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

frechet spaces with nuclear kothe quotients each separable frechet non banach space x with a continuous norm is shown to have a quotient y with a continuous norm and a basis if in addition y can be chosen to be nuclear we say that x has a nuclear kothe quotient we obtain a slightly technical characterization of those separable frechet spaces with nuclear kothe quotients in particular separable reflexive frh secondary 46a12 46a14 46a20 46a25 46a35 research partially supported by the national science foundation 01982 american mathematical society 0002 9947 82 0000 0090 05 00

a new upper bound on the ml decoding error probability of linear binary block codes in awgn interference performance evaluation of maximum likelihood ml soft decision decoded binary block codes is usually carried out using bounding techniques many tight upper bounds on the error probability of binary codes are based on the so called gallager s first bounding technique gfbt the tangential sphere bound tsb of poltyrev which has been believed for many years to offer the tightest bound developed for binary block codes is an example within the framework of the tsb and gfbt we apply a new method referred to as the added hyper plane ahp technique to the decomposition of the error probability this results in a bound developed upon the application of two stages of the gfbt with two different gallager regions culminating in a tightened upper bound beyond the tsb the proposed bound is simple and only requires the spectrum of the binary code

on the error exponents of improved tangential sphere bounds the performance of maximum likelihood ml decoded binary linear block codes over the additive white gaussian noise awgn channel is addressed via the tangential sphere bound tsb and two of its recent improved versions the correspondence is focused on the derivation of the error exponents of these bounds although it was shown that some recent improvements of the tsb tighten this bound for finite length codes it is demonstrated in this correspondence that their error exponents coincide for an arbitrary ensemble of binary linear block codes the common value of these error exponents is explicitly expressed in terms of the asymptotic growth rate of the average distance spectrum

performance bounds for erasure list and decision feedback schemes with linear block codes a message independence property and some new performance upper bounds are derived in this work for erasure list and decision feedback schemes with linear block codes transmitted over memoryless symmetric channels similar to the classical work of forney this work is focused on the derivation of some gallager type bounds on the achievable tradeoffs for these coding schemes where the main novelty is the suitability of the bounds for both random and structured linear block codes or code ensembles the bounds are applicable to finite length codes and to the asymptotic case of infinite block length and they are applied to low density parity check code ensembles

new performance bounds for turbo codes we derive a new upper bound on the word and bit error probabilities of turbo codes with maximum likelihood decoding by using the gallager bound since the derivation of the bound for a given interleaver is intractable we assume uniform interleaving as in the derivation of the standard union bound for turbo codes the result is a generalization of the transfer function bound and remains useful for a wider range of signal to noise ratios particularly for some range below the channel cutoff rate the new bound is also applicable to other linear codes

coded modulation using superimposed binary codes in this correspondence we investigate in a comprehensive fashion a one layer coding shaping scheme resembling a perfectly cooperated multiple access system at the transmitter binary data are encoded by either single level or multilevel codes the coded bits are first randomly interleaved and then entered into a signal mapper at each time the signal mapper accepts as input multiple binary digits and delivers as output an amplitude signal where the input are first independently mapped into 2 pam signals possibly having different amplitudes and then superimposed to form the output the receiver consists of an iterative decoding demapping algorithm with an entropy based stopping criterion in the special cases when all the 2 pam signals have equal amplitudes based on an irregular trellis we propose an optimal soft input soft output siso demapping algorithm with quadratic rather than exponential complexity in the general cases when multilevel codes are employed we propose power allocation strategies to facilitate the iterative decoding dempaping algorithm using the unequal power allocations and the gaussian approximation based suboptimal demapping algorithm with linear complexity coded modulation with high bandwidth efficiency can be implemented

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

braided block codes a new class of binary iteratively decodable codes with good decoding performance is presented these codes called braided block codes bbcs operate on continuous data streams and are constructed by interconnection of two component block codes bbcs can be considered as convolutional or sliding version of either elias product codes or expander codes in this paper we define bbcs describe methods of their construction analyze code properties and study asymptotic iterative decoding performance

tightened upper bounds on the ml decoding error probability of binary linear block codes the performance of maximum likelihood ml decoded binary linear block codes is addressed via the derivation of tightened upper bounds on their decoding error probability the upper bounds on the block and bit error probabilities are valid for any memoryless binary input and output symmetric communication channel and their effectiveness is exemplified for various ensembles of turbo like codes over the additive white gaussian noise awgn channel an expurgation of the distance spectrum of binary linear block codes further tightens the resulting upper bounds

hypercyclic behaviour of operators in a hypercyclic c 0 semigroup abstract let t t t 0 be a hypercyclic strongly continuous semigroup of operators then each t t t 0 is hypercyclic as a single operator and it shares the set of hypercyclic vectors with the semigroup this answers in the affirmative a natural question concerning hypercyclic c 0 semigroups the analogous result for frequent hypercyclicity is also obtained

convergence of arithmetic means of operators in frechet spaces abstract every kothe echelon frechet space x that is montel and not isomorphic to a countable product of copies of the scalar field admits a power bounded continuous linear operator t such that i t does not have closed range but the sequence of arithmetic means of the iterates of t converges to 0 uniformly on the bounded sets in x on the other hand if x is a frechet space which does not have a quotient isomorphic to a nuclear kothe echelon space with a continuous norm then the sequence of arithmetic means of the iterates of any continuous linear operator t for which 1 n t n converges to 0 on the bounded sets converges uniformly on the bounded subsets of x i e t is uniformly mean ergodic if and only if the range of i t is closed this result extends a theorem due to lin for such operators on banach spaces the connection of browder s equality for power bounded operators on frechet spaces to their uniform mean ergodicity is exposed an analysis of the mean ergodic properties of the classical cesaro operator on banach sequence spaces is also given

non power bounded generators of strongly continuous semigroups abstract it is folklore that a power bounded operator on a sequentially complete locally convex space generates a uniformly continuous c 0 semigroup which is given by the corresponding power series representation recently domanski asked if in this result the assumption of being power bounded can be relaxed we employ conditions introduced by elazko to give a weaker but still sufficient condition for generation and apply our results to operators on classical function and sequence spaces

new techniques for upper bounding the ml decoding performance of binary linear codes in this paper new techniques are presented to either simplify or improve most existing upper bounds on the maximum likelihood ml decoding performance of the binary linear codes over additive white gaussian noise awgn channels firstly the recently proposed union bound using truncated weight spectrum by ma et al is re derived in a detailed way based on gallager s first bounding technique gfbt where the good region is specified by a sub optimal list decoding algorithm the error probability caused by the bad region can be upper bounded by the tail probability of a binomial distribution while the error probability caused by the good region can be upper bounded by most existing techniques secondly we propose two techniques to tighten the union bound on the error probability caused by the good region the first technique is based on pair wise error probabilities the second technique is based on triplet wise error probabilities which can be upper bounded by the fact that any three bipolar vectors form a non obtuse triangle the proposed bounds improve the conventional union bounds but have a similar complexity since they involve only the q function the proposed bounds can also be adapted to bit error probabilities

windowed decoding of spatially coupled codes spatially coupled codes have been of interest recently owing to their superior performance over memoryless binary input channels the performance is good both asymptotically since the belief propagation thresholds approach the shannon limit as well as for finite lengths since degree 2 variable nodes that result in high error floors can be completely avoided however to realize the promised good performance one needs large blocklengths this in turn implies a large latency and decoding complexity for the memoryless binary erasure channel we consider the decoding of spatially coupled codes through a windowed decoder that aims to retain many of the attractive features of belief propagation while trying to reduce complexity further we characterize the performance of this scheme by defining thresholds on channel erasure rates that guarantee a target erasure rate we give analytical lower bounds on these thresholds and show that the performance approaches that of belief propagation exponentially fast in the window size we give numerical results including the thresholds computed using density evolution and the erasure rate curves for finite length spatially coupled codes

blocks of group algebras are derived simple a derived version of maschke s theorem for finite groups is proved the derived categories bounded or unbounded of all blocks of the group algebra of a finite group are simple in the sense that they admit no nontrivial recollements this result is independent of the characteristic of the base field

bounds on the decoding error probability of binary linear codes via their spectra bounds on the error probability of maximum likelihood decoding of a binary linear code are considered the bounds derived use the weight spectrum of the code and they are tighter than the conventional union bound in the case of large noise in the channel the bounds derived are applied to a code with an average spectrum and the result is compared to the random coding exponent the author shows that the bound considered for the binary symmetrical channel case coincides asymptotically with the random coding bound for the case of awgn channel the author shows that berlekamp s 1980 tangential bound can be improved but even this improved bound does not coincide with the random coding bound although it can be very close to it

on weighted inductive limits of spaces of continuous functions the general problem of a projective description of weighted inductive limits of spaces of continuous functions was raised by bierstedt eise and summers in 3 similar questions were studied in 4 in the setting of sequence spaces on a general index set i the study was recently continued by bierstedt and meise in 5 the aim of the present article is to treat some questions proposed to the author by k d bierstedt on this subject our notation for locally convex spaces 1 c s is standard and we refer the reader to 13 and 14 if e is a 1 c s cs e denotes the family of all the continuous seminorms on e following floret 10 a 1 c s is said to satisfy the countable neighbourhood property c n p if given p ces e there are a 0 and p cs e such that p x 0 such that u 0 there is v v such that avi 0 i e if given any x x there is v v with v x 0 then v will be refered to as a nachbin family on x in what follows if e is a 1 c s c x e indicates the collection of all continuous e valued functions on x and cc x e the subspace of c x e of all the functions with compact support corresponding to a nachbin family v and to a 1 c s e we associate the following nachbin spaces

finite representability of operators abstract we introduce operator local supportability as a new type of operator finite representability that generalizes bellenot finite representability we prove that local supportability and local representability are mutually independent new examples of both types of finite representability are given for instance for every operator t we prove that t u is locally supportable in t u we also prove that given an operator t with range in c 0 1 t is locally representable in t l 1

cross parity check convolutional codes a class of convolutional codes called cross parity check cpc codes which are useful for the protection of data stored on magnetic tape is described and analyzed cpc codes are first explained geometrically their construction is described in terms of constraining data written onto a tape in such a way that when lines of varying slope are drawn across the tape the bits falling on those lines sum to zero modulo two this geometric interpretation is then formalized by the construction of canonical parity check matrices and systematic generator matrices for cpc codes and by computing their constraint lengths the distance properties of cpc codes are analyzed and it is shown that these codes are maximum distance separable convolutional codes in addition examples are given of both error and erasure decoding algorithms that take advantage of the geometric regularity of cpc codes the technique of parity check matrix reduction which is useful for reducing the inherent decoding delay of cpc codes is described the technique consists of dividing each term of the parity check matrix by some polynomial and retaining only the remainder a class of polynomials that are particularly attractive for this purpose if identified

low complexity x ems algorithms for nonbinary ldpc codes the extended min sum ems algorithm is redescribed as a reduced search trellis algorithm called m ems algorithm two variants of the m ems algorithm called t ems algorithm and d ems algorithm are presented simulation results show that these three algorithms referred to as x ems algorithms for convenience combined with factor correction techniques perform almost as well as the q ary sum product algorithm qspa but with a much lower complexity

an achievable rate region for the multiple access channel with feedback an achievable rate region r 1 leq i x 1 y x 2 u r 2 leq i x 2 y x 1 u r 1 r 2 leq i x 1 x 2 y where p u x l x 2 y p u p x l u p x 2 u p y x l x 2 is established for the multiple access channel with feedback time sharing of these achievable rates yields the rate region of this paper this region generally exceeds the achievable rate region without feedback and exceeds the rate point found by gaarder and wolf for the binary erasure multiple access channel with feedback the presence of feedback allows the independent transmitters to understand each other s intended transmissions before the receiver has sufficient information to achieve the desired decoding this allows the transmitters to cooperate in the transmission of information that resolves the residual uncertainty of the receiver at the same time independent information from the transmitters is superimposed on the cooperative correction information the proof involves list codes and block markov encoding

improved upper bounds on the ml decoding error probability of parallel and serial concatenated turbo codes via their ensemble distance spectrum the ensemble performance of parallel and serial concatenated turbo codes is considered where the ensemble is generated by a uniform choice of the interleaver and of the component codes taken from the set of time varying recursive systematic convolutional codes following the derivation of the input output weight enumeration functions of the ensembles of random parallel and serial concatenated turbo codes the tangential sphere upper bound is employed to provide improved upper bounds on the block and bit error probabilities of these ensembles of codes for the binary input additive white gaussian noise awgn channel based on coherent detection of equi energy antipodal signals and maximum likelihood decoding the influence of the interleaver length and the memory length of the component codes is investigated the improved bounding technique proposed here is compared to the conventional union bound and to a alternative bounding technique by duman and salehi 1998 which incorporates modified gallager bounds the advantage of the derived bounds is demonstrated for a variety of parallel and serial concatenated coding schemes with either fixed or random recursive systematic convolutional component codes and it is especially pronounced in the region exceeding the cutoff rate where the performance of turbo codes is most appealing these upper bounds are also compared to simulation results of the iterative decoding algorithm

examples of non archimedean frechet spaces without nuclear kothe quotients let k be a spherically complete non archimedean valued field we prove that the dual space l of the banach space c0 has a total strongly non norming subspace m using this subspace m we construct a non normable frechet space f of countable type with a continuous norm such that its strong dual fb is a strict lb space next we show that f has no nuclear kothe quotient

homological theory of recollements of abelian categories abstract we investigate several homological aspects of recollements of abelian categories in particular we study how various homological invariants and dimensions of the categories involved in a recollement situation are related and when recollements of abelian categories induce recollements at the level of the bounded derived categories finally we give applications to global finitistic and representation dimension of rings and artin algebras and to rouquier s dimension of triangulated categories

strongly continuous semigroups on some frechet spaces abstract we prove that for a strongly continuous semigroup t on the frechet space of all scalar sequences its generator is a continuous linear operator a l and that for all x and t 0 the series exp t a x k 0 t k k a k x converges to t t x this solves a problem posed by conejero moreover we improve recent results of albanese bonet and ricker about semigroups on strict projective limits of banach spaces

block markov superposition transmission construction of big convolutional codes from short codes a construction of big convolutional codes from short codes called block markov superposition transmission bmst is proposed the bmst is very similar to superposition block markov encoding sbme which has been widely used to prove multiuser coding theorems the bmst codes can also be viewed as a class of spatially coupled codes where the generator matrices of the involved short codes referred to as basic codes are coupled the encoding process of bmst can be as fast as that of the basic code while the decoding process can be implemented as an iterative sliding window decoding algorithm with a tunable delay more importantly the performance of bmst can be simply lower bounded in terms of the transmission memory given that the performance of the short code is available numerical results show that 1 the lower bounds can be matched with a moderate decoding delay in the low bit error rate ber region implying that the iterative sliding window decoding algorithm is near optimal 2 bmst with repetition codes and single parity check codes can approach the shannon limit within 0 5 db at the ber of 10 5 for a wide range of code rates and 3 bmst can also be applied to nonlinear codes

principle of local reflexivity respecting subspaces abstract we obtain a strengthening of the principle of local reflexivity in a general form the added strength makes local reflexivity operators respect given subspaces applications are given to bounded approximation properties of pairs consisting of a banach space and its subspace

on the voronoi neighbor ratio for binary linear block codes soft decision decoding of block codes is regarded as the geometrical problem of identifying the voronoi region within which a given input vector lies a measure called the neighbor ratio is proposed to characterize how many facets a voronoi region has theory and algorithms are presented to determine the neighbor ratio for binary linear block codes and results are given for several types of codes an asymptotic analysis for long codes reveals that the neighbor ratio depends on whether the code rate is less than 1 2 or not for rates below this threshold all pairs of codewords tend to share a voronoi facet for higher rates a relatively small fraction of them do

generalized tangential sphere bound on the ml decoding error probability of linear binary block codes in awgn interference the error probability of maximum likelihood ml soft decision decoded binary block codes rarely accepts nice closed forms in addition for long codes ml decoding becomes prohibitively complex nevertheless bounds on the performance of ml decoded systems provide insight into the effect of system parameters on the overall system performance as well as a measure of goodness of the suboptimum decoding methods used in practice using the so called gallager s first bounding technique involving a so called gallager region and within the framework of tangential sphere bound tsb of poltyrev we develop a general bound referred to as the generalized tsb gtsb the gallager region is chosen to be a general hyper surface of revolution hsr which is optimized to tighten the bound the search for the optimal gallager region is a classical problem dating back to gallager s thesis in the early 1960s for the random coding case gallager provided the optimal solution in a closed form while for the nonrandom case the problem has been an active area of research in information theory for many years we prove that for a sphere code the optimal hsr within the proposed gtsb is a hyper cone this will climax to the tsb of poltyrev one of the tightest bounds ever developed for binary block codes and therefore terminates the search for a better gallager region in the groundwork of the gtsb

iterative decoding threshold analysis for ldpc convolutional codes an iterative decoding threshold analysis for terminated regular ldpc convolutional ldpcc codes is presented using density evolution techniques the convergence behavior of an iterative belief propagation decoder is analyzed for the binary erasure channel and the awgn channel with binary inputs it is shown that for a terminated ldpcc code ensemble the thresholds are better than for corresponding regular and irregular ldpc block codes

another mean ergodic characterization of quasi reflexive spaces abstract in this note we characterize quasi reflexive barrelled and complete locally convex hausdorff spaces with a basis in terms of mean ergodicity of c 0 semigroups of operators acting on these spaces

minimum distance and trapping set analysis of protograph based ldpc convolutional codes low density parity check ldpc convolutional codes have been shown to be capable of achieving capacity approaching performance with iterative message passing decoding in the first part of this paper using asymptotic methods to obtain lower bounds on the free distance to constraint length ratio we show that several ensembles of regular and irregular ldpc convolutional codes derived from protograph based ldpc block codes have the property that the free distance grows linearly with respect to the constraint length i e the ensembles are asymptotically good in particular we show that the free distance to constraint length ratio of the ldpc convolutional code ensembles exceeds the minimum distance to block length ratio of the corresponding ldpc block code ensembles a large free distance growth rate indicates that codes drawn from the ensemble should perform well at high signal to noise ratios under maximum likelihood decoding when suboptimal decoding methods are employed there are many factors that affect the performance of a code recently it has been shown that so called trapping sets are a significant factor affecting decoding failures of ldpc codes over the additive white gaussian noise channel with iterative message passing decoding in the second part of this paper we study the trapping sets of the asymptotically good protograph based ldpc convolutional codes considered earlier by extending the theory presented in part one and using similar bounding techniques we show that the size of the smallest non empty trapping set grows linearly with the constraint length for these ensembles

perpendicular categories with applications to representations and sheaves this paper is concerned with the omnipresence of the formation of the subcategories right left perpendicular to a subcategory of objects in an abelian category we encounter these subcategories in various contexts the formation of quotient categories with respect to localizing subcategories cf section 2 the deletion of vertices and shrinking of arrows see 37 in the representation theory of finite dimensional algebras cf section 5 the comparison of the representation theories of different extended dynkin quivers cf section 10 the theory of tilting cf sections 4 and 6 the study of homological epimorphisms of rings cf section 4 the passage from graded modules to coherent sheaves on a possibly weighted projective variety or scheme cf section 7 and 21 the study of maximal cohen macaulay modules over surface singularities cf sections the comparison of weighted projective lines for different weight sequences cf section 9 the formation of affine and local algebras attached to path algebras of extended dynkin quivers canonical algebras and weighted projective lines cf section 11 and 21 and the concept of universal localization in 40 formation of the perpendicular category has many aspects in common with localization and allows one to dispose of localization techniques in situations not accessible to any of the classical concepts of localization this applies in particular to applications in the domain of finite dimensional algebras and their representations several applications of the methods presented in this paper are already in existence partly published or appearing in print in the near future see for instance 40 39 4 45 26 49 46 and have show the versatility of the notion of a perpendicular category it seems that right perpendicular categories first appeared as the subcategories of so called closed objects in the process of the formation of the quotient cateogory of an abelian category with respect to a localizing serre subcategory see 18 47 34 another natural occurrence is encountered in commutative algebra forming the possibly infinitely generated modules of depth 2 cf section 7 the concept and some of the central applications were first presented in a talk given by the first author at the honnef meeting in january 1985 we also note that the perfectly matching nomination perpendicular category was coined by a schofield who discovered independently the usefulness of this concept in dealing with hereditary algebras see 39 cf also section 7 the authors further acknowledge the support of the deutsche forschungsgemeinschaft spp darstellungstheorie von endlichen gruppen und endlichdimensionalen algebren throughout this paper rings are associative with unit and modules are unitary right modules mod r respectively mod r denotes the category of all respectively all finitely presented right r modules

voronoi regions for binary linear block codes the voronoi regions of a block code govern many aspects of the code s performance on a gaussian channel and they are fundamental instruments in for example error probability analysis and soft decision decoding the article presents an efficient method for finding the boundaries of the voronoi regions for an arbitrary binary linear block code two theoretical results together lead to the voronoi regions first it is shown that the question of the voronoi neighborship can be reduced into testing a simpler relation called the gabriel neighborship second a fast method of recognising gabriel neighbors is proposed these results are finally employed to describe the voronoi regions for the golay codes and several bch codes including hamming codes

a new class of multiple rate codes based on block markov superposition transmission the hadamard transform ht over the binary field provides a natural way to implement multiple rate codes referred to as ht coset codes where the code length n 2 p is fixed but the code dimension k can be varied from 1 to n 1 by adjusting the set of frozen bits the ht coset codes including reed muller rm codes and polar codes as typical examples can share the same fundamental encoder decoder architecture with the implementation complexity of order o n log n however to guarantee that all codes with designated rates perform well ht coset coding usually requires a sufficiently large code length which in turn causes difficulties in the determination of which bits are better for being frozen in this paper we propose to transmit short ht coset codes in the so called block markov superposition transmission bmst manner at the transmitter signals are spatially coupled via superposition resulting in long codes at the receiver these coupled signals are recovered by a sliding window iterative soft successive cancellation decoding algorithm most importantly the performance around or below the bit error rate ber of 10 5 can be predicted by a simple genie aided lower bound both these bounds and simulation results show that the bmst of short ht coset codes performs well within one db away from the corresponding shannon limits in a wide range of code rates

variations on the gallager bounds connections and applications there has been renewed interest in deriving tight bounds on the error performance of specific codes and ensembles based on their distance spectrum we discuss many reported upper bounds on the maximum likelihood ml decoding error probability and demonstrate the underlying connections that exist between them in addressing the gallager bounds and their variations we focus on the duman and salehi see ieee trans commun vol 46 p 717 723 1998 variation which originates from the standard gallager bound a large class of efficient bounds or their chernoff versions is demonstrated to be a special case of the generalized second version of the duman and salehi bounds implications and applications of these observations are pointed out including the fully interleaved fading channel resorting to either matched or mismatched decoding the proposed approach can be generalized to geometrically uniform nonbinary codes finite state channels bit interleaved coded modulation systems and it can be also used for the derivation of upper bounds on the conditional decoding error probability

accumulate repeat accumulate codes in this paper we propose an innovative channel coding scheme called accumulate repeat accumulate ara codes this class of codes can be viewed as serial turbo like codes or as a subclass of low density parity check ldpc codes and they have a projected graph or protograph representation this allows for high speed iterative decoding implementation using belief propagation an ara code can be viewed as precoded repeat accumulate ra code with puncturing or as precoded irregular repeat accumulate ira code where simply an accumulator is chosen as the precoder the amount of performance improvement due to the precoder will be called precoding gain using density evolution on their associated protographs we find some rate 1 2 ara codes with a maximum variable node degree of 5 for which a minimum bit snr as low as 0 08 db from channel capacity threshold is achieved as the block size goes to infinity such a low threshold cannot be achieved by ra ira or unstructured irregular ldpc codes with the same constraint on the maximum variable node degree furthermore by puncturing the inner accumulator we can construct families of higher rate ara codes with thresholds that stay close to their respective channel capacity thresholds uniformly iterative decoding simulation results are provided and compared with turbo codes in addition to iterative decoding analysis we analyzed the performance of ara codes with maximum likelihood ml decoding by obtaining the weight distribution of these codes and through existing tightest bounds we have shown that the ml snr threshold of ara codes also approaches very closely to that of random codes these codes have better interleaving gain than turbo codes

a general coding scheme for the two way channel a general coding scheme for the nonrestricted memoryless discrete two way channel is presented based on the introduction of auxiliary random variables forming a stationary markov process the coding scheme yields an achievable rate region which exceeds the inner bound of shannon in the general case a finite cardinality bound for the auxiliary random variables is given showing that the region is computable finally the capacity region for the memoryless gaussian two way channel is established

boundedness and periodicity of modules over qf rings be a minimal projective resolution m is called bounded if the lengths of the kernels ker di have a common upper bound and m is called periodic if some ker di is isomorphic to m in 1 p 7761 alperin posed the following question is every bounded kg module periodic alperin answered this question in the affirmative in case k is algebraic over its prime field l theorem 11 without any restriction on k eisenbud settled alperin s question in 6 theorem 9 21 in section 1 of the present paper we look at an arbitrary qf ring r instead of a group algebra and we give a condition which forces a bounded r module m to be periodic before we will formulate our result we make some preliminary remarks let m and x be modules over a ring r with the yoneda product for multiplication the abelian group extx m m oi extk m m becomes a graded ring which we call the extension ring of m and the abelian group ext m x becomes a graded right extr m m module we recall some classical examples of extension rings let g be a finite group and let k be a field on which g acts trivially then the cohomology ring h g k is the extension ring ext k k more generally let m be a finitely generated kg module then the ring hom m m is a g ring and the cohomology ring h g hom m m with cup product is canonically isomorphic to the extension ring extx m m with yoneda product these rings have been intensively studied see carlson 3 for details and for further references other well known exampes of extension rings arise in the homological algebra of local rings namely the extension

c0 semigroups and mean ergodic operators in a class of frechet spaces abstract it is shown that the generator of every exponentially equicontinuous uniformly continuous c 0 semigroup of operators in the class of quojection frechet spaces x which includes properly all countable products of banach spaces is necessarily everywhere defined and continuous if in addition x is a grothendieck space with the dunford pettis property then uniform continuity can be relaxed to strong continuity two results one of m lin and one of h p lotz both concerned with uniformly mean ergodic operators in banach spaces are also extended to the class of frechet spaces mentioned above they fail to hold for arbitrary frechet spaces

path partitions and forward only trellis algorithms this is a semitutorial paper on trellis based algorithms we argue that most decoding detection algorithms described on trellises can be formulated as path partitioning algorithms with proper definitions of mappings from subsets of paths to metrics of subsets thereby the only two operations needed are path concatenation and path collection which play the roles of multiplication and addition respectively furthermore we show that the trellis structure permits the path partitioning algorithms to be formulated as forward only algorithms with structures resembling the viterbi 1967 algorithm thus eliminating the need for backward computations regardless of what task needs to be performed on the trellis while all of the actual decoding detection algorithms presented here are rederivations of variations of previously known methods we believe that the exposition of the algorithms in a unified manner as forward only path partitioning algorithms is the most intuitive manner in which to generalize the viterbi algorithm we also believe that this approach may in fact influence the practical implementation of the algorithms as well as influence the construction of other forward only algorithms e g byte wise forward only detection algorithms

local reflexivity of normed spaces operators and frechet spaces a new and stronger proof of the principle of local reflexivity is given applications to normed spaces local reflexivity of operators and frechet spaces are given a partial answer to a question of pelczynski is obtained the proof is based on an unpublished proof of r c james

threshold saturation via spatial coupling why convolutional ldpc ensembles perform so well over the bec convolutional low density parity check ldpc ensembles introduced by felstrom and zigangirov have excellent thresholds and these thresholds are rapidly increasing functions of the average degree several variations on the basic theme have been proposed to date all of which share the good performance characteristics of convolutional ldpc ensembles we describe the fundamental mechanism that explains why convolutional like or spatially coupled codes perform so well in essence the spatial coupling of individual codes increases the belief propagation bp threshold of the new ensemble to its maximum possible value namely the maximum a posteriori map threshold of the underlying ensemble for this reason we call this phenomenon threshold saturation this gives an entirely new way of approaching capacity one significant advantage of this construction is that one can create capacity approaching ensembles with an error correcting radius that is increasing in the blocklength although we prove the threshold saturation only for a specific ensemble and for the binary erasure channel bec empirically the phenomenon occurs for a wide class of ensembles and channels more generally we conjecture that for a large range of graphical systems a similar saturation of the dynamical threshold occurs once individual components are coupled sufficiently strongly this might give rise to improved algorithms and new techniques for analysis

homological embeddings for preprojective algebras for a fixed finite dimensional algebra a we study representation embeddings of the form mod b rightarrow mod a such an embedding is called homological if it induces an isomorphism on all ext groups and weakly homological if only ext 1 is preserved in case a is a preprojective algebra of dynkin type we give an explicit classification of all weakly homological and homological embeddings furthermore we show that for self injective algebras a classification of homological embeddings becomes accessible once these algebras fulfil the tachikawa conjecture

spatially coupled ensembles universally achieve capacity under belief propagation we investigate spatially coupled code ensembles for transmission over the binary erasure channel it was recently shown that spatial coupling increases the belief propagation threshold of the ensemble to essentially the maximum a priori threshold of the underlying component ensemble this explains why convolutional ldpc ensembles originally introduced by felstrom and zigangirov perform so well over this channel we show that the equivalent result holds true for transmission over general binary input memoryless output symmetric channels more precisely given a desired error probability and a gap to capacity we can construct a spatially coupled ensemble that fulfills these constraints universally on this class of channels under belief propagation decoding in fact most codes in this ensemble have this property the quantifier universal refers to the single ensemble code that is good for all channels but we assume that the channel is known at the receiver the key technical result is a proof that under belief propagation decoding spatially coupled ensembles achieve essentially the area threshold of the underlying uncoupled ensemble we conclude by discussing some interesting open problems

quasi reflexive frechet spaces and mean ergodicity abstract we characterize quasi reflexive frechet spaces with a basis in terms of the properties of this basis as a consequence we prove that a frechet space with a basis is quasi reflexive of order one if and only if for every power bounded operator t either t or t is mean ergodic

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

two layer coded spatial modulation with block markov superposition transmission this paper is concerned with the spatial modulation sm a multiple input multiple output mimo transmission technique that maps information bits not only into the conventional two dimensional signal points but also into the indices of active transmit antennas we present a two layer coded sm scheme in which the spatial bits carried by the antenna indices and the signal bits carried by the conventional signals are protected separately by two error correction codes for the ease of decoding process the code rates are allocated according to the chain rule of the mutual information we choose block markov superposition transmission bmst codes for each layer since they are easily designed for any given code rate with a predictable performance lower bound an iterative sliding window decoding algorithm is also presented by exchanging messages iteratively between the two bmst decoders and the soft in soft out siso demapper of the sm to reduce the computational complexity we propose to implement the siso demapping algorithm by employing only partial soft inputs numerical results show that the bmst sm system performs well over uncorrelated rayleigh fading channels

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

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

techniques of bounding the probability of decoding error for block coded modulation structures two techniques for upper bounding the average probability of decoding error in coded modulation structures are presented the first bound which is applicable to the additive white gaussian noise awgn channel is tighter than the well known union bound and the minimum distance bound especially for low signal to noise ratio it is shown that for the leech lattice this upper bound is very close to a sphere lower bound for the second upper bound which is applicable to any memoryless channel not necessarily awgn a method of random coset coding is presented for the awgn channel a tighter upper bound is obtained by employing the method of random coset coding for calculating the average spectrum of distances of the code which is required for the computation of the first upper bound

mean ergodic operators and reflexive frechet lattices connections between positive mean ergodic operators acting in banach lattices and properties of the underlying lattice itself are well understood see the works of emel yanov wolff and zaharopol for frechet lattices or more general locally convex solid riesz spaces there is virtually no information available for a frechet lattice e it is shown here that amongst other things every power bounded linear operator on e is mean ergodic if and only if e is reflexive if and only if e is dedekind complete and every positive power bounded operator on e is mean ergodic if and only if every positive power bounded operator in the strong dual e no longer a frechet lattice is mean ergodic an important technique is to develop criteria that detect when e admits a positively complemented lattice copy of c0 l1 or l