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

linear block codes over cyclic groups the main building block for the construction of a geometrically uniform coded modulation scheme is a subgroup of g sup i where g is a group generating a low dimensional signal constellation and i is an index set in this paper we study the properties of these subgroups when g is cyclic we exploit the fact that any cyclic group of q elements is isomorphic to the additive group of z sub q the ring of integers modulo q so that we can make use of concepts related to linearity our attention is focused mainly on indecomposable cyclic groups i e of prime power order since they are the elementary building blocks of any abelian group in analogy with the usual construction of linear codes over fields we define a generator matrix and a parity check matrix trellis construction and bounds on the minimum euclidean distance are also investigated some examples of coded modulation schemes based on this theory are also exhibited and their performance evaluated

analysis and enumeration of absorbing sets for non binary graph based codes in this work we first provide the definition of absorbing sets for linear channel codes over non binary alphabets in a graphical representation of a non binary channel code an absorbing set can be described by a collection of topological and edge labeling conditions in the non binary case the equations relating neighboring variable and check nodes are over a non binary field and the edge weights are given by the non zero elements of that non binary field as a consequence it becomes more difficult for a given structure to satisfy the absorbing set constraints compared to the binary case this observation in part explains the superior performance of non binary codes over their binary counterparts we show that the conditions in the non binary absorbing set definition can be simplified in the case of non binary elementary absorbing sets based on these simplified conditions we provide design guidelines for finite length non binary codes free of small non binary elementary absorbing sets these guidelines demonstrate that even under the preserved topology the performance of a non binary graph based code in the error floor region can be substantially improved by manipulating edge weights so as to avoid small absorbing sets our various simulation results suggest that the proposed non binary absorbing set definition is useful for a range of code constructions and decoders finally by using both insights from graph theory and combinatorial techniques we establish the asymptotic distribution of non binary elementary absorbing sets for regular code ensembles

rational cohomology of witt groups let k be an algebraically closed field of characteristic p and for each n 0 let w n denote the group of witt vectors of length n w n is a commutative algebraic group for reference see jacobson 2 serre 6 one of the important properties of the witt groups is the following every commutative algebraic k group whose underlying variety is an affine space is a homomorphic image of products of w n we compute the rational cohomology of w n for n 2 h w n k s v n 1 1 l e rn 1 l

intersecting codes and independent families a binary intersecting code is a linear code with the property that any two nonzero codewords have intersecting supports these codes appear in a wide variety of contexts and applications e g multiple access cryptography and information theory this paper is devoted partly to the study of intersecting codes and partly to their use in constructing large t independent families of binary vectors the latter subject has by now been extensively studied and has application in vlsi testing defect correction e biased probability spaces and derandomization by concatenation methods we construct codes with the highest known fate asymptotically we then generalize the concept to t wise intersecting codes we give bounds on the achievable rate of such codes both existential and constructive we show how t wise intersecting codes can be used to obtain t 1 independent families with this method we obtain improved asymptotical constructions of t independent families complexity issues are discussed

cohomology of infinitesimal and discrete groups an infinitesimal algebraic group over a field k is a connected algebraic group scheme whose coordinate ring is a finite dimension al k algebra if one begins with a reductive algebraic group g defined over fp then one has a close relationship between the representation theory of g that of its infinitesimal subgroups g defined as the kernels of the frobenius endomorphisms rr g g and that of its finite chevalley subgroups g fq of fq rational points q pd a basic theorem of cline et al i 9 theorem 6 6 relates the cohomology of g in a rational representation to that of the finite groups g fq related work of cline et al 6 and of the authors 1 16 relates the cohomology of g to that of its infinitesimal subgroups a determination of the cohomology of the first infinitesimal subgroup g 1 of g with trivial coefficients has enabled the authors in 14 to make certain explicit cohomology calculations for the finite groups g fq with certain nontrivial coefficients in this paper we continue our study of infinitesimal subgroups of simple algebraic groups by proving various qualitative properties of their cohomology and by extending our computational knowledge in particular we provide a computation which applies in a range of degrees to chevalley groups over the prime field we also investigate the cohomology of infinitesimal subgroups of nilpotent groups enabling us for example to determine the cohomology of their lie algebras in certain cases we apply this lie algebra cohomology computation to determine the integral cohomology of various torsion free nilpotent groups modulo small prime torsion thereby extending a theorem of dwyer 12 in more detail the paper begins in sect 1 with a spectral sequence converging to the cohomology of the first infinitesimal subgroup g which was previously studied in 14 from this spectral sequence it easily follows that h g1 m is a noetherian h g1 k module for each finite dimensional rational gl module m we then apply the work of andersen and jantzen 1 on the gl cohomology of induced modules to obtain the analogous result for the second infinitesimal subgroup gz our analysis also permits a determination of the generic

minimality and observability of group systems abstract group systems are a generalization of willems type linear systems that are useful in error control coding it is shown that the basic ideas of willems s treatment of linear systems are easily generalized to linear systems over arbitrary rings and to group systems the interplay between systems behaviors and trellises evolution laws is discussed with respect to completeness minimality controllability and observability it is pointed out that for trellises of group systems and willems type linear systems minimality is essentially the same as observability the development is universal algebraic in nature and holds unconditionally for linear systems over the real numbers

asymptotic improvement of the gilbert varshamov bound on the size of binary codes given positive integers n and d let a sub 2 n d denote the maximum size of a binary code of length n and minimum distance d the well known gilbert varshamov bound asserts that a sub 2 n d spl ges 2 sup n v n d l where v n d spl sigma sub i 0 sup d sub i sup n is the volume of a hamming sphere of radius d we show that in fact there exists a positive constant c such that a sub 2 n d spl ges c2 sup n v n d 1 log sub 2 v n d 1 whenever d n spl les 0 499 the result follows by recasting the gilbert varshamov bound into a graph theoretic framework and using the fact that the corresponding graph is locally sparse generalizations and extensions of this result are briefly discussed

some structural properties of convolutional codes over rings convolutional codes over rings have been motivated by phase modulated signals some structural properties of the generator matrices of such codes are presented successively stronger notions of the invertibility of generator matrices are studied and a new condition for a convolutional code over a ring to be systematic is given and shown to be equivalent to a condition given by massey and mittelholzer 1990 it is shown that a generator matrix that can be decomposed into a direct sum is basic minimal and noncatastrophic if and only if all generator matrices for the constituent codes are basic minimal and noncatastrophic respectively it is also shown that if a systematic generator matrix can be decomposed into a direct sum then all generator matrices of the constituent codes are systematic but that the converse does not hold some results on convolutional codes over z p sup e are obtained

distributed source coding using abelian group codes a new achievable rate distortion region a distributed source coding problem with a joint distortion criterion that depends on the sources and the reconstruction is considered in this work while the prevalent trend in information theory has been to prove achievability results using shannon s random coding arguments using structured random codes offer rate gains over unstructured random codes for many problems motivated by this a new achievable rate distortion region an inner bound to the performance limit is presented for this problem for discrete memoryless sources based on good structured random nested codes built over abelian groups for certain sources and distortion functions the new rate region is shown to be strictly bigger than the berger tung rate region which has been the best known achievable rate region for this problem till now this is done using numerical plots achievable rates for single user source coding using abelian group codes are also obtained as a corollary of the main coding theorem it is shown that nested linear codes achieve the shannon rate distortion function in the arbitrary discrete memoryless case

new lower bounds for asymmetric and unidirectional codes new lower bounds on the sizes of asymmetric codes and unidirectional codes are presented various methods are used three of them of special interest the first is a partitioning method that is a modification of a method used to construct constant weight codes the second is a combining codes method that is used to obtain a new code from a few others the third method is shortening by weights that is applied on symmetric codes or on codes generated by the combining codes method tables for the sizes of codes of length n

linear universal decoding for compound channels over discrete memoryless channels dmc linear decoders maximizing additive metrics afford several nice properties in particular if suitable encoders are employed the use of decoding algorithms with manageable complexities is permitted for a compound dmc decoders that perform well without the channel s knowledge are required in order to achieve capacity several such decoders have been studied in the literature however there is no such known decoder which is linear hence the problem of finding linear decoders achieving capacity for compound dmc is addressed and it is shown that under minor concessions such decoders exist and can be constructed a geometric method based on the very noisy transformation is developed and used to solve this problem

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

group codes outperform binary coset codes on nonbinary symmetric memoryless channels typical minimum distances and error exponents are analyzed on the 8 psk gaussian channel for two capacity achieving code ensembles with different algebraic structure it is proved that the typical group code over the cyclic group of order eight achieves both the gilbert varshamov bound and the expurgated error exponent on the other hand the typical binary coset codes under any labeling is shown to be bounded away both from the gilbert varshamov bound at any rate and the expurgated exponent at low rates the reason for this phenomenon is shown to rely on the symmetry structure of the 8 psk constellation which is known to match the cyclic group of order eight but not the direct product of three copies of the binary group the presented results indicate that designing group codes matching the symmetry of the channel guarantees better typical code performance than designing codes whose algebraic structure does not match the channel this contrasts the well known fact that the average binary coset code achieves both the capacity and the random coding error exponent of any discrete memoryless channel

generalized hamming weights of linear codes the generalized hamming weight d sub r c of a binary linear code c is the size of the smallest support of any r dimensional subcode of c the parameter d sub r c determines the code s performance on the wire tap channel of type ii bounds on d sub r c and in some cases exact expressions are derived in particular a generalized griesmer bound for d sub r c is presented and examples are given of codes meeting this bound with equality

abelian group codes for channel coding and source coding in this paper we study the asymptotic performance of abelian group codes for the channel coding problem for arbitrary discrete finite alphabet memoryless channels as well as the lossy source coding problem for arbitrary discrete finite alphabet memoryless sources for the channel coding problem we find the capacity characterized in a single letter information theoretic form this simplifies to the symmetric capacity of the channel when the underlying group is a field for the source coding problem we derive the achievable rate distortion function that is characterized in a single letter information theoretic form when the underlying group is a field it simplifies to the symmetric rate distortion function we give several illustrative examples due to the nonsymmetric nature of the sources and channels considered our analysis uses a synergy of information theoretic and group theoretic tools

multilevel channel polarization for arbitrary discrete memoryless channels it is shown that polar codes with their original u u v kernel achieve the symmetric capacity of discrete memoryless channels with arbitrary input alphabet sizes it is shown that in general channel polarization happens in several rather than only two levels so that the synthesized channels are either useless perfect or partially perfect any subset of the channel input alphabet which is closed under addition induces a coset partition of the alphabet through its shifts for any such partition of the input alphabet there exists a corresponding partially perfect channel whose outputs uniquely determine the coset to which the channel input belongs by a slight modification of the encoding and decoding rules it is shown that perfect transmission of certain information symbols over partially perfect channels is possible our result is general regarding both the cardinality and the algebraic structure of the channel input alphabet i e we show that for any channel input alphabet size and any abelian group structure on the alphabet polar codes are optimal due to the modifications we make to the encoding rule of polar codes the constructed codes fall into a larger class of structured codes called nested group codes

geometrically uniform tcm codes over groups based on l spl times mpsk constellations the theory of geometrically uniform signal sets and codes over groups is applied to the case of l spl times mpsk constellations conditions for rotational invariance of group codes are discussed the tables of geometrically uniform partitions found in benedetto et al 1993 are used to construct good geometrically uniform trellis codes over nonbinary abelian groups the present authors consider l spl times 4psk and l spl times 8psk constellations used to transmit information rates of 1 and 2 bit two dimensions respectively and present tables of good codes over generating groups z4 sup l and z8 sup l for l ranging from 1 to 4 in most cases they improve the tables of codes known so far moreover the geometrical uniformity of codes allows a very easy performance evaluation so that the authors also present a complete set of curves of error event probability for the obtained codes

the dynamics of group codes dual abelian group codes and systems fundamental results concerning the dynamics of abelian group codes behaviors and their duals are developed duals of sequence spaces over locally compact abelian lca groups may be defined via pontryagin duality dual group codes are orthogonal subgroups of dual sequence spaces the dual of a complete code or system is finite and the dual of a laurent code or system is anti laurent if c and c sup spl perp are dual codes then the state spaces of c act as the character groups of the state spaces of c sup spl perp the controllability properties of c are the observability properties of c sup spl perp in particular c is strongly controllable if and only if c sup spl perp is strongly observable and the controller memory of c is the observer memory of c sup spl perp the controller granules of c act as the character groups of the observer granules of c sup spl perp examples of minimal observer form encoder and syndrome former constructions are given finally every observer granule of c is an end around controller granule of c

signal sets matched to groups recently linear codes over z sub m the ring of integers mod m have been presented that are matched to m ary phase modulation the general problem of matching signal sets to generalized linear algebraic codes is addressed based on these codes a definition is given for the notion of matching it is shown that any signal set in n dimensional euclidean space that is matched to an abstract group is essentially what d slepian 1968 called a group code for the gaussian channel if the group is commutative this further implies that any such signal set is equivalent to coded phase modulation with linear codes over z sub m some further results on such signal sets are presented and the signal sets matched to noncommutative groups and the linear codes over such groups are discussed

on the kazhdan lusztig polynomials for affine weyl groups in 191 kazhdan and lusztig associated to a coxeter group w certain polynomials p y w e w in the case where w is a weyl group these polynomials or their values at 1 p l give multiplicities of irreducible constituents in the jordan holder series of verma modules for a complex semisimple lie algebra corresponding to w see 3 4 also in the case where w is an affine weyl group it has been conjectured by lusztig lo that these polynomials or their values p l enter the character formula of irreducible rational representations for a semisimple algebraic group corresponding to w over an algebraically closed field of prime characteristic the purpose of this paper is to give some formula 4 2 see also 4 10 concerning the kazhdan lusztig polynomials p for affine weyl groups which appear in this lusztig conjecture namely p for y w dominant this is a generalization of the q analogue of kostant s weight multiplicity formula in 8 as a corollary of the formula we shall show that the lusztig conjecture above is consistent with the steinberg tensor product theorem 15 1 to be more precise let p be the weight lattice of a root system r ii r a positive root system the weyl group w of r acts on p hence we can define the semidirect product of w by p p w d p the element of corresponding to a e p is denoted by t the group contains an afline weyl group w w d q as a normal subgroup where q is the root lattice although is not a coxeter group in general we can define the bruhat order the length function i p la and the kazhdan lusztig polynomials p y w e i v as natural extensions of those for w the main result of this paper under the specialization q 1 is stated as follows corollary 4 10

on the equivalence between kernel quadrature rules and random feature expansions we show that kernel based quadrature rules for computing integrals can be seen as a special case of random feature expansions for positive definite kernels for a particular decomposition that always exists for such kernels we provide a theoretical analysis of the number of required samples for a given approximation error leading to both upper and lower bounds that are based solely on the eigenvalues of the associated integral operator and match up to logarithmic terms in particular we show that the upper bound may be obtained from independent and identically distributed samples from a specific non uniform distribution while the lower bound if valid for any set of points applying our results to kernel based quadrature while our results are fairly general we recover known upper and lower bounds for the special cases of sobolev spaces moreover our results extend to the more general problem of full function approximations beyond simply computing an integral with results in l2 and l norm that match known results for special cases applying our results to random features we show an improvement of the number of random features needed to preserve the generalization guarantees for learning with lipshitz continuous losses

the ml decoding performance of ldpc ensembles over z sub q we derive the asymptotic spectra of low density parity check ldpc ensembles over z sub q we consider two ensembles of ldpc matrices one is binary and the other q ary we also show that for modulo additive noise channels both ensembles achieve the random coding error exponent for graphs with sufficiently large connectivity

non binary protograph based ldpc codes enumerators analysis and designs this paper provides a comprehensive analysis of nonbinary low density parity check ldpc codes built out of protographs we consider both random and constrained edge weight labeling and refer to the former as the unconstrained nonbinary protograph based ldpc codes u nbpb codes and to the latter as the constrained nonbinary protograph based ldpc codes c nbpb codes equipped with combinatorial definitions extended to the nonbinary domain ensemble enumerators of codewords trapping sets stopping sets and pseudocodewords are calculated the exact enumerators are presented in the finite length regime and the corresponding growth rates are calculated in the asymptotic regime an exit chart tool for computing the iterative decoding thresholds of protograph based ldpc codes is presented followed by several examples of finite length u nbpb and c nbpb codes with high performance throughout this paper we provide accompanying examples which demonstrate the advantage of nonbinary protograph based ldpc codes over their binary counterparts and over random constructions the results presented in this paper advance the analytical toolbox of nonbinary graph based codes

cohomology of infinitesimal algebraic groups in this paper we study the cohomology h br k of a higher frobenius kernel br r 1 of a borel subgroup b of a reductive algebraic group g over an algebraically closed field k of a positive characteristic p for the first frobenius kernel b 1 the cohomology has been completely determined by h h andersen and j c jantzen 3 in case p h the coxeter number of g in low degrees the cohomology is independent of r 1 and their results allow us to compute the first different case h zp i br k for s lt i 3 6 in higher degrees however we have to resort to a different method although h b 2 k has already been computed for sl z also in 3 it is a considerably harder problem for sl 3 we apply algebraic steenrod operations and the transgression theorem to the lyndon hochschildserre or cartan eilenberg spectral sequence arising from a central series of the unipotent radical u of b to find h k for sl 3 5 1 the method we employ here was originally developed by m tezuka and n yagita 21 in order to investigate h u fq k q a power ofp in fact the results of sections 4 and 5 were first conceived from the corresponding results of h u fq k aside from the interests in relation to the cohomology of finite groups the work was also motivated by a close connection of the bl cohomology to g lusztig s conjectural character formula for the irreducible modules of the reductive group g which is described in section 2 hoping that the representation theory may shed more light on some topological problems we intend to take up the steenrod algebra and the a algebras in a subsequent paper

an inversion formula for the kazhdan lusztig polynomials for affine weyl groups let w be a coxeter group in 10 kazhdan and lusztig defined for each y w e w certain polynomials py w over 7 they conjectured that in the case where w is the weyl group corresponding to a complex semi simple lie algebra 5 these polynomials determine the characters of the irreducible is modules this conjecture has now been proved see 3 4 later g lusztig 11 formulated a corresponding conjecture for the irreducible modules for a semi simple algebraic group g defined over an algebraically closed field of prime characteristic p involving the same polynomials except that w is now the affine weyl group corresponding to g elementary properties of the polynomials py w show that there exist polynomials q x z w such that the inversion formula

convolutional codes over groups the basic algebraic structure theory of convolutional codes and their trellises is developed simultaneously for codes over groups rings and fields the first part which covers fundamental notions such as minimality and observability is semi tutorial in that most definitions are already standard within the modern behavioral theory as are some of the formally stated results however some of the pivotal results emphasizing the role of observability as the basic well behavedness condition for codes are new and several previous results are given simplified proofs the usefulness of the behavioral approach even for convolutional codes over fields is demonstrated by a new minimality test for encoders as well as by the straightforward derivation of some known minimality criteria for generator matrices from the basic minimality criteria for group trellises the second part of the paper deals with issues that are specific to codes over rings and groups the main result is a concise characterization the first such of those groups that can appear as the branch group of any group trellis it is further shown how such groups are presented by shift registers a new large class of noncommutative convolutional codes is also given

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

bounds on distance distributions in codes of known size we treat the problem of bounding components of the possible distance distributions of codes given the knowledge of their size and possibly minimum distance using the beckner inequality from harmonic analysis we derive upper bounds on distance distribution components which are sometimes better than earlier ones due to ashikhmin barg and litsyn we use an alternative approach to derive upper bounds on distance distributions in linear codes as an application of the suggested estimates we get an upper bound on the undetected error probability for an arbitrary code of given size we also use the new bounds to derive better upper estimates on the covering radius as well as a lower bound on the error probability threshold as a function of the code s size and minimum distance

on nilpotent commuting varieties and cohomology of frobenius kernels abstract the paper studies the dimensions of irreducible components of commuting varieties of restricted nilpotent r tuples in a classical lie algebra g defined over an algebraically closed field k as applications we obtain some new results on the structure of the even cohomology ring of frobenius kernels g r for each r 1 where g is the simply connected simple algebraic group such that lie g g explicit calculations for rank two groups are also presented

on a characterization of operators from lq into a banach space of type p with some applications to eigenvalue problems abstract let s be an operator admitting a factorization where d is a diagonal operator and t an arbitrary operator with the image in a banach space of type p we shall characterize these operators by entropy numbers using a well known inequality between eigenvalues and entropy numbers we establish some results about distributions of eigenvalues

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

on the generalized hamming weights of cyclic codes we prove several results on the generalized hamming weights ghw s of linear codes particularly for cyclic codes based on these and previously known results we give some efficient algorithms for computing ghw hierarchy of cyclic codes we give complete weight hierarchy for each of the binary cyclic codes of odd lengths spl les 31 a table of second and third ghw s of binary cyclic codes of odd lengths spl les 57 is also presented we have also computed the second ghw of all binary cyclic codes of length 63 and the third ghw of one code from each dimension

eigenvalues of compact operators with applications to integral operators abstract we give summability results for the eigenvalues of certain types of compact operators that are then applied to study integral operators

quasi structured codes for multi terminal communications a new class of structured codes called quasi group codes qgcs is introduced a qgc is a subset of a group code in contrast with the group codes qgcs are not closed under group addition the parameters of the qgc can be chosen such that the size of mathcal c mathcal c is equal to any number between mathcal c and mathcal c 2 we analyze the performance of a specific class of qgcs this class of qgcs is constructed by assigning single letter distributions to the indices of the codewords in a group code then the qgc is defined as the set of codewords whose index is in the typical set corresponding to these single letter distributions the asymptotic performance limits of this class of qgcs are characterized using single letter information quantities corresponding covering and packing bounds are derived it is shown that the point to point channel capacity and optimal rate distortion function are achievable using qgcs coding strategies based on qgcs are introduced for three fundamental multi terminal problems the korner marton problem for modulo prime power sums computation over the multiple access channel mac and mac with distributed states for each problem a single letter achievable rate region is derived it is shown through examples that the coding strategies improve upon the previous strategies based on the unstructured codes linear codes and group codes

distributed structure joint expurgation for the multiple access channel in this paper we obtain an improved lower bound on the error exponent of the memoryless multiple access channel via the use of linear codes thus demonstrating that structure can be beneficial even when capacity may be achieved via random codes we show that if the multiple access channel is additive over a finite field then any error probability and hence any error exponent achievable by a linear code for the associated single user channel is also achievable for the multiple access channel in particular linear codes allow to attain joint expurgation and hence attain the single user expurgated exponent of the single user channel whenever the latter is achieved by a uniform distribution thus for additive channels at low rates where expurgation is needed our approach strictly improves performance over previous results where expurgation was used for at most one of the users even when the multiple access channel is not additive it may be transformed into such a channel while the transformation is information lossy we show that the distributed structure gain in some nearly additive cases outweighs the loss finally we apply a similar approach to the gaussian multiple access channel while we believe that due to the power constraints it is impossible to attain the single user error exponent we do obtain an improvement over the best known achievable error exponent given by gallager for certain parameters this is accomplished using a nested lattice triplet with judiciously chosen parameters

design and analysis of nonbinary ldpc codes for arbitrary discrete memoryless channels we present an analysis under the iterative decoding of coset low density parity check ldpc codes over gf q designed for use over arbitrary discrete memoryless channels particularly nonbinary and asymmetric channels we use a random coset analysis to produce an effect that is similar to output symmetry with binary channels we show that the random selection of the nonzero elements of the gf q parity check matrix induces a permutation invariance property on the densities of the decoder messages which simplifies their analysis and approximation we generalize several properties including symmetry and stability from the analysis of binary ldpc codes we show that under a gaussian approximation the entire q 1 dimensional distribution of the vector messages is described by a single scalar parameter like the distributions of binary ldpc messages we apply this property to develop extrinsic information transfer exit charts for our codes we use appropriately designed signal constellations to obtain substantial shaping gains simulation results indicate that our codes outperform multilevel codes at short block lengths we also present simulation results for the additive white gaussian noise awgn channel including results within 0 56 db of the unrestricted shannon limit i e not restricted to any signal constellation at a spectral efficiency of 6 bits s hz

regular and irregular progressive edge growth tanner graphs we propose a general method for constructing tanner graphs having a large girth by establishing edges or connections between symbol and check nodes in an edge by edge manner called progressive edge growth peg algorithm lower bounds on the girth of peg tanner graphs and on the minimum distance of the resulting low density parity check ldpc codes are derived in terms of parameters of the graphs simple variations of the peg algorithm can also be applied to generate linear time encodeable ldpc codes regular and irregular ldpc codes using peg tanner graphs and allowing symbol nodes to take values over gf q q 2 are investigated simulation results show that the peg algorithm is a powerful algorithm to generate good short block length ldpc codes

performance bounds for nonbinary linear block codes over memoryless symmetric channels the performance of nonbinary linear block codes is studied in this paper via the derivation of new upper bounds on the block error probability under maximum likelihood ml decoding the transmission of these codes is assumed to take place over a memoryless and symmetric channel the new bounds which are based on the gallager bounds and their variations are applied to the gallager ensembles of nonbinary and regular low density parity check ldpc codes these upper bounds are also compared with sphere packing lower bounds this study indicates that the new upper bounds are useful for the performance evaluation of coded communication systems which incorporate nonbinary coding techniques

finite length scaling for iteratively decoded ldpc ensembles we investigate the behavior of iteratively decoded low density parity check ldpc codes over the binary erasure channel in the so called ldquowaterfall region rdquo we show that the performance curves in this region follow a simple scaling law we conjecture that essentially the same scaling behavior applies in a much more general setting and we provide some empirical evidence to support this conjecture the scaling law together with the error floor expressions developed previously can be used for a fast finite length optimization

decoding algorithms for nonbinary ldpc codes over gf q in this letter we address the problem of decoding nonbinary low density parity check ldpc codes over finite fields gf q with reasonable complexity and good performance in the first part of the letter we recall the original belief propagation bp decoding algorithm and its fourier domain implementation we show that the use of tensor notations for the messages is very convenient for the algorithm description and understanding in the second part of the letter we introduce a simplified decoder which is inspired by the min sum decoder for binary ldpc codes we called this decoder extended min sum ems we show that it is possible to greatly reduce the computational complexity of the check node processing by computing approximate reliability measures with a limited number of values in a message by choosing appropriate correction factors or offsets we show that the ems decoder performance is quite good and in some cases better than the regular bp decoder the optimal values of the factor and offset correction are obtained asymptotically with simulated density evolution our simulations on ultra sparse codes over very high order fields show that nonbinary ldpc codes are promising for applications which require low frame error rates for small or moderate codeword lengths the ems decoder is a good candidate for practical hardware implementations of such codes

geometrically uniform partitions of l mpsk constellations and related binary trellis codes the theory of geometrically uniform trellis codes is applied to the case of multidimensional psk phase shift keying constellations the symmetry group of an l mpsk m ary psk constellation is completely characterized conditions for rotational invariance of geometrically uniform partitions of a signal constellation are given through suitable algorithms geometrically uniform partitions of l mpsk m 4 8 16 and l 1 2 3 4 constellations are found which present good characteristics in terms of the set of distances at a given partition level the maximum obtainable rotational invariance and the isomorphism of the quotient group associated with the partition these partitions are used as starting points in a search for good geometrically uniform trellis codes based on binary convolutional codes

the dynamics of group codes state spaces trellis diagrams and canonical encoders a group code c over a group g is a set of sequences of group elements that itself forms a group under a component wise group operation a group code has a well defined state space sigma sub k at each time k each code sequence passes through a well defined state sequence the set of all state sequences is also a group code the state code of c the state code defines an essentially unique minimal realization of c the trellis diagram of c is defined by the state code of c and by labels associated with each state transition the set of all label sequences forms a group code the label code of c which is isomorphic to the state code of c if c is complete and strongly controllable then a minimal encoder in controller canonical feedbackfree form may be constructed from certain sets of shortest possible code sequences called granules the size of the state space sigma sub k is equal to the size of the state space of this canonical encoder which is given by a decomposition of the input groups of c at each time k if c is time invariant and nu controllable then mod sigma sub k mod pi sub 1

system theoretic properties of convolutional codes over rings convolutional codes over rings are particularly suitable for representing codes over phase modulation signals in order to develop a complete structural analysis of this class of codes it is necessary to study rational matrices over rings which constitutes the generator matrices encoders for such convolutional codes noncatastrophic minimal systematic and basic generator matrices are introduced and characterized by using a canonical form for polynomial matrices over rings finally some classes of convolutional codes defined according to the generator matrix they admit are introduced and analyzed from a system theoretic point of view

the threshold probability of a code we define and estimate the threshold probability spl theta of a linear code using a theorem of margulis 1974 originally conceived for the study of the probability of disconnecting a graph we then apply this concept to the study of the erasure and z channels for which we propose linear coding schemes that admit simple decoding we show that spl theta is particularly relevant to the erasure channel since linear codes achieve a vanishing error probability as long as p spl lesspl theta where p is the probability of erasure in effect spl theta can be thought of as a capacity notion designed for codes rather than for channels binomial codes haven the highest possible spl theta and achieve capacity as for the z channel a subcapacity is derived with respect to the linear coding scheme for a transition probability in the range log 3 2 1 we show how to achieve this subcapacity as a by product we obtain improved constructions and existential results for intersecting codes linear sperner families which are used in our coding schemes

labelings and encoders with the uniform bit error property with applications to serially concatenated trellis codes the well known uniform error property for signal constellations and codes is extended to encompass information bits we introduce a class of binary labelings for signal constellations called bit geometrically uniform bgu labelings for which the uniform bit error property holds i e the bit error probability does not depend on the transmitted signal strong connections between the symmetries of constellations and binary hamming spaces are involved for block coded modulation bcm and trellis coded modulation tcm euclidean space codes bgu encoders are introduced and studied the properties of bgu encoders prove quite useful for the analysis and design of codes aimed at minimizing the bit rather than symbol error probability applications to the analysis and the design of serially concatenated trellis codes are presented together with a case study which realizes a spectral efficiency of 2 b s hz

generalized hamming weights for linear codes motivated by cryptographical applications the algebraic structure of linear codes from a new perspective is studied by viewing the minimum hamming weight as a certain minimum property of one dimensional subcodes a generalized notion of higher dimensional hamming weights is obtained these weights characterize the code performance on the wire tap channel of type ii basic properties of generalized weights are derived the values of these weights for well known classes of codes are determined and lower bounds on code parameters are obtained several open problems are also listed

ldpc codes over rings for psk modulation this paper describes the design and analysis of low density parity check ldpc codes over rings and shows how these codes when mapped onto appropriate signal constellations can be used to effect bandwidth efficient modulation specifically ldpc codes are constructed over the integer rings spl zopf sub m and g sub m sup 2 and mapped onto phase shift keying psk type signal sets to yield geometrically uniform signal space codes this paper identifies and addresses the design issues that affect code performance examples of codes over spl zopf sub 8 and g sub 64 mapped onto 8 ary and 64 ary signal sets at a spectral efficiency of 1 5 and 2 0 bits per second per hertz b s hz illustrate the approach simulation of these codes over the additive white gaussian noise awgn channel demonstrates that this approach is a good alternative to bandwidth efficient techniques based on binary ldpc codes e g bit interleaved coded modulation

the homotopy groups of the localized toda smith spectrum 1 at the prime 3 in this paper we try to compute the homotopy groups of the l2 localized toda smith spectrum v 1 at the prime 3 by using the adamsnovikov spectral sequence and have almost done so this computation involves non trivial differentials d5 and dg of the adams novikov spectral sequence different from the case p 3 we also determine the homotopy groups of some l2 localized finite spectra relating to v 1 we further show some of the non trivial differentials on elements relating so called 3 elements in the adams novikov spectral sequence for 7r s0

geometrically uniform codes a signal space code c is defined as geometrically uniform if for any two code sequences in c there exists an isometry that maps one sequence into the other while leaving the code c invariant geometrical uniformity a strong kind of symmetry implies such properties as a the distance profiles from code sequences in c to all other code sequences are all the same and b all voronoi regions of code sequences in c have the same shape it is stronger than ungerboeck zehavi wolf symmetry or calderbank sloane regularity nonetheless most known good classes of signal space codes are shown to be generalized coset codes and therefore geometrically uniform including a lattice type trellis codes based on lattice partitions lambda lambda such that z sup n lambda lambda 4z sup n is a lattice partition chain and b phase shift keying psk type trellis codes based on up to four way partitions of a 2 sup n psk signal set

some results on map decoding of non binary ldpc codes over the bec in this paper the transmission over the binary erasure channel bec using non binary ldpc nbldpc codes is considered the concept of peeling decoder and stopping sets is generalized to nbldpc codes using these generalizations a combinatorial characterization of decoding failures of nbldpc codes is given under assumption that the belief propagation bp decoder is used then the residual ensemble of codes resulted by the bp decoder is defined and the design rate and the expectation of total number of codewords of the residual ensemble are computed the decoding failure criterion combined with the density evolution analysis helps us to compute the asymptotic residual degree distribution for nbldpc codes our approach to compute the residual degree distribution on the check node side is not efficient as it is based on enumeration of all the possible connections on the check node side which satisfy the decoding failure criterion so the computation of the asymptotic check node side residual degree distribution and further part of our analysis is performed for nbldpc codes over gf2m with m 2 in order to show that asymptotically almost every code in such ldpc ensemble has a rate equal to the design rate we generalize the argument of the maxwell construction to nbldpc codes defined over gf22 it is also observed that like in the binary setting the maxwell construction relating the performance of map and bp decoding holds in this setting

minimal and systematic convolutional codes over finite abelian groups abstract convolutional codes over abelian groups provide an effective theoretical framework for the analysis of some classes of tcm codes the encoder synthesis for this class of codes is not as simple as in the binary case since minimal encoders in the group case might be necessarily nonlinear in this contribution an algorithmic method testing whether a convolutional code over an abelian group admits a systematic or a minimal homomorphic encoder is provided this test consists in verifying whether a subgroup splits in a group through this method the class of codes admitting systematic encoders and the class of codes admitting minimal encoders can be compared finally this test is applied to some examples of practical relevance

maxwell construction the hidden bridge between iterative and maximum a posteriori decoding there is a fundamental relationship between belief propagation and maximum a posteriori decoding a decoding algorithm which is called the maxwell decoder is introduced and provides a constructive description of this relationship both the algorithm itself and the analysis of the new decoder are reminiscent of the maxwell construction in thermodynamics this paper investigates in detail the case of transmission over the binary erasure channel while the extension to general binary memoryless channels is discussed in a companion paper

intersecting goppa codes bounds on the rates of intersecting goppa codes are derived and compared with previous bounds on intersecting codes intersecting self intersecting and highly intersecting goppa codes are discussed in detail it is shown that asymptotically good intersecting goppa codes exist and lower bounds are given on their rates six theorems for goppa codes are also presented

random coding techniques for nonrandom codes this work provides techniques to apply the channel coding theorem and the resulting error exponent which was originally derived for totally random block code ensembles to ensembles of codes with less restrictive randomness demands as an example the random coding technique can even be applied for an ensemble that contains a single code for a specific linear code we get an upper bound for the error probability which equals gallager s 1968 random coding bound up to a factor determined by the maximum ratio between the weight distribution of the code and the expected random weight distribution

the capacity of finite abelian group codes over symmetric memoryless channels the capacity of finite abelian group codes over symmetric memoryless channels is determined for certain important examples such as m psk constellations over additive white gaussian noise awgn channels with m a prime power it is shown that this capacity coincides with the shannon capacity i e there is no loss in capacity using group codes this had previously been known for binary linear codes used over binary input output symmetric memoryless channels on the other hand a counterexample involving a three dimensional geometrically uniform constellation is presented in which the use of abelian group codes leads to a loss in capacity the error exponent of the average group code is determined and it is shown to be bounded away from the random coding error exponent at low rates for finite abelian groups not admitting galois field structure

the combinatorics of steenrod operations on the cohomology of grassmannians abstract the study of the action of the steenrod algebra on the mod p cohomology of spaces has many applications to the topological structure of those spaces in this paper we present combinatorial formulas for the action of steenrod operations on the cohomology of grassmannians both in the borel and the schubert picture we consider integral lifts of steenrod operations which lie in a certain hopf algebra of differential operators the latter has been considered recently as a realization of the landweber novikov algebra in complex cobordism theory it also has connections with the action of the virasoro algebra on the boson fock space our formulas for steenrod operations are based on methods which have not been used before in this area namely hammond operators and the combinatorics of schur functions we also discuss applications of our formulas to the geometry of grassmannians