a transform theory for a class of group invariant codes a binary cyclic code of length of n can be defined by a set of parity check equations that is invariant under the additive cyclic group generated by a i to i 1 modulo n a class of quasicyclic codes that have parity check equations invariant under the group generated by a subgroup of a and a subgroup of the multiplicative group m i to 2 sup k i modulo n is studied in detail the fourier transform transforms the action of the additive group and a linearized polynomial transform transforms the action of the multiplicative group the form of the transformed code equations permits a bch like bound on the minimum distance of the code the techniques are illustrated by the construction of several codes that are decodable by the berlekamp massey algorithm and that have parameters that approach or surpass those of the best codes known

two constructions of asymptotically optimal codebooks according to the welch bound abstract in a number of applications the codebook that minimizes the maximal cross correlation amplitude imax is often desirable there are two major ingredients in this paper these two contents are to propose two constructions of codebooks the codebooks generated by these constructions asymptotically meet the welch bound what we find is the codebook constructed in section 2 can also asymptotically meet the levenshtein bound the parameters of these codebooks are new

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

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

deriving good ldpc convolutional codes from ldpc block codes low density parity check ldpc convolutional codes are capable of achieving excellent performance with low encoding and decoding complexity in this paper we discuss several graph cover based methods for deriving families of time invariant and time varying ldpc convolutional codes from ldpc block codes and show how earlier proposed ldpc convolutional code constructions can be presented within this framework some of the constructed convolutional codes significantly outperform the underlying ldpc block codes we investigate some possible reasons for this convolutional gain and we also discuss the mostly moderate decoder cost increase that is incurred by going from ldpc block to ldpc convolutional codes

on the dynamics of the error floor behavior in regular ldpc codes it is shown that dominant trapping sets of regular low density parity check ldpc codes so called absorption sets undergo a two phased dynamic behavior in the iterative message passing decoding algorithm using a linear dynamic model for the iteration behavior of these sets it is shown that they undergo an initial geometric growth phase which stabilizes in a final bit flipping behavior where the algorithm reaches a fixed point this analysis is shown to lead to very accurate numerical calculations of the error floor bit error rates down to error rates that are inaccessible by simulation the topology of the dominant absorption sets of an example code the ieee 802 3an 2048 1723 regular ldpc code are identified and tabulated using topological relationships in combination with search algorithms

characterizations of pseudo codewords of low density parity check codes an important property of low density parity check codes is the existence of highly efficient algorithms for their decoding many of the most efficient recent graph based algorithms e g message passing iterative decoding and linear programming decoding crucially depend on the efficient representation of a code in a graphical model in order to understand the performance of these algorithms we argue for the characterization of codes in terms of a so called fundamental cone in euclidean space this cone depends upon a given parity check matrix of a code rather than on the code itself we give a number of properties of this fundamental cone derived from its connection to unramified covers of the graphical models on which the decoding algorithms operate for the class of cycle codes these developments naturally lead to a characterization of the fundamental cone as the newton polyhedron of the hashimoto edge zeta function of the underlying graph

adaptive methods for linear programming decoding detectability of failures of linear programming lp decoding and the potential for improvement by adding new constraints motivate the use of an adaptive approach in selecting the constraints for the underlying lp problem in this paper we make a first step in studying this method and show that by starting from a simple lp problem and adaptively adding the necessary constraints the complexity of lp decoding can be significantly reduced in particular we observe that with adaptive lp decoding the sizes of the lp problems that need to be solved become practically independent of the density of the parity check matrix we further show that adaptively adding extra constraints such as constraints based on redundant parity checks can provide large gains in the performance

a dual ascent approach for steiner tree problems on a directed graph the steiner tree problem on a directed graph stdg is to find a directed subtree that connects a root node to every node in a designated node setv we give a dual ascent procedure for obtaining lower bounds to the optimal solution value the ascent information is also used in a heuristic procedure for obtaining feasible solutions to the stdg computational results indicate that the two procedures are very effective in solving a class of stdg s containing up to 60 nodes and 240 directed 120 undirected arcs

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

construction of quasi cyclic ldpc codes for awgn and binary erasure channels a finite field approach in the late 1950s and early 1960s finite fields were successfully used to construct linear block codes especially cyclic codes with large minimum distances for hard decision algebraic decoding such as bose chaudhuri hocquenghem bch and reed solomon rs codes this paper shows that finite fields can also be successfully used to construct algebraic low density parity check ldpc codes for iterative soft decision decoding methods of construction are presented ldpc codes constructed by these methods are quasi cyclic qc and they perform very well over the additive white gaussian noise awgn binary random and burst erasure channels with iterative decoding in terms of bit error probability block error probability error floor and rate of decoding convergence collectively particularly they have low error floors since the codes are qc they can be encoded using simple shift registers with linear 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

two new rate 2 p binary quasi cyclic codes a class of rate 2 p quasi cyclic codes can be characterized in terms of m spl times m circulant matrices in this correspondence two new codes with parameters 80 10 35 and 95 10 42 are presented which improve the known lower bound on the maximum possible minimum distance the former code can be extended with an even parity check bit to an 81 10 36 code that establishes that d sub 2 81 10 36

a unified approach to whiteman s and ding helleseth s generalized cyclotomy over residue class rings the theory of cyclotomy dates back to gauss and has a number of applications in combinatorics coding theory and cryptography cyclotomy over a residue class ring bbzv can be divided into classical cyclotomy or generalized cyclotomy depending on v prime or composite in this paper we introduce a generalized cyclotomy of order d over bbzp1e1p2e2 pnen which includes whiteman s and ding helleseth s generalized cyclotomy as special cases here p1 p2 pn are pairwise distinct odd primes satisfying d pi 1 for all 1 i n and e1 e2 en are positive integers we derive some basic properties of the corresponding cyclotomic numbers and obtain a general formula to compute them via classical cyclotomic numbers as applications we completely solve an open problem and a conjecture on whiteman s generalized cyclotomy of order four over bbzp1p2 besides we also construct an infinite series of near optimal codebooks over bbzp1p2 as well as some infinite series of asymptotically optimal difference systems of sets over bbzp1e1p2e2 pnen

new constructions of asymptotically optimal codebooks with multiplicative characters in practical applications such as direct spread code division multiple access communications space time codes and compressed sensing and codebooks with small inner product correlation are required it is extremely difficult to construct codebooks achieving the levenshtein bound in this paper two new constructions of infinitely many codebooks with multiplicative characters of finite fields are presented these constructions produce complex codebooks asymptotically achieving the levenshtein bound and codebooks asymptotically achieving the welch bound the codebooks presented in this paper have new parameters

a factoring approach for the steiner tree problem in undirected networks in communication networks many applications such as video on demand and video conferencing must establish a communications tree that spans a subset k in a vertex set the source node can then send identical data to all nodes in set k along this tree this kind of communication is known as multicast communication a network optimization problem called the steiner tree problem stp is presented to find a least cost multicasting tree in this paper an o e algorithm is presented to find a minimum steiner tree for series parallel graphs where e is the number of edges based on this algorithm we proposed an o 2 2 c e algorithm to solve the steiner tree problem for general graphs where c is the number of applied factoring procedures the c value is strongly related to the topology of a given graph this is quite different from other algorithms with exponential time complexities in k

two classes of codebooks nearly meeting the welch bound in this paper the value of imax c for codebooks c c d constructed by certain almost difference sets d in fqx is determined and expressed in terms of jacobi sums from which it shows that such codebooks nearly meet the welch bound this result is an answer of a question raised by c ding and t feng in 2 we also present another series of codebooks which nearly meet the welch bound where the codebook is constructed by a subset r fq1 fq2 when q1 q2 de is an almost difference set of r

a family of alltop functions that are ea inequivalent to the cubic function sequences with optimal correlation properties are much sought after for applications in communication systems in 1980 alltop ieee trans inf theory 26 3 350 354 1980 described a set of sequences based on a cubic function and showed that these sequences were optimal with respect to the known bounds on auto and crosscorrelation subsequently these sequences were used to construct mutually unbiased bases mubs a structure of importance in quantum information theory the key feature of this cubic function is that its difference function is a planar function functions with planar difference functions have been called alltop functions this paper provides a new family of alltop functions and establishes the use of alltop functions for construction of sequence sets and mubs

new families of codebooks achieving the levenstein bound in this paper a construction of codebooks based on a set of bent functions satisfying certain conditions is introduced it includes some earlier constructions of codebooks meeting the levenstein bound as special cases with this construction two new families of codebooks achieving the levenstein bound are obtained the codebooks constructed in this paper could have a very small alphabet size

on the stopping distance of array code parity check matrices for q an odd prime and 1 les m les q we study two binary qm times q2 parity check matrices for binary array codes for both parity check matrices we determine the stopping distance and the minimum distance of the associated code for 2 les m les 3 and for m q 4 5 in the case m q 4 7 the stopping distance and the related minimum distance are also determined for one of the given parity check matrices moreover we give a lower bound on the stopping distances for m 3 and q 3

constructions of type ii qc ldpc codes with girth eight from sidon sequence in this paper we consider the constructions of type ii quasi cyclic qc low density parity check ldpc codes with girth eight from a sidon sequence we first derive the necessary and sufficient conditions guaranteeing a girth eight type ii qc ldpc code by combining these conditions with the concept of the sidon sequence three classes of type ii qc ldpc codes are subsequently proposed with girth eight to the best of our knowledge the second and the third classes we proposed are the first series of systematic and algebraic constructions for the girth eight type ii qc ldpc codes that have rates being greater than a half and at the same time have distance upper bounds being not limited to 12 via simulations we show the promising performance of the proposed type ii qc ldpc codes with girth eight in addition a set of general bounds and explicit random constructions without using a sidon sequence are also presented for the type ii qc ldpc codes with girth six or eight we observe that the proposed codes perform better than or equally well as the random qc ldpc codes from peg and quadr congr methods while the novel codes possess both highly structured parity check matrices and very flexible choices in circulant size

enumerators for protograph based ensembles of ldpc and generalized ldpc codes protograph based ldpc and generalized ldpc g ldpc codes have the advantages of a simple design procedure and highly structured encoders and decoders the design of such protograph based codes relies on what is effectively a computer based search as such following gallager it is prudent to restrict the search to a good ensemble for example an ensemble whose minimum distance grows linearly with codeword length a good ensemble can also mean one with good stopping set trapping set or pseudocodeword properties in this paper ensemble codeword weight enumerators for finite length ldpc and g ldpc codes based on protographs were derived and then the asymptotic case was considered the asymptotic results allow us to determine whether or not the typical relative minimum distance in the ensemble grows linearly with codeword length then the codeword weight enumerator technique is adapted to yield ensemble stopping set trapping set and pseudocodeword enumerators for protograph ldpc and g ldpc codes in this case the asymptotic results allow us to determine whether or not the typical relative smallest stopping set size trapping set size and pseudoweight grows linearly with codeword length trapping set enumerators for g ldpc code ensembles represent a more complex problem which we do not consider here

a matrix theoretic approach for analyzing quasi cyclic low density parity check codes a matrix theoretic approach for studying quasi cyclic codes based on matrix transformations via fourier transforms and row and column permutations is developed these transformations put a parity check matrix in the form of an array of circulant matrices into a diagonal array of matrices of the same size over an extension field the approach is amicable to the analysis and construction of quasi cyclic low density parity check codes since it takes into account the specific parity check matrix used for decoding with iterative message passing algorithms based on this approach the dimension of the codes and parity check matrices for the dual codes can be determined several algebraic and geometric constructions of quasi cyclic codes are presented as applications along with simulation results showing their performance over additive white gaussian noise channels decoded with iterative message passing algorithms

pseudocodewords of tanner graphs this paper presents a detailed analysis of pseudocodewords of tanner graphs pseudocodewords arising on the iterative decoder s computation tree are distinguished from pseudocodewords arising on finite degree lifts lower bounds on the minimum pseudocodeword weight are presented for the bec bsc and awgn channel some structural properties of pseudocodewords are examined and pseudocodewords and graph properties that are potentially problematic with min sum iterative decoding are identified an upper bound on the minimum degree lift needed to realize a particular irreducible lift realizable pseudocodeword is given in terms of its maximal component and it is shown that all irreducible lift realizable pseudocodewords have components upper bounded by a finite value t that is dependent on the graph structure examples and different tanner graph representations of individual codes are examined and the resulting pseudocodeword distributions and iterative decoding performances are analyzed the results obtained provide some insights in relating the structure of the tanner graph to the pseudocodeword distribution and suggest ways of designing tanner graphs with good minimum pseudocodeword weight

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

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

on finding bipartite graphs with a small number of short cycles and large girth the problem of finding bipartite tanner graphs with given degree sequences that have large girth and few short cycles is of great interest in many applications including construction of good low density parity check ldpc codes in this paper we prove that for given integers alpha beta and gamma and degree sequences pi and pi the problem of determining whether there exists a simple bipartite graph with degree sequences pi pi that has at most alpha beta and gamma cycles of length four six and eight respectively is np complete this is proved by a two step polynomial time reduction from the 3 partition problem on the other hand using connections to linear hypergraphs we prove that given the degree sequence pi a polynomial time algorithm can be devised to determine whether there exists a bipartite graph whose degree sequence on one side of the bipartition is pi and has a girth of at least six in addition to these complexity results we devise a quasi polynomial time algorithm that can construct a bipartite graph with given irregular degree sequences and a girth that increases logarithmically with the size of the graph compared to the well known progressive edge growth peg algorithm the proposed method in some cases results in tanner graphs with larger girth particularly for scenarios where the degree sequences of the graph are strictly enforced

pseudocodeword performance analysis for ldpc convolutional codes message passing iterative decoders for low density parity check ldpc block codes are known to be subject to decoding failures due to so called pseudocodewords these failures can cause the large signal to noise ratio snr performance of message passing iterative decoding to be worse than that predicted by the maximum likelihood ml decoding union bound in this paper we address the pseudocodeword problem from the convolutional code perspective in particular we compare the performance of ldpc convolutional codes with that of their wrapped quasi cyclic block versions and we show that the minimum pseudoweight of an ldpc convolutional code is at least as large as the minimum pseudoweight of an underlying quasi cyclic code this result which parallels a well known relationship between the minimum hamming weight of convolutional codes and the minimum hamming weight of their quasi cyclic counterparts is due to the fact that every pseudocodeword in the convolutional code induces a pseudocodeword in the block code with pseudoweight no larger than that of the convolutional code s pseudocodeword this difference in the weight spectra leads to improved performance at low to moderate snrs for the convolutional code a conclusion supported by simulation results

quasi cyclic ldpc codes based on pre lifted protographs quasi cyclic low density parity check qc ldpc codes based on protographs are of great interest to code designers because analysis and implementation are facilitated by the protograph structure and the use of circulant permutation matrices for protograph lifting however these restrictions impose undesirable fixed upper limits on important code parameters such as minimum distance and girth in this paper we consider an approach to constructing qc ldpc codes that uses a two step lifting procedure based on a protograph and by following this method instead of the usual one step procedure we obtain improved minimum distance and girth properties we also present two new design rules for constructing good qc ldpc codes using this two step lifting procedure and in each case we obtain a significant increase in minimum distance and achieve a certain guaranteed girth compared with one step circulant based liftings the expected performance improvement is verified by simulation results

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

controlling the error floor in ldpc decoding the error floor of ldpc is revisited as an effect of dynamic message behavior in the so called absorbing sets of the code it is shown that if the signal growth in the absorbing sets is properly balanced by the growth of set external messages the error floor can be lowered to essentially arbitrarily low levels importance sampling techniques are discussed and used to verify the analysis as well as to discuss the impact of iterations and message quantization on the code performance in the ultra low ber error floor regime

pseudo codeword analysis of tanner graphs from projective and euclidean planes we consider coded data transmission over a binary input output symmetric memoryless channel using a binary linear code in order to understand the performance of maximum likelihood ml decoding one studies the codewords in particular the minimal codewords and their hamming weights in the context of linear programming lp decoding one s attention needs to be shifted to the pseudo codewords in particular to the minimal pseudo codewords and their pseudo weights in this paper we investigate some families of codes that have good properties under lp decoding namely certain families of low density parity check ldpc codes that are derived from projective and euclidean planes we study the structure of their minimal pseudo codewords and give lower bounds on their pseudo weight besides this main focus we also present some results that hold for pseudo codewords and minimal pseudo codewords of any tanner graph and we highlight how the importance of minimal pseudo codewords under lp decoding varies depending on which binary input output symmetric memoryless channel is used

algebraic design and implementation of protograph codes using non commuting permutation matrices random lifts of graphs or equivalently random permutation matrices have been used to construct good families of codes known as protograph codes an algebraic analog of this approach was recently presented using voltage graphs and it was shown that many existing algebraic constructions of graph based codes that use commuting permutation matrices may be seen as special cases of voltage graph codes voltage graphs are graphs that have an element of a finite group assigned to each edge and the assignment determines a specific lift of the graph in this paper we discuss how assignments of permutation group elements to the edges of a base graph affect the properties of the lifted graph and corresponding codes and present a construction method of ldpc code ensembles based on non commuting permutation matrices we also show encoder and decoder implementations for these codes

reed solomon based quasi cyclic ldpc codes designs girth cycle structure and reduction of short cycles designs and constructions of quasi cyclic qc ldpc codes for the awgn channel are presented the codes are constructed based on the conventional parity check matrices of reed solomon rs codes and are referred to as rs qc ldpc codes several classes of rs qc ldpc codes are given cycle structural properties of the tanner graphs of codes in these classes are analyzed and specific methods for constructing codes with girth at least eight and reducing their short cycles are presented the designed codes perform well in both waterfall and low error rate regions

six new binary quasi cyclic codes six new quasi cyclic codes are presented which improve the lower bounds on the minimum distance for a binary code a local exhaustive search is used to find these codes and many other quasi cyclic codes which attain the lower bounds

a construction of codebooks associated with binary sequences an n k codebook is a set of n unit norm code vectors in a k dimensional vector space for its applications it is desired that the maximum magnitude of inner products between a pair of distinct code vectors should be as small as possible meeting the welch bound equality strictly or asymptotically in this paper an n k codebook is constructed from a k n partial matrix with k n where each code vector is equivalent to a column of the matrix to obtain the k n matrix k rows are selected from a j n matrix associated with a binary sequence of length j and hamming weight k where a set of the selected row indices is equivalent to the index set of nonzero entries of the binary sequence it is then discovered that the maximum magnitude of inner products between a pair of distinct code vectors is determined by the maximum magnitude of transform of the binary sequence thus constructing a codebook with small magnitude of inner products is equivalent to finding a binary sequence where the maximum magnitude of its transform is as small as possible from the discovery new classes of codebooks with nontrivial bounds on the maximum inner products are constructed from fourier and hadamard matrices associated with binary sequences

optimized design of finite length separable circulant based spatially coupled codes an absorbing set based analysis in this paper we characterize the finite length performance of separable circulant based spatially coupled scb sc ldpc codes for transmission over the additive white gaussian noise channel for a general class of finite length graph based codes it is known that the existence of small absorbing sets causes a performance degradation in the error floor regime we first present the mathematical conditions for the existence of absorbing sets in binary scb sc codes this analysis enables us to find the exact number of absorbing sets as a function of the design parameters in particular our results show that the choice of the cutting vector affects the number of absorbing sets and therefore the error floor performance of the code for a fixed column weight we find provably optimal cutting vectors that result in the least number of absorbing sets furthermore we extend our analysis to nonbinary scb sc codes where we show that the choice of the cutting vector is not as critical as in the binary case we provide an algorithm which provably removes the problematic nonbinary absorbing sets from nonbinary scb sc codes by informed selection of edge labels our simulation results show the superior error floor performance of our designed binary and nonbinary scb sc codes compared with binary unstructured and nonbinary quasi cyclic sc codes available in the open literature

complex codebooks from combinatorial designs codebooks also called signal sets meeting the welch bounds are desirable in code division multiple access cdma systems in 2003 binary codebooks meeting welch s bounds were constructed using difference sets by ding golin and kloslashve recently a generic construction of codebooks with cyclic difference sets meeting welch s bound on the maximum cross correlation amplitude was developed by xia in this correspondence the idea of xia is extended and related constructions of optimal codebooks with both cyclic and noncyclic difference sets are presented these codebooks are optimal in the sense that they also meet this welch bound in addition complex codebooks that almost meet welch s bound on the maximum cross correlation amplitude are also constructed with almost difference sets

searching for voltage graph based ldpc tailbiting codes with large girth the relation between parity check matrices of quasi cyclic qc low density parity check ldpc codes and biadjacency matrices of bipartite graphs supports searching for powerful ldpc block codes using the principle of tailbiting compact representations of bipartite graphs based on convolutional codes can be found bounds on the girth and the minimum distance of ldpc block codes constructed in such a way are discussed algorithms for searching iteratively for ldpc block codes with large girth and for determining their minimum distance are presented constructions based on all one matrices steiner triple systems and qc block codes are introduced finally new qc regular ldpc block codes with girth up to 24 are given

optimal codebooks from binary codes meeting the levenshtein bound in this paper a generic construction of codebooks based on binary codes is introduced with this generic construction a few previous constructions of optimal codebooks are extended and a new class of codebooks almost meeting the levenshtein bound is presented exponentially many codebooks meeting or almost meeting the levenshtein bound from binary codes are obtained in this paper the codebooks constructed in this paper have alphabet size 4 as a byproduct three bounds on the parameters of binary codes are derived

a beast for prowling in trees when searching for convolutional codes and tailbiting codes of high complexity it is of vital importance to use fast algorithms for computing their weight spectra which corresponds to finding low weight paths in their code trellises this can be efficiently done by a combined search in both forward and backward code trees a bidirectional efficient algorithm for searching such code trees beast is presented for large encoder memories it is shown that beast is significantly more efficient than comparable algorithms beast made it possible to find new convolutional and tailbiting codes that have larger free minimum distances than the previously best known codes with the same parameters tables of such codes are presented

coherence optimization and best complex antipodal spherical codes vector sets with optimal coherence according to the welch bound cannot exist for all pairs of dimension and cardinality if such an optimal vector set exists it is an equiangular tight frame and represents the solution to a grassmannian line packing problem best complex antipodal spherical codes bcascs are the best vector sets with respect to the coherence by extending methods used to find best spherical codes in the real valued euclidean space the proposed approach aims to find bcascs and thereby a complex valued vector set with minimal coherence there are many applications demanding vector sets with low coherence examples are not limited to several techniques in wireless communication or to the field of compressed sensing within this contribution existing analytical and numerical approaches for coherence optimization of complex valued vector spaces are summarized and compared to the proposed approach the numerically obtained coherence values improve previously reported results the drawback of increased computational effort is addressed and a faster approximation is proposed which may be an alternative for time critical cases

bounds on the minimum distance of punctured quasi cyclic ldpc codes recent work by divsalar has shown that properly designed protograph based low density parity check codes typically have minimum hamming distance linearly increasing with block length this fact rests on ensemble arguments over all possible expansions of the base protograph however when implementation complexity is considered the expansions are frequently selected from a smaller class of structured expansions for example protograph expansion by cyclically shifting connections generates a quasi cyclic qc code other recent work by smarandache and vontobel has provided upper bounds on the minimum distance of qc codes in this paper we generalize these bounds to punctured qc codes and then show how to tighten these for certain classes of codes we then evaluate these upper bounds for the family of protograph codes known as ar4ja codes that have been recommended for use in deep space communications in a standard established by the consultative committee for space data systems at block lengths larger than 4400 bits these upper bounds fall well below the ensemble lower bounds

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

a new approach to constructing quadratic pseudo planar functions over mathbb f _ 2 n planar functions over finite fields give rise to finite projective planes they were also used in the constructions of des like iterated ciphers error correcting codes and codebooks they were originally defined only in finite fields with odd characteristic but recently zhou introduced pesudo planar functions in even characteristic which yields similar applications all known pesudo planar functions are quadratic and hence they give presemifields in this paper a new approach to constructing quadratic pseudo planar functions is given then five explicit families of pseudo planar functions are constructed one of which is a binomial two of which are trinomials and the other two are quadrinomials all known pesudo planar functions are revisited some of which are generalized these functions not only lead to projective planes relative difference sets and presemifields but also give optimal codebooks meeting the levenstein bound complete sets of mutually unbiased bases and compressed sensing matrices with low coherence

lowering the error floor of ldpc codes using cyclic liftings cyclic liftings are proposed to lower the error floor of low density parity check ldpc codes the liftings are designed to eliminate dominant trapping sets of the base code by removing the short cycles which are part of the trapping sets we derive a necessary and sufficient condition for the cyclic permutations assigned to the edges of a cycle of length l in the base graph such that the inverse image of in the lifted graph consists of only cycles of length strictly larger than l the proposed method is universal in the sense that it can be applied to any ldpc code over any channel and for any iterative decoding algorithm it also preserves important properties of the base code such as degree distributions and in some cases the code rate the constructed codes are quasi cyclic and thus attractive from a practical point of view the proposed method is applied to both structured and random codes over the binary symmetric channel bsc the error floor improves consistently by increasing the lifting degree and the results show significant improvements in the error floor compared to the base code a random code of the same degree distribution and block length and a random lifting of the same degree similar improvements are also observed when the codes designed for the bsc are applied to the additive white gaussian noise awgn channel

from cages to trapping sets and codewords a technique to derive tight upper bounds on the minimum size of trapping sets and minimum distance of ldpc codes cages defined as regular graphs with minimum number of nodes for a given girth are well studied in graph theory trapping sets are graphical structures responsible for error floor of low density parity check ldpc codes and are well investigated in coding theory in this paper we make connections between cages and trapping sets in particular starting from a cage or a modified cage we construct a trapping set in multiple steps based on the connection between cages and trapping sets we then use the available results in graph theory on cages and derive tight upper bounds on the size of the smallest trapping sets for variable regular ldpc codes with a given variable degree and girth the derived upper bounds in many cases meet the best known lower bounds and thus provide the actual size of the smallest trapping sets considering that non zero codewords are a special case of trapping sets we also derive tight upper bounds on the minimum weight of such codewords i e the minimum distance of variable regular ldpc codes as a function of variable degree and girth

quasicyclic low density parity check codes from circulant permutation matrices in this correspondence the construction of low density parity check ldpc codes from circulant permutation matrices is investigated it is shown that such codes cannot have a tanner graph representation with girth larger than 12 and a relatively mild necessary and sufficient condition for the code to have a girth of 6 8 10 or 12 is derived these results suggest that families of ldpc codes with such girth values are relatively easy to obtain and consequently additional parameters such as the minimum distance or the number of redundant check sums should be considered to this end a necessary condition for the codes investigated to reach their maximum possible minimum hamming distance is proposed

quasi cyclic low density parity check codes with girth larger than 12 a quasi cyclic qc low density parity check ldpc code can be viewed as the protograph code with circulant permutation matrices or circulants in this correspondence we find all the subgraph patterns of protographs of qc ldpc codes having inevitable cycles of length 2i i 6 7 8 9 10 i e the cycles that always exist regardless of the shift values of circulants it is also derived that if the girth of the protograph is 2g g 2 its protograph code cannot have the inevitable cycles of length smaller than 6g based on these subgraph patterns we propose new combinatorial construction methods of the protographs whose protograph codes can have girth larger than or equal to 14 or 18 we also propose a couple of shift value assigning rules for circulants of a qc ldpc code guaranteeing the girth 14

near optimal partial hadamard codebook construction using binary sequences obtained from quadratic residue mapping in this paper a new class of n k near optimal partial hadamard codebooks is proposed the construction of the proposed codebooks from hadamard matrices is based on binary row selection sequences which are generated by quadratic residue mapping of p ary m sequences the proposed codebooks have parameters n p n and k p 1 2p n sqrt n 1 for an odd prime p and an even positive integer n we prove that the maximum magnitude of inner products between the code vectors of the proposed codebooks asymptotically achieves the welch bound equality for sufficiently large p and derive their inner product distribution

on meeting the peak correlation bounds in this paper we study the problem of meeting peak periodic or aperiodic correlation bounds for complex valued sets of sequences to this end the welch levenstein and exponential bounds on the peak inner product of sequence sets are considered and used to provide compound peak correlation bounds in both periodic and aperiodic cases the peak aperiodic correlation bound is further improved by using the intrinsic dimension deficiencies associated with its formulation in comparison to the compound bound the new aperiodic bound contributes an improvement of more than 35 for some specific values of the sequence length n and set cardinality m we study the tightness of the provided bounds by using both analytical and computational tools in particular novel algorithms based on alternating projections are devised to approach a given peak periodic or aperiodic correlation bound several numerical examples are presented to assess the tightness of the provided correlation bounds as well as to illustrate the effectiveness of the proposed methods for meeting these bounds

robust rate compatible punctured ldpc convolutional codes a family of robust rate compatible rc punctured low density parity check convolutional codes ldpc ccs is derived from a time invariant ldpc cc mother code by periodically puncturing encoded bits variable nodes with respect to several criteria 1 ensuring the recoverability of punctured variable nodes 2 minimizing the number of completely punctured cycle trapping sets cpctss and 3 minimizing the number of punctured variable nodes involved in short cycles the influence of 1 and 3 on iterative decoding performance is felt most strongly in the waterfall region of the bit error rate ber curve while 2 has a larger effect in the error floor or high signal to noise ratio snr region we show that the length of the puncturing period is an important parameter when designing high rate punctured codes and moreover that extending the puncturing period can improve the decoding performance and extend the range of compatible rates as examples we obtain families of rc ldpc ccs from several time invariant ldpc cc mother codes with monomial and binomial entries in their polynomial syndrome former matrices

quasi cyclic representation and vector representation of rs ldpc codes rs ldpc codes constructed based on the codewords of reed solomon rs codes with two information symbols are an important class of ldpc codes in this paper we present two representations namely quasi cyclic qc representation and vector representation for rs ldpc codes under the first representation most part of the parity check matrix of a full length rs ldpc code consists of circulant permutation matrices and zero matrices as a result the class of codes can enjoy the advantages in hardware implementation as qc ldpc codes in addition the base matrix under the qc representation of an rs ldpc code can be explicitly given such that the rank of its parity check matrix can be analyzed combinatorially under the second representation each permutation matrix in the parity check matrix of an rs ldpc code is defined by a nonbinary vector whose entries are a permutation of entries in the field from which the rs code is constructed then the affine invariance property is proved for full length rs ldpc codes which can facilitate the structural analysis of the codes

zeta functions of finite graphs and coverings iii abstract a graph theoretical analog of brauer siegel theory for zeta functions of number fields is developed using the theory of artin l functions for galois coverings of graphs from parts i and ii in the process we discuss possible versions of the riemann hypothesis for the ihara zeta function of an irregular graph

some best rate 1 p and rate p 1 p systematic quasi cyclic codes tables of 1 p and rate p 1 p binary quasi cyclic qc codes that extend previously published results are presented many of these codes attain bounds given by t verhoeff 1987 who composed a table of bounds on the maximum possible minimum distance of binary linear codes many of the codes presented meet or improve these bounds a best code is considered to be one which has the largest possible minimum distance for the given code dimensions n and k and class of error correcting codes a good code has the maximum known minimum distance for the class of codes an optimal code is one that achieves the maximum possible minimum distance for a linear code with the same dimensions binary power residue codes are found and used to construct qc codes

uep constructions of quasi cyclic low density parity check codes via masking in this paper the algebraic constructions of quasi cyclic low density parity check qc ldpc codes with the unequal error protection uep property are considered a criterion for constructing such codes via the masking technique is proposed based on which explicit conditions on the base parity check matrices and masking matrices to achieve uep are provided we also give three specific constructions of uep qc ldpc codes furthermore a sufficient condition to ensure strict uep is presented simulation results demonstrate the superiority of our constructed codes over time sharing schemes the constructed codes also have competitive error performance against randomly constructed equal error protection ldpc codes and irregular uep ldpc codes designed based on the degree distribution

optimal reduction of two terminal directed acyclic graphs algorithms for series parallel graphs can be extended to arbitrary two terminal dags if node reductions are used along with series and parallel reductions a node reduction contracts a vertex with unit in degree out degree into its sole incoming outgoing neighbor this paper gives an o n 2 5 algorithm for minimizing node reductions based on vertex cover in a transitive auxiliary graph applications include the analysis of pert networks dynamic programming approaches to network problems and network reliability for np hard problems one can obtain algorithms that are exponential only in the minimum number of node reductions rather than the number of vertices this gives improvements if the underlying graph is nearly series parallel

new constructions of codebooks nearly meeting the welch bound with equality an n k codebook c is a set of n unit norm complex vectors in bbck optimal codebooks meeting the welch bound with equality are desirable in a number of areas however it is very difficult to construct such optimal codebooks there have been a number of attempts to construct codebooks nearly meeting the welch bound with equality i e the maximal cross correlation amplitude imax c is slightly higher than the welch bound equality but asymptotically achieves it for large enough n in this paper using difference sets and the product of abelian groups we propose new constructions of codebooks nearly meeting the welch bound with equality our methods yield many codebooks with new parameters in some cases our constructions are comparable to known constructions

design of multiple edge protographs for qc ldpc codes avoiding short inevitable cycles there have been lots of efforts on the construction of quasi cyclic qc low density parity check ldpc codes with large girth however most of them focus on protographs with single edges and little research has been done for the construction of qc ldpc codes lifted from protographs with multiple i e parallel edges compared to single edge protographs multiple edge protographs have benefits such that qc ldpc codes lifted from them can potentially have larger minimum hamming distance in this paper all subgraph patterns of multiple edge protographs which prevent qc ldpc codes from having large girth by inducing inevitable cycles are fully investigated based on a graph theoretic approach by using combinatorial designs a systematic construction method of multiple edge protographs is proposed for regular qc ldpc codes with girth at least 12 and another method is proposed for regular qc ldpc codes with girth at least 14 moreover a construction algorithm of qc ldpc codes based on certain liftings of multiple edge protographs is proposed and it is shown that the resulting qc ldpc codes have larger upper bounds on the minimum hamming distance than those lifted from single edge protographs simulation results are provided to compare the performance of the proposed qc ldpc codes with progressive edge growth peg ldpc codes and with peg qc ldpc codes

quasi cyclic ldpc codes influence of proto and tanner graph structure on minimum hamming distance upper bounds quasi cyclic qc low density parity check ldpc codes are an important instance of proto graph based ldpc codes in this paper we present upper bounds on the minimum hamming distance of qc ldpc codes and study how these upper bounds depend on graph structure parameters like variable degrees check node degrees girth of the tanner graph and of the underlying proto graph moreover for several classes of proto graphs we present explicit qc ldpc code constructions that achieve or come close to the respective minimum hamming distance upper bounds because of the tight algebraic connection between qc codes and convolutional codes we can state similar results for the free hamming distance of convolutional codes in fact some qc code statements are established by first proving the corresponding convolutional code statements and then using a result by tanner that says that the minimum hamming distance of a qc code is upper bounded by the free hamming distance of the convolutional code that is obtained by unwrapping the qc code

eliminating trapping sets in low density parity check codes by using tanner graph covers we discuss error floor asympotics and present a method for improving the performance of low density parity check ldpc codes in the high snr error floor region the method is based on tanner graph covers that do not have trapping sets from the original code the advantages of the method are that it is universal as it can be applied to any ldpc code channel decoding algorithm and it improves performance at the expense of increasing the code length without losing the code regularity without changing the decoding algorithm and under certain conditions without lowering the code rate the proposed method can be modified to construct convolutional ldpc codes also the method is illustrated by modifying tanner mackay and margulis codes to improve performance on the binary symmetric channel bsc under the gallager b decoding algorithm decoding results on awgn channel are also presented to illustrate that optimizing codes for one channel decoding algorithm can lead to performance improvement on other channels

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

on the minimum distance of array codes as ldpc codes for a prime q and an integer j spl les q the code c q j is a class of low density parity check ldpc codes from array codes which has a nice algebraic structure in this correspondence we investigate the minimum distance d q j of the code in an algebraic way we first prove that the code is invariant under a doubly transitive group of affine permutations then we show that d 5 4 8 d 7 4 8 and d q 4 spl ges 10 for any prime q 7 in addition we also analyze the codewords of weight 6 in the case of j 3 and the codewords of weight 8 in c 5 4 and c 7 4

a link between quasi cyclic codes and convolutional codes given a quasi cyclic code with minimum hamming distance d a set of convolutional codes is derived with free distance equal to d it is shown that an increase in the rate of these codes results in a decrease in the memory length the connection between these codes is illustrated with several well known quasi cyclic codes the free distance of some partial unit memory convolutional codes can be determined using the results in this correspondence