizing decision thresholds plusmnt for the class of row column diagonal rcd array codes and their generalizations on the binary symmetric channel with erasures bsc e where decoding is via an extended bounded distance decoder e bdd this model applies to very high data rate systems where soft decision iterative decoding is impractical we demonstrate that substantial wer performance gain can be obtained especially at higher channel signal to noise ratio csnr csnr delta 10 log10 1 2sigma2 values by employing two optimal thresholds plusmnt chosen to minimize wer we study the wer improvement and signal to noise ratio snr performance gain as a function of code length n for fixed minimum distance dmin and as a function of dmin for fixed n a distinct difference in the behavior of wer versus the threshold t and the optimal threshold t versus snr is observed between codes with odd and even dmin with the even dmin codes exhibiting larger performance improvements when employing a bsc e instead of a bsc

mean field and mixed mean field iterative decoding of low density parity check codes in this paper the mean field mf and mixed mean field mmf algorithms for decoding low density parity check ldpc codes are considered the mf principle is well established in statistical physics and artificial intelligence instead of using a single completely factorized approximated distribution as in the mf approach the mixed mf algorithm forms a weighted average of several mf distributions as an approximation of the true posterior probability distribution the mf decoding algorithm for linear block codes is derived and shown to be an approximation of the a posteriori probability app decoding algorithm the mf approach is then developed in the context of iterative decoding and presented as an approximation of the popular belief propagation decoding method these results are extended to iterative decoding with the mmf algorithm simulation results show that the mf and mmf decoding algorithms yield a good performance complexity tradeoff especially when employed for decoding ldpc codes based on finite geometries

on the covering radius of codes the covering radius r of a code is the maximal distance of any vector from the code this work gives a number of new results concerning t n k the minimal covering radius of any binary code of length n and dimension k for example t n 4 and t n 5 are determined exactly and reasonably tight bounds on t n k are obtained for any k when n is large these results are found by using several new constructions for codes with small covering radius one construction the amalgamated direct sum involves a quantity called the norm of a code codes with norm leq 2 r 1 are called normal and may be combined efficiently the paper concludes with a table giving bounds on t n k for n leq 64

new lower bounds for binary covering codes g d chen et al ibid vol it 32 p 680 94 1986 presented two new lower bounds for k n r where k n r denotes the minimum cardinality of a binary code of length n and covering radius r the author shows that a slight modification gives further improvements and some examples are given to confirm the argument codes that have a certain partitioning property are considered

cyclic and quasi cyclic ldpc codes on constrained parity check matrices and their trapping sets this paper is concerned with construction and structural analysis of both cyclic and quasi cyclic codes particularly low density parity check ldpc codes it consists of three parts the first part shows that a cyclic code given by a parity check matrix in circulant form can be decomposed into descendant cyclic and quasi cyclic codes of various lengths and rates some fundamental structural properties of these descendant codes are developed including the characterization of the roots of the generator polynomial of a cyclic descendant code the second part of the paper shows that cyclic and quasi cyclic descendant ldpc codes can be derived from cyclic finite geometry ldpc codes using the results developed in the first part of the paper this enlarges the repertoire of cyclic ldpc codes the third part of the paper analyzes the trapping set structure of regular ldpc codes whose parity check matrices satisfy a certain constraint on their rows and columns several classes of finite geometry and finite field cyclic and quasi cyclic ldpc codes with large minimum distances are shown to have no harmful trapping sets of size smaller than their minimum distances consequently their error floor performances are dominated by their minimum distances

constructions of nonlinear covering codes constructions of nonlinear covering codes are given using any nonlinear starting code of covering radius r spl ges 2 these constructions form an infinite family of codes with the same covering radius a nonlinear code is treated as a union of cosets of a linear code new infinite families of nonlinear covering codes are obtained concepts of r l objects r l partitions and r l length are described for nonlinear codes

new classes of partial geometries and their associated ldpc codes the use of partial geometries to construct parity check matrices for binary low density parity check ldpc codes has resulted in the design of successful codes with a probability of error on the awgn channel close to the shannon capacity at bit error rate down to 10 15 such considerations have motivated this further investigation a new and simple construction of a type of partial geometries with a quasi cyclic qc structure is given and their properties are investigated two new classes of this type of partial geometries one based on prime fields and the other based on cyclic subgroups of prime orders of finite fields are constructed qc ldpc codes with good error performances are constructed based on these two new classes of partial geometries the trapping sets of the partial geometry codes were previously considered using the geometric aspects of the underlying structure to derive information on the size of allowable trapping sets this topic is further considered here finally there is a natural relationship between partial geometries and strongly regular graphs the eigenvalues of the adjacency matrices of such graphs are well known and it is of interest to determine if any of the tanner graphs derived from the partial geometries are good expanders for certain parameter sets since it can be argued that codes with good geometric and expansion properties might perform well on the awgn channel under message passing decoding

algebraic construction of quasi cyclic ldpc codes for the awgn and erasure channels this paper is concerned with construction of quasi cyclic qc low density parity check ldpc codes for three different types of channels the additive white gaussian noise the binary random erasure and the binary burst erasure channels two algebraic methods for systematic construction of qc ldpc codes are presented codes constructed perform well over all three types of channels

new combinatorial construction techniques for low density parity check codes and systematic repeat accumulate codes this paper presents several new construction techniques for low density parity check ldpc and systematic repeat accumulate ra codes based on specific classes of combinatorial designs the improved code design focuses on high rate structured codes with constant column weights 3 and higher the proposed codes are efficiently encodable and exhibit good structural properties experimental results on decoding performance with the sum product algorithm show that the novel codes offer substantial practical application potential for instance in high speed applications in magnetic recording and optical communications channels

reduced complexity iterative decoding of low density parity check codes based on belief propagation two simplified versions of the belief propagation algorithm for fast iterative decoding of low density parity check codes on the additive white gaussian noise channel are proposed both versions are implemented with real additions only which greatly simplifies the decoding complexity of belief propagation in which products of probabilities have to be computed also these two algorithms do not require any knowledge about the channel characteristics both algorithms yield a good performance complexity trade off and can be efficiently implemented in software as well as in hardware with possibly quantized received values

optimal erasure selection of m ary pam signaling for errors and erasures decoding algorithms erasure selection for errors and erasures decoding algorithms is investigated and analyzed where declaring erasures provides additional gain for hard decision decoding to derive an analytical criterion for optimal erasure selection we first investigate erasure effects on decoding performance by which a metric function for erasure selection is derived from the derived metric a suboptimal threshold is defined which can be used for errors and erasures decoding of large code distance channel codes moreover application to m ary m 2 pam constellation is also described for verification of the proposed erasure threshold simulation results with popular channel codes are included

more binary covering codes are normal it is shown that every optimal binary code with covering radius r 1 is normal this parity proves a conjecture of cohen lobstein and sloane 1986 it is also proved that codes with minimal distance 2r or 2r 1 are normal a generalization of frankl s construction 1987 of abnormal codes is given

further results on the covering radius of codes a number of upper and lower bounds are obtained for k n r the minimal number of codewords in any binary code of length n and covering radius r several new constructions are used to derive the upper bounds including an amalgamated direct sum construction for nonlinear codes this construction works best when applied to normal codes and we give some new and stronger conditions which imply that a linear code is normal an upper bound is given for the density of a covering code over any alphabet and it is shown that k n 2 r 1 leq k n r holds for sufficiently large n

upper bounds for q ary covering codes new methods for constructing q ary covering codes are presented the author introduces the concepts of p seminormal and strongly p seminormal codes and shows how seminormal codes and punctured hamming codes can be combined to construct new covering codes using these methods upper bounds for ternary covering codes are improved the new bounds are k sub 3

on the pseudocodeword redundancy of binary linear codes for a binary linear code the pseudocodeword redundancy with respect to the additive white gaussian noise channel the binary symmetric channel or the max fractional weight is defined to be the smallest number of rows in a parity check matrix such that the corresponding minimum pseudoweight is equal to the minimum hamming distance of the code it is shown that most codes do not have a finite pseudocodeword redundancy also upper bounds on the pseudocodeword redundancy for some families of codes including codes based on designs are provided the pseudocodeword redundancies for all codes of small length at most 9 are computed furthermore comprehensive results are provided on the cases of cyclic codes of length at most 250 for which the eigenvalue bound of vontobel and koetter is sharp

a lattice based systematic recursive construction of quasi cyclic ldpc codes this paper presents a low complexity recursive and systematic method to construct good well structured low density parity check ldpc codes the method is based on a recursive application of a partial kronecker product operation on a given gamma x q q ges 3 a prime integer lattice l gamma x q the n 1 fold product of l gamma x q by itself denoted ln gamma x q represents a regular quasi cyclic qc ldpc code denoted see pdf of high rate and girth 6 the minimum distance of see pdf is equal to that of the core code see pdf introduced by l gamma x q the support of the minimum weight codewords in see pdf are characterized by the support of the same type of codewords in see pdf from performance perspective the constructed codes compete with the pseudorandom ldpc codes

lattice solitary wave solutions of discrete nonlinear wave equations using a direct method it is shown that the real exponential approach to the solutions of nonlinear evolution and wave equations can also he applied to discrete systems as examples the lattice solitary wave and kink solutions of the discrete modified kdv dmkdv equation and some modified versions of this equation are derived using this approach

efficient encoding of quasi cyclic low density parity check codes efficient encoding of quasi cyclic low density parity check codes quasi cyclic qc low density parity check ldpc codes form an important subclass of ldpc codes these codes have encoding advantage over other types of ldpc codes this paper addresses the issue of efficient encoding of qc ldpc codes two methods are presented to find the generator matrices of qc ldpc codes in systematic circulant form from their parity check matrices given in circulant form based on the systematic circulation form of the generator matrix of a qc ldpc code various types of encoding circuits using simple shift registers are devised it is shown that the encoding complexity of a qc ldpc code is linearly proportional to the number of parity bits of the code for serial encoding and to the length of the code for high speed parallel encoding

iterative decoding of binary block and convolutional codes iterative decoding of two dimensional systematic convolutional codes has been termed turbo de coding using log likelihood algebra we show that any decoder can be used which accepts soft inputs including a priori values and delivers soft outputs that can be split into three terms the soft channel and a priori inputs and the extrinsic value the extrinsic value is used as an a priori value for the next iteration decoding algorithms in the log likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code the iteration is controlled by a stop criterion derived from cross entropy which results in a minimal number of iterations optimal and suboptimal decoders with reduced complexity are presented simulation results show that very simple component codes are sufficient block codes are appropriate for high rates and convolutional codes for lower rates less than 2 3 any combination of block and convolutional component codes is possible several interleaving techniques are described at a bit error rate ber of 10 sup 4 the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block convolutional component codes interleaver sizes less than 1000 and for three to six iterations

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

new binary covering codes obtained by simulated annealing new binary covering codes of radius 1 obtained by simulated annealing are presented these constructions establish that k 9 1 spl les 62 and k 12 1 spl les 380 the article is concerned with finding upper bounds on k n r the minimum cardinality of any binary code of length n and with covering radius r

near optimum universal belief propagation based decoding of low density parity check codes in this paper we propose a belief propagation bp based decoding algorithm which utilizes normalization to improve the accuracy of the soft values delivered by a previously proposed simplified bp based algorithm the normalization factors can be obtained not only by simulation but also importantly theoretically this new bp based algorithm is much simpler to implement than bp decoding as it requires only additions of the normalized received values and is universal i e the decoding is independent of the channel characteristics some simulation results are given which show this new decoding approach can achieve an error performance very close to that of bp on the additive white gaussian noise channel especially for low density parity check ldpc codes whose check sums have large weights the principle of normalization can also be used to improve the performance of the max log map algorithm in turbo decoding and some coding gain can be achieved if the code length is long enough

an explicit expression of the dark n soliton solution of the mkdv equation by means of the darboux transformation a darboux transformation is developed for generating dark multi soliton solutions of the mkdv equation the darboux matrices are found explicitly in recursive manner and a system of linear algebraic equations is derived for determining the dark n soliton solutions by means of the binet cauchy formula an explicit expression of the dark n soliton solution is obtained

construction of solutions of the sine gordon equation by means of fredholm determinants abstract a new straightforward and rigorous approach to the theory of the sine gordon equation is presented it provides a simple and unified view of the three known methods for solution namely hirota s method of generating n soliton solutions the inverse scattering transform and backlund transformations in addition new types of solutions corresponding to multiple poles in the scattering data can be easily obtained the principal tool used is the theory of fredholm determinants generalisations to other soliton equations seem to be possible

crc handbook of combinatorial designs balanced incomplete block designs and t designs2 v k l designs of small orderbibds with small block sizet designs t 3steiner systemssymmetric designsresolvable and near resolvable designslatin squares mols and orthogonal arrayslatin squaresmutually orthogonal latin squares mols incomplete molsorthogonal arrays of index more than oneorthogonal arrays of strength more than twopairwise balanced designspbds and gdds the basicspbds recursive constructionspbd closurepairwise balanced designs as linear spacespbds and gdds of higher indexpbds frames and resolvabilityother combinatorial designsassociation schemesbalanced part ternary designsbalanced tournament designsbhaskar rao designscomplete mappings and sequencings of finite groupsconfigurationscostas arrayscoveringscycle systemsdifference familiesdifference matricesdifference sets abeliandifference sets nonabeliandifference triangle setsdirected designsd optimal matricesembedding partial quasigroupsequidistant permutation arraysfactorial designsfrequency squaresgeneralized quadranglesgraph decompositions and designsgraphical designshadamard matrices and designshall triple systemshowell designsmaximal sets of molsmendelsohn designsthe oberwolfach problemordered designs and perpendicular arraysorthogonal designsorthogonal main effect planspackingspartial geometriespartially balanced incomplete block designsquasigroupsquasi symmetric designs r l designsroom squaresself orthogonal latin squares sols sols with a symmetric orthogonal mate solssom sequences with zero autocorrelationskolem sequencesspherical t designsstarterstrades and defining sets t m s netstuscan squarest wise balanced designsuniformly resolvable designsvector space designsweighing matrices and conference matriceswhist tournamentsyouden designs generalizedyouden squaresapplicationscodescomputer science selected applicationsapplications of designs to cryptographyderandomizationoptimality and efficiency comparing block designsgroup testingscheduling a tournamentwinning the lotteryrelated mathematics and computational methodsfinite groups and designsnumber theory and finite fieldsgraphs and multigraphsfactorizations of graphsstrongly regular graphstwo graphsclassical geometriesprojective planes nondesarguesiancomputational methods in design theoryindex

bounds on the expansion properties of tanner graphs this work focuses on the expansion properties of a tanner graph because they are known to be related to the performance of associated iterative message passing algorithms over various channels by analyzing the eigenvalues and corresponding eigenvectors of the normalized incidence matrix representing a tanner graph lower bounds on these expansion properties are derived specifically for the binary erasure channel these results lead to two lower bounds on stopping distance for any given binary linear code and an upper bound on stopping redundancy for the family of difference set codes type i 2 d projective geometry low density parity check ldpc codes

reduced complexity decoding of ldpc codes various log likelihood ratio based belief propagation llr bp decoding algorithms and their reduced complexity derivatives for low density parity check ldpc codes are presented numerically accurate representations of the check node update computation used in llr bp decoding are described furthermore approximate representations of the decoding computations are shown to achieve a reduction in complexity by simplifying the check node update or symbol node update or both in particular two main approaches for simplified check node updates are presented that are based on the so called min sum approximation coupled with either a normalization term or an additive offset term density evolution is used to analyze the performance of these decoding algorithms to determine the optimum values of the key parameters and to evaluate finite quantization effects simulation results show that these reduced complexity decoding algorithms for ldpc codes achieve a performance very close to that of the bp algorithm the unified treatment of decoding techniques for ldpc codes presented here provides flexibility in selecting the appropriate scheme from performance latency computational complexity and memory requirement perspectives

iterative algorithms for decoding a class of two step majority logic decodable cyclic codes codes constructed based on finite geometries form a large class of cyclic codes with large minimum distances which can be decoded with simple majority logic decoding in one or multiple steps in 2001 kou lin and fossorier showed that the one step majority logic decodable finite geometry codes form a class of cyclic ldpc codes whose tanner graphs are free of cycles of length 4 these cyclic finite geometry ldpc codes perform very well over the awgn channel using iterative decoding based on belief propagation idbp and have very low error floors however the standard idbp is not effective for decoding other cyclic finite geometry codes because their tanner graphs contain too many short cycles of length 4 which severely degrade the decoding performance this paper investigates iterative decoding of two step majority logic decodable finite geometry codes three effective algorithms for decoding these codes are proposed these algorithms are devised based on the orthogonal structure of the parity check matrices of the codes to avoid or reduce the degrading effect of the short cycles of length 4 these decoding algorithms provide significant coding gains over the standard idbp using either the sum product or the min sum algorithms

construction of structured regular ldpc codes a design theoretic approach a new combinatorial technique for constructing girth 6 structured binary regular low density parity check ldpc codes based on special types of t designs is given a very large number of well known t designs can be used by this method for code construction based on this method a t v k design d x b can be exploited for code construction if it satisfies the following three conditions 1 b1 b2 t for any two blocks b1 b2 b and b1 b2 2 1 and 3 k t though the technique works for any t design satisfying these conditions we focus only on the utilization of simple triple systems super simple bibds steiner systems and large sets lss of t designs we also construct binary and non binary girth 6 qc ldpc codes from the t designs satisfying these conditions by using matrix dispersion method experimental results show that the constructed non binary qc ldpc codes can provide good practical performance under iterative decoding using the fast fourier transform based q ary sum product algorithm fft qspa and they can achieve acceptable coding gains over random like codes of comparable parameters decoded with sum product algorithm spa

a new binary code of length 10 and covering radius 1 a mixed code of covering radius 1 that has 60 codewords is constructed this code is then used to show that k 10 1

iterative list decoding of some ldpc codes we present an iterative list decoding algorithm for low density parity check ldpc codes in particular we apply this decoder to a class of ldpc codes from finite geometries and show that the 73 45 10 projective geometry code can be maximum likelihood ml decoded with low complexity moreover the list decoding approach enables us to give a theoretical analysis of the performance we also consider list bit flipping bf decoding of longer ldpc codes

bounds for abnormal binary codes with covering radius one the normality of binary codes is studied the minimum cardinality of a binary code of length n with covering radius r is denoted by k n r it is assumed that c is an n m r code that is a binary code of length n with m codewords and covering radius r it is shown that if c is an n m 1 code then it is easy to find a normal n m 1 code by changing c in a suitable way and that all the optimal n m 1 codes i e those for which m k n 1 are normal and their every coordinate is acceptable it is shown that if c is an abnormal n m code then n or 9 and an abnormal 9118 1 code which is the smallest abnormal code known at present is constructed lower bounds on the minimum cardinality of a binary abnormal code of length n with covering radius 1 are derived and it is shown that if an n m 1 code is abnormal then m or 96

on normal and subnormal q ary codes the authors extend to the q ary case the notions of a normal code a subnormal code and the amalgamated direct sum construction in order to investigate problems related to the covering radius of codes for example the authors prove that every nonbinary nontrivial perfect code is absubnormal they also include some linear programming lower bounds on ternary codes with covering radius 2 or 3

iterative decoding algorithms for a class of non binary two step majority logic decodable cyclic codes this paper presents two iterative decoding algorithms for a class of non binary two step majority logic nb ts mlg decodable cyclic codes a partial parallel decoding scheme is also introduced to provide a balanced trade off between decoding speed and storage requirements unlike non binary one step mlg decodable cyclic codes the tanner graphs of which are 4 cycle free nb ts mlg decodable cyclic codes contain a large number of short cycles of length 4 which tend to degrade decoding performance the proposed algorithms utilize the orthogonal structure of the parity check matrices of the codes to resolve the degrading effects of the short cycles of length 4 simulation results demonstrate that the nb ts mlg decodable cyclic codes decoded with the proposed algorithms offer coding gains as much as 2 5 db over reed solomon codes of the same lengths and rates decoded with either hard decision or algebraic soft decision decoding

performance limits of reed solomon coded cdma with orthogonal signaling in a rayleigh fading channel the asymptotic performance of reed solomon rs coded m ary orthogonal signaling with ratio threshold test rtt type demodulation in a rayleigh fading channel is considered we show that the minimum e sub b n sub 0 needed for error free communication is eln2 2 75 db with rtt and 4 79 6 8 db with hard decisions the optimum code rate that minimizes the required e sub b n sub 0 is e sup 1 with rtt and 0 46 with hard decision and the optimum ratio threshold approaches 1 for large m next we investigate the fundamental limit in a direct sequence spread spectrum multiple access ds ssma system employing an m ary orthogonal code of length n mm which is obtained by spreading every row of an m spl times m hadamard matrix with a user specific random sequence of length n we derive the minimum e sub b n sub 0 for error free communication as a function of the number of users the optimum code rate that minimizes e sub b n sub 0 and the maximum limit on the total information transmission rate then we consider a multirate ds ssma system where a population of users simultaneously transmit at different power levels a variety of traffic types of different information rates we derive the minimum required e sub b n sub 0 and the optimum code rate for each traffic type

gradient descent bit flipping algorithms for decoding ldpc codes a novel class of bit flipping bf algorithm for decoding low density parity check ldpc codes is presented the proposed algorithms which are referred to as gradient descent bit flipping gdbf algorithms can be regarded as simplified gradient descent algorithms the proposed algorithms exhibit better decoding performance than known bf algorithms such as the weighted bf algorithm or the modified weighted bf algorithm for several ldpc codes

unveiling turbo codes some results on parallel concatenated coding schemes a parallel concatenated coding scheme consists of two simple constituent systematic encoders linked by an interleaver the input bits to the first encoder are scrambled by the interleaver before entering the second encoder the codeword of the parallel concatenated code consists of the input bits to the first encoder followed by the parity check bits of both encoders this construction can be generalized to any number of constituent codes parallel concatenated schemes employing two convolutional codes as constituent codes in connection with an iterative decoding algorithm of complexity comparable to that of the constituent codes have been previously shown to yield remarkable coding gains close to theoretical limits they have been named and are known as turbo codes we propose a method to evaluate an upper bound to the bit error probability of a parallel concatenated coding scheme averaged over all interleavers of a given length the analytical bounding technique is then used to shed some light on some crucial questions which have been floating around in the communications community since the proposal of turbo codes

an improved decoding algorithm for finite geometry ldpc codes in this letter an improved bit flipping decoding algorithm for high rate finite geometry low density parity check fg ldpc codes is proposed both improvement in performance and reduction in decoding delay are observed by flipping multiple bits in each iteration our studies show that the proposed algorithm achieves an appealing tradeoff between performance and complexity for fg ldpc codes

lower bounds for q ary covering codes the authors prove combinatorial lower bounds for k sub q n r the minimal cardinality of any q ary code of length n and covering radius r tables of lower bounds for k sub q n r are presented for q 3 4 5

hierarchical and high girth qc ldpc codes we present an approach to designing capacity approaching high girth low density parity check ldpc codes that are friendly to hardware implementation and compatible with some desired input code structure defined using a protograph the approach is based on a mapping of any class of codes defined using a protograph into a family of hierarchical quasi cyclic hqc ldpc codes whereas the parity check matrices of standard quasi cyclic qc ldpc codes are composed of circulant submatrices those of hqc ldpc codes are composed of a hierarchy of circulant submatrices that are in turn constructed from circulant submatrices and so on through some number of levels next we present a girth maximizing algorithm that optimizes the degrees of freedom within the family of codes to yield a high girth hqc ldpc code subject to bounds imposed by the fact that hqc codes are still quasi cyclic finally we discuss how certain characteristics of a code protograph will lead to inevitable short cycles and show that these short cycles can be eliminated using a squashing procedure that results in a high girth qc ldpc code although not a hierarchical one we illustrate our approach with three design examples of qc ldpc codes two girth 10 codes of rates 1 3 and 0 45 and one girth 8 code of rate 0 7 all of which are obtained from protographs of one sided spatially coupled codes

approximations to large amplitude solitary waves on nonlinear electrical lattices abstract in this paper we describe an approximate method to characterise solitary wave solutions of nonlinear lattice equations it is based upon one and two point pade approximations to a series of the real exponential travelling wave solutions of the underlying dispersive system the theory is applied to an example of a lattice system which models an experimental nonlinear transmission line and the results obtained are consistent with numerical simulations even for relatively large amplitude solitary waves the speed amplitude relation is investigated and compared with the derived using quasi continuum methods