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

entrainment control in a noisy neural system the open plus closed loop opcl entrainment control put forth by jackson and grosu physica d 85 1995 is applied to an effective neuron system as a way to extract stable limit cycles from a chaotic attractor analogous to the retrieval of memories from a memory searching state additive gaussian white noise representing the natural noise inherent in any real dynamical system is added to the entrainment control mechanism moderate levels of additive noise have little effect on successful entrainment as reflected in phase space plots and lyapunov exponents all three lyapunov exponents are negative which suggests parallels between opcl control and chaotic synchronization al 2

two neuron dynamics and adiabatic elimination abstract we consider an effective neuron network obtained from the hopfield model by adiabatic elimination the validity of the slaving procedure is first numerically examined the self interaction of the effective neuron network may lead to non zero attractors far from the origin ui 0 the qualitative structure of the attractors for this two effective neuron model with j12j21 0 or

low ber performance estimation of ldpc codes via application of importance sampling to trapping sets we introduce an importance sampling is method that successfully simulates the performance of low density parity check ldpc codes in an awgn channel at very low bit error rates bers by effectively finding and biasing bit node combinations that are the dominant sources of error events called trapping sets the developed technique provokes more frequent decoder failures consequently fewer simulation runs and higher simulation gains are achieved

circular shift linear network codes with arbitrary odd block lengths circular shift linear network coding lnc is a class of vector lnc with low encoding and decoding complexities and with local encoding kernels chosen from cyclic permutation matrices when l is a prime with primitive root 2 it was recently shown that a scalar linear solution over gf 2 l 1 induces an l dimensional circular shift linear solution at rate l 1 l in this paper we prove that for arbitrary odd l every scalar linear solution over gf 2 m l where m l refers to the multiplicative order of 2 modulo l can induce an l dimensional circular shift linear solution at a certain rate based on the generalized connection we further prove that for such l with m l beyond a threshold every multicast network has an l dimensional circular shift linear solution at rate phi l l where phi l is the euler s totient function of l an efficient algorithm for constructing such a solution is designed finally we prove that every multicast network is asymptotically circular shift linearly solvable

ldpc codes based on latin squares cycle structure stopping set and trapping set analysis it is well known that certain combinatorial structures in the tanner graph of a low density parity check ldpc code exhibit a strong influence on its performance under iterative decoding these structures include cycles stopping trapping sets and parameters such as the diameter of the code in general it is very hard to find a complete characterization of such configurations in an arbitrary code and even harder to understand the intricate relationships that exist between these entities it is therefore of interest to identify a simple setting in which all the described combinatorial structures can be enumerated and studied within a joint framework one such setting is developed in this paper for the purpose of analyzing the distribution of short cycles and the structure of stopping and trapping sets in tanner graphs of ldpc codes based on idempotent and symmetric latin squares the parity check matrices of ldpc codes based on latin squares have a special form that allows for connecting combinatorial parameters of the codes with the number of certain subrectangles in the latin squares subrectangles of interest can be easily identified and in certain instances completely enumerated this study can be extended in several different directions one of which is concerned with modifying the code design process in order to eliminate or reduce the number of configurations bearing a negative influence on the performance of the code another application of the results includes determining to which extent a configuration governs the behavior of the bit error rate curve in the waterfall and error floor regions

the inverse galois problem for mathrm psl _ 2 mathbb f _ p we show that the simple group psl2 fp occurs as the galois group of an extension of the rationals for all primes p 5 we obtain our galois extensions by studying the galois action on the second etale cohomology groups of a specific elliptic surface

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

codes on finite geometries new algebraic methods for constructing codes based on hyperplanes of two different dimensions in finite geometries are presented the new construction methods result in a class of multistep majority logic decodable codes and three classes of low density parity check ldpc codes decoding methods for the class of majority logic decodable codes and a class of codes that perform well with iterative decoding in spite of having many cycles of length 4 in their tanner graphs are presented most of the codes constructed can be either put in cyclic or quasi cyclic form and hence their encoding can be implemented with linear shift registers

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

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

vector network coding based on subspace codes outperforms scalar linear network coding this paper considers vector network coding solutions based on rank metric codes and subspace codes the main result of this paper is that vector solutions can significantly reduce the required alphabet size compared to the optimal scalar linear solution for the same multicast network the multicast networks considered in this paper have one source with h messages and the vector solution is over a field of size q with vectors of length t for a given network let the smallest field size for which the network has a scalar linear solution be q s then the gap in the alphabet size between the vector solution and the scalar linear solution is defined to be q s q t in this contribution the achieved gap is q h 2 t 2 h o t for any q geq 2 and any even h geq 4 if h geq 5 is odd then the achieved gap of the alphabet size is q h 3 t 2 h 1 o t previously only a gap of size one had been shown for networks with a very large number of messages these results imply the same gap of the alphabet size between the optimal scalar linear and some scalar nonlinear network coding solution for multicast networks for three messages we also show an advantage of vector network coding while for two messages the problem remains open several networks are considered all of them are generalizations and modifications of the well known combination networks the vector network codes that are used as solutions for those networks are based on subspace codes particularly subspace codes obtained from rank metric codes some of these codes form a new family of subspace codes which poses a new research problem

integral crystalline cohomology over very ramified valuation rings in this paper we intend to show how fontaine s comparison theory fol fo2 between crystalline and p adic etale cohomology can be extented to very ramified base rings so far it had been developed either for qp coefficients or zp coefficients if the base is unramified over the witt vectors see fa2 for example it states that for any p adic discrete valuation ring v there exists a ring b v with the following property if x denotes a smooth and proper v scheme the p adic etale cohomology of the generic fiber x xv k is related to the crystalline cohomology of x v0 vo defined below by an isomorphism

bifurcation and stabilization of oscillations in ring neural networks with inertia effects of inertia on oscillations in ring networks of unidirectionally coupled sigmoidal neurons are studied it is known that ring neural networks without inertia are bistable and the duration of transient oscillations increases exponentially with the number of neurons in this paper a kinematical description of traveling waves in the oscillations in the networks is extended to networks with inertia when the inertia is below a critical value and the state of each neuron is over damped properties of the networks are qualitatively the same as those without inertia the duration of the transient oscillations then increases with inertia and the increasing rate of the logarithm of the duration becomes more than double when the inertia exceeds a critical value and the state of each neuron becomes under damped properties of the networks qualitatively change the periodic solution is then stabilized through the pitchfork bifurcation as inertia increases more bifurcations occur so that various periodic solutions are generated and the stability of the periodic solutions changes alternately further stable oscillations generated with inertia are observed in an experiment on an analog circuit

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

networks matroids and non shannon information inequalities we define a class of networks called matroidal networks which includes as special cases all scalar linearly solvable networks and in particular solvable multicast networks we then present a method for constructing matroidal networks from known matroids we specifically construct networks that play an important role in proving results in the literature such as the insufficiency of linear network coding and the unachievability of network coding capacity we also construct a new network from the vamos matroid which we call the vamos network and use it to prove that shannon type information inequalities are in general not sufficient for computing network coding capacities to accomplish this we obtain a capacity upper bound for the vamos network using a non shannon type information inequality discovered in 1998 by zhang and yeung and then show that it is smaller than any such bound derived from shannon type information inequalities this is the first application of a non shannon type inequality to network coding we also compute the exact routing capacity and linear coding capacity of the vamos network finally using a variation of the vamos network we prove that shannon type information inequalities are insufficient even for computing network coding capacities of multiple unicast networks

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

stability and chaos in an inertial two neuron system abstract inertia is added to a continuous time hopfield 1984 effective neuron system we explore the effects on the stability of the fixed points of the system a two neuron system with one or two inertial terms added is shown to exhibit chaos the chaos is confirmed by lyapunov exponents power spectra and phase space plots

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

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

on vector linear solvability of multicast networks vector linear network coding lnc is a generalization of the conventional scalar lnc such that the data unit transmitted on every edge is an l dimensional vector of data symbols over a base field gf q vector lnc enriches the choices of coding operations at intermediate nodes and there is a popular conjecture on the benefit of vector lnc over scalar lnc in terms of alphabet size of data units there exist single source multicast networks that are vector linearly solvable of dimension l over gf q but not scalar linearly solvable over any field of size q leq q l this paper introduces a systematic way to construct such multicast networks and subsequently establish explicit instances to affirm the positive answer of this conjecture for infinitely many alphabet sizes p l with respect to an arbitrary prime p on the other hand this paper also presents explicit instances with the special property that they do not have a vector linear solution of dimension l over gf 2 but have scalar linear solutions over gf q for some q where q can be odd or even this discovery also unveils that over a given base field a multicast network that has a vector linear solution of dimension l does not necessarily have a vector linear solution of dimension l l

conjecture de l inertie moderee de serre on considere k un corps complet pour une valuation discrete de caracteristique nulle et dont le corps residuel est suppose parfait de caracteristique p on appelle e l indice de ramification absolue de k mathcal o k son anneau des entiers et overline k une cloture algebrique soit xk un schema propre et lisse sur k admettant un modele propre et semi stable x sur mathcal o k generalisant les resultats de breuil valables pour e 1 on demontre dans cet article un isomorphisme de periodes reliant le r ieme groupe de cohomologie etale de x overline k a coefficients dans pn et un r ieme groupe de cohomologie log cristalline de la fibre speciale de x nous avons toutefois les restrictions er 1

mean square exponential stability of stochastic inertial neural networks by introducing some parameters perturbed by white noises we propose a class of stochastic inertial neural networks in random environments to describe dynamics behaviours of this stochastic system

dynamics of simple electronic neural networks abstract we examine systems of one and two nonlinear threshold switching elements neurons of the kind used in electronic neural networks characteristics of these systems which deviate from standard ideal models are found to induce complex dynamics when the neurons possess a finite frequency response or a transfer characteristic with a time delay underdamped transients and instability leading to oscillation can occur inertia in the neuron connections is found to cause ringing about fixed points convoluted basin boundaries instability and spontaneous oscillation and chaotic behavior when driven furthermore the collective behavior of a network of multiple neurons can be underdamped even when the individual connections are overdamped these results imply that care should be exercised in implementing networks with electronic devices or when adding inertia to enhance the performance of optimizing networks

low floor decoders for ldpc codes one of the most significant impediments to the use of ldpc codes in many communication and storage systems is the error rate floor phenomenon associated with their iterative decoders the error floor has been attributed to certain subgraphs of an ldpc code s tanner graph induced by so called trapping sets we show in this paper that once we identify the trapping sets of an ldpc code of interest a sum product algorithm spa decoder can be custom designed to yield floors that are orders of magnitude lower than floors of the the conventional spa decoder we present three classes of such decoders 1 a bi mode decoder 2 a bit pinning decoder which utilizes one or more outer algebraic codes and 3 three generalized ldpc decoders we demonstrate the effectiveness of these decoders for two codes the rate 1 2 2640 1320 margulis code which is notorious for its floors and a rate 0 3 640 192 quasi cyclic code which has been devised for this study although the paper focuses on these two codes the decoder design techniques presented are fully generalizable to any ldpc code

quasi cyclic ldpc codes an algebraic construction this paper presents two new large classes of qc ldpc codes one binary and one non binary codes in these two classes are constructed by array dispersions of row distance constrained matrices formed based on additive subgroups of finite fields experimental results show that codes constructed perform very well over the awgn channel with iterative decoding based on belief propagation codes of a subclass of the class of binary codes have large minimum distances comparable to finite geometry ldpc codes and they offer effective tradeoff between error performance and decoding complexity when decoded with low complexity reliability based iterative decoding algorithms such as binary message passing decoding algorithms non binary codes decoded with a fast fourier transform based sum product algorithm achieve significantly large coding gains over reed solomon codes of the same lengths and rates decoded with either the hard decision berlekamp massey algorithm or the algebraic soft decision kotter vardy algorithm they have potential to replace reed solomon codes in some communication or storage systems where combinations of random and bursts of errors or erasures occur

construction of regular and irregular ldpc codes geometry decomposition and masking two algebraic methods for systematic construction of structured regular and irregular low density parity check ldpc codes with girth of at least six and good minimum distances are presented these two methods are based on geometry decomposition and a masking technique numerical results show that the codes constructed by these methods perform close to the shannon limit and as well as random like ldpc codes furthermore they have low error floors and their iterative decoding converges very fast the masking technique greatly simplifies the random like construction of irregular ldpc codes designed on the basis of the degree distributions of their code graphs

insufficiency of linear coding in network information flow it is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite field alphabet it is also known that this result does not generalize to arbitrary networks there are several examples in the literature of solvable networks with no scalar linear solution over any finite field however each example has a linear solution for some vector dimension greater than one it has been conjectured that every solvable network has a linear solution over some finite field alphabet and some vector dimension we provide a counterexample to this conjecture we also show that if a network has no linear solution over any finite field then it has no linear solution over any finite commutative ring with identity our counterexample network has no linear solution even in the more general algebraic context of modules which includes as special cases all finite rings and abelian groups furthermore we show that the network coding capacity of this network is strictly greater than the maximum linear coding capacity over any finite field exactly 10 greater so the network is not even asymptotically linearly solvable it follows that even for more general versions of linearity such as convolutional coding filter bank coding or linear time sharing the network has no linear solution

linear network coding over rings part i scalar codes and commutative alphabets linear network coding over finite fields is a well studied problem we consider the more general setting of linear coding for directed acyclic networks with finite commutative ring alphabets our results imply that for scalar linear network coding over commutative rings fields can always be used when the alphabet size is flexible but other rings may be needed when the alphabet size is fixed we prove that if a network has a scalar linear solution over some finite commutative ring then the unique smallest such commutative ring is a field we also show that fixed size commutative rings are quasi ordered such that all the scalar linearly solvable networks over any given ring are also scalar linearly solvable over any higher ordered ring we study commutative rings that are maximal with respect to this quasi order as they may be considered the best commutative rings of a given size we prove that a commutative ring is maximal if and only if some network is scalar linearly solvable over the ring but not over any other commutative ring of the same size furthermore we show that maximal commutative rings are direct products of certain fields specified by the integer partitions of the prime factor multiplicities of the ring s size finally we prove that there is a unique maximal commutative ring of size m if and only if each prime factor of m has multiplicity in 1 2 3 4 6 as consequences 1 every finite field is such a maximal ring and 2 for each prime p some network is scalar linearly solvable over a commutative ring of size p k but not over the field of the same size if and only if k not in 1 2 3 4 6

design of ldpc decoders for improved low error rate performance quantization and algorithm choices many classes of high performance low density parity check ldpc codes are based on parity check matrices composed of permutation submatrices we describe the design of a parallel serial decoder architecture that can be used to map any ldpc code with such a structure to a hardware emulation platform high throughput emulation allows for the exploration of the low bit error rate ber region and provides statistics of the error traces which illuminate the causes of the error floors of the 2048 1723 reed solomon based ldpc rs ldpc code and the 2209 1978 array based ldpc code two classes of error events are observed oscillatory behavior and convergence to a class of non codewords termed absorbing sets the influence of absorbing sets can be exacerbated by message quantization and decoder implementation in particular quantization and the log tanh function approximation in sum product decoders strongly affect which absorbing sets dominate in the errorfloor region we show that conventional sum product decoder implementations of the 2209 1978 array based ldpc code allow low weight absorbing sets to have a strong effect and as a result elevate the error floor dually quantized sum product decoders and approximate sum product decoders alleviate the effects of low weight absorbing sets thereby lowering the error floor

some bounds for ramification of p n torsion semi stable representations let p be an odd prime k a finite extension of q p g k gal kbar k its absolute galois group and e e k q p its absolute ramification index suppose that t is a p n torsion representation of g k that is isomorphic to a quotient of g k stable z p lattices in a semi stable representation with hodge tate weights in 0 r we prove that there exists a constant mu depending only on n e and r such that the upper numbering ramification group g k mu acts on t trivially

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

a class of non linearly solvable networks for each positive composite integer m a network is constructed which is solvable over an alphabet of size m but is not solvable over any smaller alphabet these networks have no linear solutions over any module alphabets and are not asymptotically linearly solvable over any finite field alphabets the networks capacities are all shown to equal one and their linear capacities are all shown to be bounded away from one for all finite field alphabets in addition if m is a non power of prime composite number then such a network is not solvable over any prime power size alphabet

polynomial time algorithms for multicast network code construction the famous max flow min cut theorem states that a source node s can send information through a network v e to a sink node t at a rate determined by the min cut separating s and t recently it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to re encode the information they receive we demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed we give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity we extend these algorithms to integer capacities and to codes that are tolerant to edge failures

linear network coding over rings part ii vector codes and non commutative alphabets in part i we studied linear network coding over finite commutative rings and made comparisons to the well studied case of linear network coding over finite fields here we consider the more general setting of linear network coding over finite possibly non commutative rings and modules we prove the following results regarding the linear solvability of directed acyclic networks over various finite alphabets for any network the following are equivalent i vector linear solvability over some field ii scalar linear solvability over some ring and iii linear solvability over some module analogously the following are equivalent a scalar linear solvability over some field b scalar linear solvability over some commutative ring and c linear solvability over some module whose ring is commutative whenever any network is linearly solvable over a module a smallest such module arises in a vector linear solution for that network over a field if a network is scalar linearly solvable over some non commutative ring but not over any commutative ring then such a non commutative ring must have size at least 16 and for some networks this bound is achieved an infinite family of networks is demonstrated each of which is scalar linearly solvable over some non commutative ring but not over any commutative ring whenever p is prime and 1 le k le 6 if a network is scalar linearly solvable over some ring of size p k then it is also k dimensional vector linearly solvable over the field mathrm gf p but the converse does not necessarily hold this result is extended to all k ge 1 when the ring is commutative

quasi cyclic ldpc codes on cyclic subgroups of finite fields a new class of quasi cyclic ldpc codes whose parity check matrices are arrays of circulant permutation matrices constructed based on cyclic subgroups of finite fields is presented this class of codes contains several known classes of algebraic quasi cyclic ldpc codes as subclasses experimental results show that the codes constructed perform very well over the awgn channel when decoded with iterative decoding based on belief propagation this class of new qc ldpc codes contains a subclass of codes which have large minimum distances combinatorial expressions for the ranks of the parity check matrices of a subclass of codes constructed based on fields of characteristic two are given

circular shift linear network coding we study a class of linear network coding lnc schemes called circular shift lnc whose encoding operations consist of only circular shifts and bit wise additions formulated as a special vector linear code over gf 2 an l dimensional circular shift linear code of degree delta restricts its local encoding kernels to be the summation of at most delta cyclic permutation matrices of size l we show that on a general network for a certain block length l every scalar linear solution over gf 2 l 1 can induce an l dimensional circular shift linear solution with 1 bit redundancy per edge transmission consequently specific to a multicast network such a circular shift linear solution of an arbitrary degree delta can be efficiently constructed which has an interesting complexity tradeoff between encoding and decoding with different choices of delta by further proving that circular shift lnc is insufficient to achieve the exact capacity of certain multicast networks we show the optimality of the efficiently constructed circular shift linear solution in the sense that its 1 bit redundancy is inevitable finally both theoretical and numerical analysis imply that with increasing l a randomly constructed circular shift linear code has linear solvability behavior comparable to a randomly constructed permutation based linear code but has shorter overheads

linear vector physical layer network coding for mimo two way relay channels design and performance analysis in this paper we propose a new linear vector physical layer network coding nc scheme for spatial multiplexing multiple input multiple output mimo two way relay channel twrc where the channel state information csi is not available at the transmitters in this scheme each user transmits m independent quadrature amplitude modulation signal streams respectively from its m antennas to the relay based on the receiver side csi the relay determines a nc generator matrix for linear vector network coding and reconstructs the associated m linear combinations of all messages we present an explicit solution for the generator matrix that minimizes the error probability at a high snr as well as an efficient algorithm to find the optimized solution we propose a novel typical error event analysis that exploits a new characterization of the deep fade events for the twrc we derive a new closed form expression for the average error probability of the proposed scheme over a rayleigh fading mimo twrc our analysis shows that the proposed scheme achieves the optimal error rate performance at a high snr numerical results show that the proposed scheme significantly outperforms existing schemes and match well with our analytical results

fixed time stabilization of fuzzy neutral type inertial neural networks with time varying delay abstract this paper addresses the problem of fixed time stabilization for a class of fuzzy neutral type inertial neural networks fntinns with time varying delay by using a novel fixed time stability theorem for dynamical systems two different feedback control laws are designed to ensure the fixed time stabilization of fntinns with time varying delay the proposed theoretical results can lead to a better upper settling time estimation compared to existing results finally three simulation examples are provided to illustrate the validity of the proposed theoretical results

a bifurcation analysis of the four dimensional generalized hopfield neural network abstract a four neuron hopfield neural network with asymmetric weights and self connection is analyzed its stable steady state and periodic attractors are identified and a complete bifurcation diagram is constructed a center manifold reduction is undertaken and by means of normal form theory the characteristics of the limit cycles are obtained the network is seen to have a large memory storage capacity with both fixed point and periodic memories coexisting the onset of chaos as reported previously is considered

on the index coding problem and its relation to network coding and matroid theory the index coding problem has recently attracted a significant attention from the research community due to its theoretical significance and applications in wireless ad hoc networks an instance of the index coding problem includes a sender that holds a set of information messages x x1 xk and a set of receivers r each receiver x h in r needs to obtain a message x x and has prior side information consisting of a subset h of x the sender uses a noiseless communication channel to broadcast encoding of messages in x to all clients the objective is to find an encoding scheme that minimizes the number of transmissions required to satisfy the demands of all the receivers in this paper we analyze the relation between the index coding problem the more general network coding problem and the problem of finding a linear representation of a matroid in particular we show that any instance of the network coding and matroid representation problems can be efficiently reduced to an instance of the index coding problem our reduction implies that many important properties of the network coding and matroid representation problems carry over to the index coding problem specifically we show that vector linear codes outperform scalar linear index codes and that vector linear codes are insufficient for achieving the optimum number of transmissions

weight cycling and serre type conjectures for unitary groups we prove that for forms of u 3 which are compact at infinity and split at places dividing a prime p in generic situations the serre weights of a mod p modular galois representation which is irreducible when restricted to each decomposition group above p are exactly those previously predicted by herzig we do this by combining explicit computations in p adic hodge theory based on a formalism of strongly divisible modules and breuil modules with descent data which we develop here with a technique that we call weight cycling

burst decoding of cyclic codes based on circulant parity check matrices an error burst correcting algorithm is developed based on a circulant parity check matrix of a cyclic code the proposed algorithm is more efficient than error trapping if the code rate is less than about 2 3 it is shown that for any n k cyclic code there is an n n circulant parity check matrix such that the algorithm applied to this matrix corrects error bursts of lengths up to the error burst correction limit of the cyclic code this same matrix can be used to efficiently correct erasure bursts of lengths up to n k the error burst correction capabilities of a class of cyclic low density parity check ldpc codes constructed from finite geometries are also considered

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

algebraic algorithms for vector network coding we develop new algebraic algorithms for scalar and vector network coding in vector network coding the source multicasts information by transmitting vectors of length l while intermediate nodes process and combine their incoming packets by multiplying them with l l coding matrices that play a similar role as coding coefficients in scalar coding we start our work by extending the algebraic framework developed for multicasting over graphs by koetter and medard to include operations over matrices we build on this generalized framework to provide a new approach for both scalar and vector code design which attempts to minimize the employed field size and employed vector length while selecting the coding operations our algorithms also lead as a special case to network code designs that employ structured matrices

design of quantum ldpc codes from quadratic residue sets we design classes of quantum low density parity check ldpc codes called quasi cyclic stabilizer qcs codes from conventional qc ldpc codes the proposed qcs codes belong to the family of non calderbank shor steane stabilizer codes the qc ldpc codes are self orthogonal with respect to the symplectic inner product sip and are constructed from submatrices of nonorthogonal latin squares via array dispersion the latin squares are constructed using quadratic non residue sets of prime modulus p where p 4n pm 1 for p 4n 1 two constructions namely type i a and type i b qcs codes are proposed based on matrix superposition and matrix concatenation respectively for p 4n 1 type ii qcs codes are proposed based on permutations of a base matrix we show that the parity check matrix for type i b and type ii qcs codes is self orthogonal with respect to the sip for all orders of circulant permutation matrix this resulting in ensembles of qcs codes characterized by a single base matrix we show that the minimum distance of type ii qcs codes can be lower bounded by the minimum distance of the qc ldpc codes simulation results show that the proposed qcs codes outperform some codes in the literature with a noteworthy low error floor below 10 7 over quantum depolarizing channels

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

network error correcting codes using small fields existing construction algorithms of block network error correcting codes require a rather large field size which grows with the size of the network and the number of sinks and thereby can be prohibitive in large networks in this work we give an algorithm which starting from a given network error correcting code can obtain another network code using a small field with the same error correcting capability as the original code an algorithm for designing network codes using small field sizes proposed recently by ebrahimi and fragouli can be seen as a special case of our algorithm the major step in our algorithm is to find a least degree irreducible polynomial which is coprime to another large degree polynomial we utilize the algebraic properties of finite fields to implement this step so that it becomes much faster than the brute force method as a result the algorithm given by ebrahimi and fragouli is also quickened

two reliability based iterative majority logic decoding algorithms for ldpc codes this paper presents two novel reliability based iterative majority logic decoding algorithms for ldpc codes both algorithms are binary message passing algorithms and require only logical operations and integer additions consequently they can be implemented with simple combinational logic circuits they either outperform or perform just as well as the existing weighted bit flipping or other reliability based iterative decoding algorithms for ldpc codes in error performance with a faster rate of decoding convergence and less decoding complexity compared to the sum product algorithm for ldpc codes they offer effective trade offs between performance and decoding complexity

quasi cyclic ldpc codes an algebraic construction rank analysis and codes on latin squares quasi cyclic ldpc codes are the most promising class of structured ldpc codes due to their ease of implementation and excellent performance over noisy channels when decoded with message passing algorithms as extensive simulation studies have shown in this paper an approach for constructing quasi cyclic ldpc codes based on latin squares over finite fields is presented by analyzing the parity check matrices of these codes combinatorial expressions for their ranks and dimensions are derived experimental results show that with iterative decoding algorithms the constructed codes perform very well over the awgn and the binary erasure channels