fault tolerant hamiltonian laceability of balanced hypercubes the balanced hypercube as a new variant of hypercube has many desirable properties such as strong connectivity high regularity and symmetry the particular property of the balanced hypercube is that each processor has a backup processor sharing the same neighborhood a hamiltonian bipartite graph g v 0 v 1 e is said to be hamiltonian laceable if there is a hamiltonian path between any two vertices x v 0 and y v 1 it has been proved that the balanced hypercube bh n is hamiltonian laceable for all n 1 in this paper we have proved that after at most 2 n 2 faulty edges occur bh n remains hamiltonian laceable for all n 2 this result is optimal with respect to the number of faulty edges can be tolerated in bh n

universal space time coding a universal framework is developed for constructing full rate and full diversity coherent space time codes for systems with arbitrary numbers of transmit and receive antennas the proposed framework combines space time layering concepts with algebraic component codes optimized for single input single output siso channels each component code is assigned to a thread in the space time matrix allowing it thus full access to the channel spatial diversity in the absence of the other threads diophantine approximation theory is then used in order to make the different threads transparent to each other within this framework a special class of signals which uses algebraic number theoretic constellations as component codes is thoroughly investigated the lattice structure of the proposed number theoretic codes along with their minimal delay allow for polynomial complexity maximum likelihood ml decoding using algorithms from lattice theory combining the design framework with the cayley transform allows to construct full diversity differential and noncoherent space time codes the proposed framework subsumes many of the existing codes in the literature extends naturally to time selective and frequency selective channels and allows for more flexibility in the tradeoff between power efficiency bandwidth efficiency and receiver complexity simulation results that demonstrate the significant gains offered by the proposed codes are presented in certain representative scenarios

common randomness in information theory and cryptography i secret sharing as the first part of a study of problems involving common randomness at distance locations information theoretic models of secret sharing generating a common random key at two terminals without letting an eavesdropper obtain information about this key are considered the concept of key capacity is defined single letter formulas of key capacity are obtained for several models and bounds to key capacity are derived for other models

geodesical refinement of mimo limited feedback in this paper we investigate a codeword refinement technique called double codeword coding in the context of mimo feedback in the proposed method the second codeword is fed back in addition to the best codeword as their interpolation at the transmitter improves the channel knowledge the derivation of the optimal interpolation parameter is presented together with the derivation of an upper bound to the interpolation parameter based on channel quality feedback available at the transmitter we show that double codeword feedback refinement has slightly lower complexity than alternative techniques and still achieves similar performance furthermore double codeword coding can be generalized to higher transmission ranks using extended multiuser mimo link simulations we verify that the double codeword feedback is able to provide significant gains in 3gpp lte context

space time block codes from orthogonal designs we introduce space time block coding a new paradigm for communication over rayleigh fading channels using multiple transmit antennas data is encoded using a space time block code and the encoded data is split into n streams which are simultaneously transmitted using n transmit antennas the received signal at each receive antenna is a linear superposition of the n transmitted signals perturbed by noise maximum likelihood decoding is achieved in a simple way through decoupling of the signals transmitted from different antennas rather than joint detection this uses the orthogonal structure of the space time block code and gives a maximum likelihood decoding algorithm which is based only on linear processing at the receiver space time block codes are designed to achieve the maximum diversity order for a given number of transmit and receive antennas subject to the constraint of having a simple decoding algorithm the classical mathematical framework of orthogonal designs is applied to construct space time block codes it is shown that space time block codes constructed in this way only exist for few sporadic values of n subsequently a generalization of orthogonal designs is shown to provide space time block codes for both real and complex constellations for any number of transmit antennas these codes achieve the maximum possible transmission rate for any number of transmit antennas using any arbitrary real constellation such as pam for an arbitrary complex constellation such as psk and qam space time block codes are designed that achieve 1 2 of the maximum possible transmission rate for any number of transmit antennas for the specific cases of two three and four transmit antennas space time block codes are designed that achieve respectively all 3 4 and 3 4 of maximum possible transmission rate using arbitrary complex constellations the best tradeoff between the decoding delay and the number of transmit antennas is also computed and it is shown that many of the codes presented here are optimal in this sense as well

lattice codes for the wiretap gaussian channel construction and analysis we consider the gaussian wiretap channel where two legitimate players alice and bob communicate over an additive white gaussian noise awgn channel while eve is eavesdropping also through an awgn channel we propose a coding strategy based on lattice coset encoding we define the secrecy gain as a design criterion for wiretap lattice codes expressed in terms of the lattice theta series which characterizes eve s confusion as a function of the channel parameters the secrecy gain is studied for even unimodular lattices and an asymptotic analysis shows that it grows exponentially in the dimension of the lattice examples of wiretap lattice codes are given

an even unimodular 72 dimensional lattice of minimum 8 an even unimodular 72 dimensional lattice having minimum 8 is constructed as a tensor product of the barnes lattice and the leech lattice over the ring of integers in the imaginary quadratic number field with discriminant 7 the automorphism group of contains the absolutely irreducible rational matrix group psl2 7 sl2 25 2

the golden code a 2 spl times 2 full rate space time code with nonvanishing determinants in this paper the golden code for a 2 spl times 2 multiple input multiple output mimo system is presented this is a full rate 2 spl times 2 linear dispersion algebraic space time code with unprecedented performance based on the golden number 1 spl radic 5 2

displacement operator based decompositions of matrices using circulants or other group matrices abstract we show how an arbitrary square matrix can be expressed as sums of products of circulant and upper or lower triangular toeplitz matrices and as sums of products of matrices derived from finite groups group matrices and matrices which are close to group matrices the results obtained are interesting from several points of view they lead to different methods for computing linear transforms using ffts or fast convolution algorithms to faster methods for solving toeplitz systems to potential methods for mapping linear transforms to parallel computer architectures with interconnection scheme given by the group graph of a finite group and possibly to matrix theoretic methods for expressing relationship between finite groups

asymptotic performance of linear receivers in mimo fading channels linear receivers are an attractive low complexity alternative to optimal processing for multiple antenna multiple input multiple output mimo communications in this paper we characterize the information theoretic performance of mimo linear receivers in two different asymptotic regimes for fixed number of antennas we investigate the limit of error probability in the high signal to noise ratio snr regime in terms of the diversity multiplexing tradeoff dmt following this we characterize the error probability for fixed snr in the regime of large but finite number of antennas as far as the dmt is concerned we report a negative result we show that both linear zero forcing zf and linear minimum mean square error mmse receivers achieve the same dmt which is largely suboptimal even in the case where outer coding and deacircnot coding is performed across the antennas we also provide an apacircnot proximate quantitative analysis of the markedly different behavior of the mmse and zf receivers at finite rate and nonasymptotic snr and show that while the zf receiver achieves poor diversity at any finite rate the mmse receiver error curve slope flattens out progressively as the coding rate increases when snr is fixed and the number of antennas becomes large we show that the mutual information at the output of an mmse or zf linear receiver has fluctuations that converge in distribution to a gaussian random variable whose mean and variance can be characterized in closed form this analysis extends to the linear reacircnot ceiver case a well known result previously obtained for the optimal receiver simulations reveal that the asymptotic analysis captures accurately the outage behavior of systems even with a moderate number of antennas

square matrix embeddable space time block codes for complex signal constellations space time block codes for providing transmit diversity in wireless communication systems are considered based on the principles of linearity and unitarity a complete classification of linear codes is given in the case when the symbol constellations are complex and the code is based on a square matrix or restriction of such by deleting columns antennas maximal rate delay optimal codes are constructed within this category the maximal rates allowed by linearity and unitarity fall off exponentially with the number of transmit antennas

an upper bound on the ratio of the probabilities of subgroups and cosets corresp through a counterexample redinbo s upper bound on the ratio of the probability of a group and that of its coset is shown to be incorrect the given example is also shown to provide a new upper bound

a classification of unimodular lattice wiretap codes in small dimensions lattice coding over a gaussian wiretap channel where an eavesdropper listens to transmissions between a transmitter and a legitimate receiver is considered a new lattice invariant called the secrecy gain is used as a code design criterion for wiretap lattice codes since it was shown to characterize the confusion that a chosen lattice can cause at the eavesdropper the higher the secrecy gain of the lattice the more confusion in this paper secrecy gains of extremal odd unimodular lattices as well as unimodular lattices in dimension n 16 n 23 are computed covering the four extremal odd unimodular lattices and all the 111 nonextremal unimodular lattices both odd and even providing thus a classification of the best wiretap lattice codes coming from unimodular lattices in dimension n 8 n 23 finally to permit lattice encoding via construction a the corresponding error correction codes of the best lattices are determined

full diversity high rate space time block codes from division algebras we present some general techniques for constructing full rank minimal delay rate at least one space time block codes stbcs over a variety of signal sets for arbitrary number of transmit antennas using commutative division algebras field extensions as well as using noncommutative division algebras of the rational field spl qopf embedded in matrix rings the first half of the paper deals with constructions using field extensions of spl qopf working with cyclotomic field extensions we construct several families of stbcs over a wide range of signal sets that are of full rank minimal delay and rate at least one appropriate for any number of transmit antennas we study the coding gain and capacity of these codes using transcendental extensions we construct arbitrary rate codes that are full rank for arbitrary number of antennas we also present a method of constructing stbcs using noncyclotomic field extensions in the later half of the paper we discuss two ways of embedding noncommutative division algebras into matrices left regular representation and representation over maximal cyclic subfields the 4 spl times 4 real orthogonal design is obtained by the left regular representation of quaternions alamouti s 1998 code is just a special case of the construction using representation over maximal cyclic subfields and we observe certain algebraic uniqueness characteristics of it also we discuss a general principle for constructing cyclic division algebras using the nth root of a transcendental element and study the capacity of the stbcs obtained from this construction another family of cyclic division algebras discovered by brauer 1933 is discussed and several examples of stbcs derived from each of these constructions are presented

applications of ldpc codes to the wiretap channel with the advent of quantum key distribution qkd systems perfect i e information theoretic security can now be achieved for distribution of a cryptographic key qkd systems and similar protocols use classical error correcting codes for both error correction for the honest parties to correct errors and privacy amplification to make an eavesdropper fully ignorant from a coding perspective a good model that corresponds to such a setting is the wire tap channel introduced by wyner in 1975 in this correspondence we study fundamental limits and coding methods for wire tap channels we provide an alternative view of the proof for secrecy capacity of wire tap channels and show how capacity achieving codes can be used to achieve the secrecy capacity for any wiretap channel we also consider binary erasure channel and binary symmetric channel special cases for the wiretap channel and propose specific practical codes in some cases our designs achieve the secrecy capacity and in others the codes provide security at rates below secrecy capacity for the special case of a noiseless main channel and binary erasure channel we consider encoder and decoder design for codes achieving secrecy on the wiretap channel we show that it is possible to construct linear time decodable secrecy codes based on low density parity check ldpc codes that achieve secrecy

secret key agreement by public discussion from common information the problem of generating a shared secret key s by two parties knowing dependent random variables x and y respectively but not sharing a secret key initially is considered an enemy who knows the random variable z jointly distributed with x and y according to some probability distribution p sub xyz can also receive all messages exchanged by the two parties over a public channel the goal of a protocol is that the enemy obtains at most a negligible amount of information about s upper bounds on h s as a function of p sub xyz are presented lower bounds on the rate h s n as n to infinity are derived for the case in which x x sub 1 x sub n y y sub 1 y sub n and z z sub 1 z sub n result from n independent executions of a random experiment generating x sub i y sub i and z sub i for i 1 n it is shown that such a secret key agreement is possible for a scenario in which all three parties receive the output of a binary symmetric source over independent binary symmetric channels even when the enemy s channel is superior to the other two channels

golden space time block coded modulation in this paper block coded modulation is used to design a 2 times 2 multiple input multiple output mimo space time code for slow fading channels the golden code is chosen as the inner code the scheme is based on a set partitioning of the golden code using two sided ideals whose norm is a power of two in this case a lower bound for the minimum determinant is given by the minimum hamming distance the description of the ring structure of the quotients suggests further optimization in order to improve the overall distribution of determinants simulation results show that the proposed schemes achieve a significant gain over the un coded golden code

efficient encoding of low density parity check codes low density parity check ldpc codes can be considered serious competitors to turbo codes in terms of performance and complexity and they are based on a similar philosophy constrained random code ensembles and iterative decoding algorithms we consider the encoding problem for ldpc codes more generally we consider the encoding problem for codes specified by sparse parity check matrices we show how to exploit the sparseness of the parity check matrix to obtain efficient encoders for the 3 6 regular ldpc code for example the complexity of encoding is essentially quadratic in the block length however we show that the associated coefficient can be made quite small so that encoding codes even of length n spl sime 100000 is still quite practical more importantly we show that optimized codes actually admit linear time encoding

almost universal codes for mimo wiretap channels despite several works on secrecy coding for fading and mimo wiretap channels from an error probability perspective the construction of information theoretically secure codes over such channels remains an open problem in this paper we consider a fading wiretap channel model where the transmitter has only partial statistical channel state information our channel model includes static channels i i d block fading channels and ergodic stationary fading with fast decay of large deviations for the eavesdropper s channel we extend the flatness factor criterion from the gaussian wiretap channel to fading and mimo wiretap channels and establish a simple design criterion where the normalized product distance minimum determinant of the lattice and its dual should be maximized simultaneously moreover we propose concrete lattice codes satisfying this design criterion which are built from algebraic number fields with constant root discriminant in the single antenna case and from division algebras centered at such number fields in the multiple antenna case the proposed lattice codes achieve strong secrecy and semantic security for all rates r where c b and c e are bob and eve s channel capacities respectively and kappa is an explicit constant gap furthermore these codes are almost universal in the sense that a fixed code is good for secrecy for a wide range of fading models finally we consider a compound wiretap model with a more restricted uncertainty set and show that rates r are achievable where bar c b is a lower bound for bob s capacity and bar c e is an upper bound for eve s capacity for all the channels in the set

application of perfect space time codes pep bounds and some practical insights the design of perfect space time codes constructed from cyclic division algebra cda on the quasi static mimo channel has received lots of attention in industry over the last few years however the recent standards that use multiple antennas terminals such as ieee 802 11n ieee 802 16e or lte are based on more realistic assumptions involving the use of outer codes and multi taps channels in this paper the question regarding the utility of using a high dimensional modulation schemes such as space time codes in a standard context is addressed for this we evaluate the performance of the perfect space time codes in term of an upper bound on the pairwise error probability and we compare it to the one of simple spatial division multiplexing sdm schemes finally simulation results considered in the context of ieee 802 11n show that the 2 2 mimo perfect code i e the golden code performs relatively similarly as the sdm scheme in the presence of a good outer code

coset codes i introduction and geometrical classification practically all known good constructive coding techniques for bandlimited channels including lattice codes and various trellis coded modulation schemes can be characterized as coset codes a coset code is defined by a lattice partition lambda lambda and by a binary encoder c that selects a sequence of cosets of the lattice lambda the fundamental coding gain of a coset code as well as other important parameters such as the error coefficient the decoding complexity and the constellation expansion factor are purely geometric parameters determined by c lambda lambda the known types of coset codes as well as a number of new classes that systematize and generalize known codes are classified and compared in terms of these parameters

an efficient zj based decoding algorithm for space time trellis codes in slow rayleigh fading channels in this paper we propose a computationally efficient decoding algorithm for space time trellis codes in slow rayleigh fading channels the proposed scheme is based on a stack algorithm with two key ideas i a variable stack size depending upon the signal to noise ratio to avoid the exhaustive search of paths and ii a normalized metric which is defined as each cumulative path metric divided by its own length in the stack to provide an appropriate comparison of the paths with different lengths simulation results demonstrate that the proposed algorithm achieves near ml performance with significant reduction in complexity compared with the conventional viterbi algorithm

counterexample to the generalized belfiore sole secrecy function conjecture for l modular lattices in this paper we show that the secrecy function conjecture that states that the maximum of the secrecy function of an l modular lattice occurs at 1 sqrt l is false by proving that the four modular lattice c 4 mathbb z oplus sqrt 2 mathbb z oplus 2 mathbb z fails to satisfy this conjecture after this we indicate how the secrecy function must be modified in the l modular case to provide a more meaningful comparison for l modular lattices and show that this new secrecy function indeed has a maximum at 1 sqrt l for various two modular lattices and for c 4 we conjecture that this must hold true for all l modular lattices

representations and routing for cayley graphs computer networks it is shown that all finite cayley graphs can be represented by generalized chordal rings gcr an example borel cayley graph is used to illustrate the generation of gcr representations a sufficient condition is given for the representation of a cayley graph as a chordal ring cr with the integer labeling of gcr representations a straightforward progressive routing algorithm based on table look up is summarized

optimal emulations by butterfly like networks the power of butterfly like networks as multicomputer interconnection networks is studied by considering how efficiently the butterfly can emulate other networks emulations are studied formally via graph embeddings so the topic here becomes how efficiently can one embed the graph underlying a given interconnection network in the graph underlying the butterfly network within this framework the slowdown incurred by an emulation is measured by the sum of the dilation and the congestion of the corresponding embedding respectively the maximum amount that the embedding stretches an edge of the guest graph and the maximum traffic across any edge of the host graph the efficiency of resource utilization in an emulation is measured by the expansion of the corresponding embedding the ratio of the sizes of the host to guest graph three main results expose a number of optimal emulations by butterfly networks call a family of graphs balanced if complete binary trees can be embedded in the family with simultaneous dilation congestion and expansion 0 1 1 the family of butterfly graphs is balanced 2 a any graph from a family of maxdegree d graphs having a recursive separator of size s x can be embedded in any balanced graph family with simultaneous dilation o log d i s 2 i g and expansion o 1 b any dilation d embedding of a maxdegree d graph in a butterfly graph can be converted to an embedding having simultaneous dilation o d and congestion o dd 3 any embedding of a planar graph g in a butterfly graph must have dilation log g g where g is the size of the smallest 1 3 2 3 node separator of g and g is the size of g s largest interior face applications of these results include 1 the n node x tree network can be emulated by the butterfly network with slowdown o log log n and expansion 0 1 no embedding has dilation smaller than log log n independent of expansion 2 every embedding of the n x n mesh in the butterfly graph has dilation log n any expansion o 1 embedding in the butterfly graph achieves dilation o log n these applications provide the first examples of networks that can be embedded more efficiently in hypercubes than in butterflies we also show that analogues of these results hold for networks that are structurally related to the butterfly network the upper bounds hold for the hypercube and the de bruijn networks possibly with altered constants the lower bounds hold at least in weakened form for the de bruijn network

on the geometric ergodicity of metropolis hastings algorithms for lattice gaussian sampling sampling from the lattice gaussian distribution has emerged as an important problem in coding decoding and cryptography in this paper the classic metropolis hastings mh algorithm in markov chain monte carlo methods is adopted for lattice gaussian sampling two mh based algorithms are proposed which overcome the limitation of klein s algorithm the first one referred to as the independent metropolis hastings klein mhk algorithm establishes a markov chain via an independent proposal distribution we show that the markov chain arising from this independent mhk algorithm is uniformly ergodic namely it converges to the stationary distribution exponentially fast regardless of the initial state moreover the rate of convergence is analyzed in terms of the theta series leading to predictable mixing time a symmetric metropolis klein algorithm is also proposed which is proven to be geometrically ergodic

super orthogonal space time trellis codes we introduce a new class of space time codes called super orthogonal space time trellis codes these codes combine set partitioning and a super set of orthogonal space time block codes in a systematic way to provide full diversity and improved coding gain over earlier space time trellis code constructions we also study the optimality of our set partitioning and provide coding gain analysis codes operating at different rates up to the highest theoretically possible rate for different number of states can be designed by using our optimal set partitioning super orthogonal space time trellis codes can provide a tradeoff between rate and coding gain simulation results show more than 2 db improvements over the codes presented in the literature while providing a systematic design methodology

space time codes from structured lattices we present constructions of space time st codes based on lattice coset coding first we focus on st code constructions for the short block length case i e when the block length is equal to or slightly larger than the number of transmit antennas we present constructions based on dense lattice packings and nested lattice voronoi shaping our codes achieve the optimal diversity multiplexing tradeoff dmt of quasi static multiple input multiple output mimo fading channels for any fading statistics and perform very well also at practical moderate values of signal to noise ratios snr then we extend the construction to the case of large block lengths by using trellis coset coding we provide constructions of trellis coded modulation tcm schemes that are endowed with good packing and shaping properties both short block and trellis constructions allow for a reduced complexity decoding algorithm based on minimum mean squared error generalized decision feedback equalizer mmse gdfe lattice decoding and a combination of this with a viterbi tcm decoder for the tcm case beyond the interesting algebraic structure we exhibit codes whose performance is among the state of the art considering codes with similar encoding decoding complexity

coset codes ii binary lattices and related codes for pt i see ibid vol 34 no 5 p 1123 51 1988 the family of barnes wall lattices including d sub 4 and e sub 8 of lengths n 2 sup n and their principal sublattices which are useful in constructing coset codes are generated by iteration of a simple construction called the squaring construction the closely related reed muller codes are generated by the same construction the principal properties of these codes and lattices are consequences of the general properties of iterated squaring constructions which also exhibit the interrelationships between codes and lattices of different lengths an extension called the cubing construction generates good codes and lattices of lengths n 3 2 sup n including the golay code and leech lattice with the use of special bases for 8 space another related construction generates the nordstrom robinson code and an analogous 16 dimensional nonlattice packing these constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum likelihood decoding algorithms

transmission subspace tracking for mimo systems with low rate feedback this paper describes a feedback algorithm for tracking the dominant subspaces of continuously time varying channels in multiantenna communication systems the nature of the problem is quantization of subspaces it is well known that subspaces can be mathematically modeled as points in a grassmann manifold we model the variations between the dominant subspaces of channels at adjacent time instants to be along geodesics in the grassmann manifold instead of quantizing the subspaces themselves we propose to quantize the geodesic trajectory connecting two subspaces more specifically we quantize a key entity that characterizes a geodesic arc the velocity matrix which resembles angular speed in a one dimensional complex space two techniques are proposed for quantizing the velocity matrix of the geodesic in the first a 1 bit feedback is utilized to indicate the preferred sign of a random velocity matrix of the geodesic in the other the velocity matrix is quantized using a gaussian vector quantization codebook numerical results show that the performance of the proposed 1 bit feedback algorithm is better than a previously proposed grassmannian subspace packing scheme at low to medium doppler frequencies and better than a gradient sign feedback scheme at all doppler frequencies in our simulations the gaussian vector quantization algorithm is always better than the 1 bit feedback algorithm

an error probability approach to mimo wiretap channels we consider mimo multiple input multiple output wiretap channels where a legitimate transmitter alice is communicating with a legitimate receiver bob in the presence of an eavesdropper eve and communication is done via mimo channels we suppose that alice s strategy is to use an infinite lattice codebook which then allows her to perform coset encoding we analyze eve s probability of correctly decoding the message alice meant to bob and from minimizing this probability we derive a code design criterion for mimo lattice wiretap codes the case of block fading channels is treated similarly and fast fading channels are derived as a particular case the alamouti code is carefully studied as an illustration of the analysis provided

zero forcing methods for downlink spatial multiplexing in multiuser mimo channels the use of space division multiple access sdma in the downlink of a multiuser multiple input multiple output mimo wireless communications network can provide a substantial gain in system throughput the challenge in such multiuser systems is designing transmit vectors while considering the co channel interference of other users typical optimization problems of interest include the capacity problem maximizing the sum information rate subject to a power constraint or the power control problem minimizing transmitted power such that a certain quality of service metric for each user is met neither of these problems possess closed form solutions for the general multiuser mimo channel but the imposition of certain constraints can lead to closed form solutions this paper presents two such constrained solutions the first referred to as block diagonalization is a generalization of channel inversion when there are multiple antennas at each receiver it is easily adapted to optimize for either maximum transmission rate or minimum power and approaches the optimal solution at high snr the second known as successive optimization is an alternative method for solving the power minimization problem one user at a time and it yields superior results in some e g low snr situations both of these algorithms are limited to cases where the transmitter has more antennas than all receive antennas combined in order to accommodate more general scenarios we also propose a framework for coordinated transmitter receiver processing that generalizes the two algorithms to cases involving more receive than transmit antennas while the proposed algorithms are suboptimal they lead to simpler transmitter and receiver structures and allow for a reasonable tradeoff between performance and complexity

on a special class of broadcast channels with confidential messages it is shown that csiszar and korner s 1978 characterization of a discrete memoryless channel dmc x spl rarr y as being less noisy than the dmc x spl rarr z is equivalent to the condition that the mutual information difference i x y i x z be a convex spl cap function of the probability distribution for x this result is used to obtain a simple determination of the capacity region of the broadcast channel with confidential messages bcc which is a dmc x spl rarr y z when the dmc x spl rarr y to the legitimate receiver is less noisy than the dmc x spl rarr z to the enemy cryptanalyst and there is a probability distribution for x having strictly positive components that achieves capacity on both these channels in particular when these dmc s are both symmetric then the secrecy capacity of the bcc is the difference of their capacities it is shown further that the less noisy condition in this result cannot be weakened to the condition that the dmc x spl rarr y be more capable than the dmc x spl rarr z in the sense of csiszar and korner

efficient embeddings of trees in hypercubes the boolean hypercube is a particularly versatile network for parallel computing it is well known that multi dimensional grid machines can be simulated on a hypercube with no communications overhead in this paper we show that every bounded degree tree can be simulated on the hypercube with constant communications overhead our proof in fact shows that every bounded degree graph with an 0 1 separator can be embedded in a hypercube of the same size with dilation and congestion both 0 1 we prove also that not all bounded degree graphs can be efficiently embedded within the hypercube

a survey of combinatorial gray codes the term combinatorial gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified small way this notion generalizes the classical binary reflected gray code scheme for listing n bit binary numbers so that successive numbers differ in exactly one bit position as well as work in the 1960s and 1970s on minimal change listings for other combinatorial families including permutations and combinations the area of combinatorial gray codes was popularized by herbert wilf in his invited address at the siam conference on discrete mathematics in 1988 and his subsequent siam monograph combinatorial algorithms an update 1989 in which he posed some open problems and variations on the theme this resulted in much recent activity in the area and most of the problems posed by wilf are now solved in this paper we survey the area of combinatorial gray codes describe recent results variations and trends and highlight some open problems

mimo systems with limited rate differential feedback in slowly varying channels in this paper an adaptive limited feedback linear precoding technique for temporally correlated multiple input multiple output mimo channels is proposed where the receiver has perfect channel knowledge but the transmitter only receives a quantized channel direction to perform adaptation to the time correlation structure we employ a differential feedback where the amount of the perturbation added to the previous precoder is determined by the statistics of the directional variation based on random matrix quantization analysis we develop a spherical cap codebook approach where the cap is centered at the previous precoder and the radius of the cap is determined proportional to the identified directional variation if the channel is highly correlated in time it is shown that the proposed differential feedback scheme achieves significant throughput improvement in the large codebook size regime the rest of the paper is devoted to developing a systematic spherical cap codebook generation method the developed approach employs a feedback scheme that uses a differential rotation of the previously used precoder our codebook adaptation is based on generating a perturbation in euclidean space and projecting the perturbation onto the unitary space simulation results show that the proposed adaptation scheme accurately tracks the channel using only a small rate of feedback

channel adaptive quantization for limited feedback mimo beamforming systems multiple input multiple output mimo wireless systems can achieve significant diversity and array gain by using transmit beamforming and receive combining techniques in the absence of full channel knowledge at the transmitter the transmit beamforming vector can be quantized at the receiver and sent to the transmitter using a low rate feedback channel in the literature quantization algorithms for the beamforming vector are designed and optimized for a particular channel distribution commonly the uncorrelated rayleigh distribution when the channel is not uncorrelated rayleigh however these quantization strategies result in a degradation of the receive signal to noise ratio snr in this paper switched codebook quantization is proposed where the codebook is dynamically chosen based on the channel distribution the codebook adaptation enables the quantization to exploit the spatial and temporal correlation inherent in the channel the convergence properties of the codebook selection algorithm are studied assuming a block stationary model for the channel in the case of a nonstationary channel it is shown using simulations that the selected codebook tracks the distribution of the channel resulting in improvements in snr simulation results show that in the case of correlated channels the snr performance of the link can be significantly improved by adaptation compared with nonadaptive quantization strategies designed for uncorrelated rayleigh fading channels

duality for the group coset probability ratio corresp from the macwilliams and assmus mattson identifies it is shown that sullivan s lower bound on the ratio of the probability of a group and a coset has a dual relation with the upper bound

characteristic polynomials of digraphs having a semi free action abstract let be a semi free action of a group g on a finite digraph the front divisor of has as vertex set the set of the vertex orbits of and there are t arcs going from u to v in if from any vertex u of the orbit u there are t arcs of going towards the vertices in v our main result is that the characteristic polynomial of is a product of the characteristic polynomial of its front divisor and polynomials associated with the free part of under this work extends earlier work of lee and kim characteristic polynomials of graphs having a semifree action linear algebra appl 307 2000 35 46

small diameter symmetric networks from linear groups a report is presented on a collection of constructions of symmetric networks that provide the largest known values for the number of nodes that can be placed in a network of a given degree and diameter some of the constructions are in the range of current potential engineering significance the constructions are cayley graphs of linear groups obtained by experimental computation

reduced feedback mimo ofdm precoding and antenna selection transmitter precoding for multiple input multiple output orthogonal frequency division multiplexing mimo ofdm is an effective way of leveraging the diversity gains afforded by a multiple transmit multiple receive antenna system in a frequency selective environment in the limited feedback scenario optimal precoder representation for narrowband mimo systems using moderately sized codebooks designed on the grassmann manifold has been shown to perform remarkably well in mimo ofdm systems precoder matrices have to be designed for all subcarriers and the amount of feedback can get prohibitively large this is especially true for next generation wireless local area networks and wireless metropolitan area networks which have a large number of subcarriers in this paper we present techniques to reduce this feedback requirement and the performance of these algorithms is numerically shown to provide improvement over existing schemes

interconnection networks based on a generalization of cube connected cycles a generalization of the cube connected cycles of preparata and vuillemin is described which retains the symmetry of these architectures while allowing for constructions of greater density and of arbitrary degree these constructions are of a type known as cayley graphs and their analysis is greatly facilitated by the applicability of methods from abstract algebra

on a conjecture by belfiore and sole on some lattices the point of this paper is to show that the secrecy function attains its maximum at y 1 on all known extremal unimodular lattices and on some possibly existing even extremal unimodular lattices this is a special case of a conjecture by belfiore and sole furthermore we will give a very simple method to verify or disprove the conjecture on any given unimodular lattice

some cyclo quaternionic lattices abstract the faithful lattices of rank 2 p 1 of the groups sl 2 p are described for small primes p these and related lattices are investigated by computer in particular a new extremal even unimodular lattice of rank 48 is constructed

class congruence property and two phase routing of borel cayley graphs dense symmetric graphs are useful interconnection models for multicomputer systems borel cayley graphs the densest degree 4 graphs for a range of diameters are attractive candidates however the group theoretic representation of these graphs makes the development of efficient routing algorithms difficult in earlier reports we showed that all degree 4 borel cayley graphs have generalized chordal ring gcr and chordal ring cr representations we present the class congruence property and use this property to develop the two phase routing algorithm for borel cayley graphs in a special gcr representation the algorithm requires a small space complexity of o p k for n p spl times k nodes although suboptimal the algorithm finds paths with length bounded by 2d where d is the diameter furthermore our computer implementation of the algorithm on networks with 1 081 and 15 657 nodes shows that the average path length is on the order of the diameter the performance of the algorithm is compared with that of existing optimal and suboptimal algorithms

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

fault tolerant ring embedding in de bruijn networks a method of embedding a ring in a d ary de bruijn multiprocessor network in the event of multiple node processor failures is presented in particular the algorithm guarantees that a 2 sup n n 1 node ring will be found in a binary de bruijn network with a single faulty node where 2 sup n is the total number of nodes in the network it is also shown that a d sup n 1 node ring can always be found in the presence of d 1 link failures when d is a prime power and the network contains d sup n nodes the latter is accomplished by constructing d 1 edge disjoint cycles each of length d sup n 1 a modification of the graph that allows it to admit a hamiltonian cycle in the event of d 1 edge failures is also discussed

group representations and lattices this paper began as an attempt to understand the euclidean lattices that were recently constructed using the theory of elliptic curves by elkies e and shioda shl sh2 from the point of view of group representations the main idea appears in a note of thompson th2 if one makes strong irreducibility hypotheses on a rational representation v of a finite group g then the g stable euclidean lattices in v are severely restricted unfortunately these hypotheses are rarely satisfied when v is absolutely irreducible over q but there are many examples where the ring endg v is an imaginary quadratic field or a definite quaternion algebra these representations allow us to construct some of the mordell weil lattices considered by elkies and shioda as well as some interesting even unimodular lattices that do not seem to come from the theory of elliptic curves in 1 we discuss lattices and hermitian forms on y7 and in 2 4 the strong irreducibility hypotheses we wish to make in 5 we show how our hypotheses imply the existence of a finite number up to isomorphism of euclidean z g lattices l in v with endg l a maximal order in endg v we give some examples with dim v 8 in 6 and in 7 9 discuss the invariants of l such as the dual lattice and theta function the rest of the paper is devoted to examples in most g is a finite group of lie type and v is obtained as an irreducible summand of the weil representation of g some of the representation theoretic problems left open by this paper are to find all examples of pairs g v satisfying the strong irreducibility hypotheses of 2 4 and to determine the invariants shortest nonzero vector theta function thompson series of the g lattices l so effortlessly constructed inside v

new trellis codes based on lattices and cosets a new technique is proposed for constructing trellis codes which provides an alternative to ungerboeck s method of set partitioning the new codes use a signal constellation consisting of points from an n dimensional lattice lambda with an equal number of points from each coset of a sublattice lambda one part of the input stream drives a generalized convolutional code whose outputs are cosets of lambda while the other part selects points from these cosets several of the new codes are better than those previously known

sparse graph codes for quantum error correction sparse graph codes appropriate for use in quantum error correction are presented quantum error correcting codes based on sparse graphs are of interest for three reasons first the best codes currently known for classical channels are based on sparse graphs second sparse graph codes keep the number of quantum interactions associated with the quantum error correction process small a constant number per quantum bit independent of the block length third sparse graph codes often offer great flexibility with respect to block length and rate we believe some of the codes we present are unsurpassed by previously published quantum error correcting codes

general nonasymptotic and asymptotic formulas in channel resolvability and identification capacity and their application to the wiretap channel several nonasymptotic formulas are established in channel resolvability and identification capacity and they are applied to the wiretap channel by using these formulas the epsi capacities of the above three problems are considered in the most general setting where no structural assumptions such as the stationary memoryless property are made on a channel as a result we solve an open problem proposed by han and verduacute moreover we obtain lower bounds of the exponents of error probability and the wiretapper s information in the wiretap channel

codes over matrix rings for space time coded modulations it is known that for transmission over quasi static mimo fading channels with n transmit antennas diversity can be obtained by using an inner fully diverse space time block code while coding gain derived from the determinant criterion comes from an appropriate outer code when the inner code has a cyclic algebra structure over a number field as for perfect space time codes an outer code can be designed via coset coding more precisely by taking the quotient of the algebra by a two sided ideal which leads to matrices over finite alphabets for the outer code in this paper we show that the determinant criterion induces various metrics on the outer code such as the hamming and bachoc distances when n 2 partitioning the 2 2 golden code by using an ideal above the prime 2 leads to consider codes over either m2 f2 or m2 f2 i both being noncommutative alphabets by identifying them as algebras over a finite field or a finite ring respectively we establish an unexpected connection with classical error correcting codes over f4 and f4 i matrix rings of higher dimension suitable for 3 3 and 4 4 perfect codes give rise to more complex examples

comments on low diameter interconnections for routing in high performance parallel systems with connections and extensions to arc coloring of coset graphs for original paper see r melhem ibid vol 56 p 502 10 2007 recently melhem presented a new class of low diameter interconnection ldi networks ieee trans computers vol 56 no 4 pp 502 510 we note that ldi networks are the same as the previously known generalized de bruijn graphs point out an error in the decomposition of ldi networks into permutations and find that the correct decomposition scheme is an instance of arc coloring for coset graphs hence we pursue a number of general results on arc coloring of coset graphs that can be applied to this particular decomposition problem as well as within many other contexts including complete arc coloring and normality of coset graphs

group action graphs and parallel architectures the authors develop an algebraic framework that exposes the structural kinship among the debruijn shujcfle exchange butterjy and cube connected cycles networks and illustrate algorithmic benefits that ensue from the exposed relationships the framework builds on two alge braically specified genres of graphs a group action graph gag for short is given by a set v of vertices and a set h of permutations of v for each v e v and each r eii there is an arc labeled r from vertex v to vertex v a cayley graph is a gag v h where v is the group gr h generated by h and where each r 6 h acts on each g 6 gr h by right multiplication the graphs gr h h and v h are called associated graphs it is shown that every gag is a quotient graph of its associated cayley graph by applying such general results the authors determine the following the butterfly network a cayley graph and the debruijn network a gag are associated graphs the cube connected cycles network a cayley graph and the shuffle exchange network a gag are associated graphs the order n instance of both the butterfly and the cube connected cycles share the same underlying group but have slightly different generator sets ii by analyzing these algebraic results it is delimited for any cayley graph g and associated gag 7 a family of leveled algorithms which run as efficiently on t as they do on the much larger g further analysis of the results yields new surprisingly efficient simulations by the shuffle oriented networks the shuffle exchange and debruijn networks of like sized butterfly oriented networks the butterfly and cube connected cycles networks an n vertex butterfly oriented network can be simulated by the smallest shuffle oriented network that is big enough to hold it with slowdown o log log n this simulation is exponentially faster than the anticipated logarithmic slowdown the mappings that underlie the simulation can be computed in linear time and they afford one an algorithmic tech nique for translating any program developed for a butterfly oriented architecture into an equivalent program for a shuffle oriented architecture the latter program incurring only the indicated slowdown factor

golden space time trellis coded modulation in this paper we present a multidimensional trellis coded modulation scheme for a high rate 2times2 multiple input multiple output mimo system over slow fading channels set partitioning of the golden code is designed specifically to increase the minimum determinant the branches of the outer trellis code are labeled with these partitions and viterbi algorithm is applied for trellis decoding in order to compute the branch metrics a sphere decoder is used the general framework for code design and optimization is given performance of the proposed scheme is evaluated by simulation and it is shown that it achieves significant performance gains over the uncoded golden code

a universal lattice code decoder for fading channels we present a maximum likelihood decoding algorithm for an arbitrary lattice code when used over an independent fading channel with perfect channel state information at the receiver the decoder is based on a bounded distance search among the lattice points falling inside a sphere centered at the received point by judicious choice of the decoding radius we show that this decoder can be practically used to decode lattice codes of dimension up to 32 in a fading environment

quantized multimode precoding in spatially correlated multiantenna channels multimode precoding where the number of independent data streams is adapted optimally can be used to maximize the achievable throughput in multiantenna communication systems motivated by standardization efforts embraced by the industry the focus of this work is on systematic precoder design with realistic assumptions on the spatial correlation channel state information csi at the transmitter and the receiver and implementation complexity for the spatial correlation of the channel matrix we assume a general channel model based on physical principles that has been verified by many recent measurement campaigns we also assume a coherent linear minimum mean square error mmse receiver and knowledge of the spatial statistics at the transmitter along with the presence of an ideal low rate feedback link from the receiver to the transmitter the reverse link is used for codebook index feedback and the goal of this work is to construct precoder codebooks adaptable in response to the statistical information such that the achievable throughput is significantly enhanced over that of a fixed nonadaptive independent and identically distributed i i d codebook design we illustrate how a codebook of semiunitary precoder matrices localized around some fixed center on the grassmann manifold can be skewed in response to the spatial correlation via low complexity maps that can rotate and scale submanifolds on the grassmann manifold the skewed codebook in combination with a low complexity statistical power allocation scheme is then shown to bridge the gap in performance between a perfect csi benchmark and an i i d codebook design

communication aspects of networks based on geometric incidence relations abstract we explore the communication properties of a family of networks based on the incidence relation of the middle subspaces of a projective space network theoretic issues as well as implementation details are discussed the networks are shown to be symmetric and nearly optimal in diameter they support a natural routing scheme with efficient implementation of optimal complexity an extremely high redundancy makes the networks robust a parallel routing algorithm is analysed and is shown to achieve running time o diameter which is also the lower bound

dimension reduction techniques for the minimization of theta functions on lattices we consider the minimization of theta functions theta lambda alpha sum p in lambda e pi alpha p 2 amongst lattices lambda subset mathbb r d by reducing the dimension of the problem following as a motivation the case d 3 where minimizers are supposed to be either the bcc or the fcc lattices a first way to reduce dimension is by considering layered lattices and minimize either among competitors presenting different sequences of repetitions of the layers or among competitors presenting different shifts of the layers with respect to each other the second case presents the problem of minimizing theta functions also on translated lattices namely minimizing l u mapsto theta l u alpha another way to reduce dimension is by considering lattices with a product structure or by successively minimizing over concentric layers the first direction leads to the question of minimization amongst orthorhombic lattices whereas the second is relevant for asymptotics questions which we study in detail in two dimensions

diagonal algebraic space time block codes we construct a new family of linear space time st block codes by the combination of rotated constellations and the hadamard transform and we prove them to achieve the full transmit diversity over a quasi static or fast fading channels the proposed codes transmit at a normalized rate of 1 symbol s when the number of transmit antennas n 1 2 or n is a multiple of four we spread a rotated version of the information symbol vector by the hadamard transform and send it over n transmit antennas and n time periods for other values of n we construct the codes by sending the components of a rotated version of the information symbol vector over the diagonal of an n spl times n st code matrix the codes maintain their rate diversity and coding gains for all real and complex constellations carved from the complex integers ring z i and they outperform the codes from orthogonal design when using complex constellations for n 2 the maximum likelihood ml decoding of the proposed codes can be implemented by the sphere decoder at a moderate complexity it is shown that using the proposed codes in a multiantenna system yields good performances with high spectral efficiency and moderate decoding complexity