uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

further asymptotic upper bounds on the minimum distance of trellis codes asymptotic upper bounds on the minimum distance of trellis codes are derived a universal bound and bounds specific to phase shift keying psk and quadrature amplitude modulation qam signal sets are obtained

multilevel trellis coded modulations for the rayleigh fading channel the authors present coded 8 phase shift keyed 8 psk modulations for the rayleigh fading channel the schemes are based on multilevel trellis coded modulation constructions and utilize maximum free hamming distance binary convolutional codes as building blocks a suboptimal multistage decoder that utilizes interstage interleaving and iterative decoding is proposed and evaluated examples are constructed to show that the proposed schemes outperform the best modified codes of the ungerboeck type due to significantly higher implicit time diversity yielding seven branches of built in time diversity whereas the ungerboeck code yields four branches of time diversity for a 64 state system the transmission delay is higher however the new schemes can provide three levels of unequal error protection when 8 psk or 8 differential phase shift keying 8 dpsk modulations are used they provide 10 14 db channel signal to noise ratio gain over uncoded 4 dpsk at a bit error rate of 10 sup 3 for a modest decoding complexity

on trellis codes with a delay processor and a signal mapper it is already known that a trellis code t which is constructed by using the encoder of a convolutional code c with short constraint length followed by a delay processor and a signal mapper is equivalent to a trellis code with large constraint length in this paper we derive a new lower bound on the free distance of t which in some cases is better than the previously derived bound moreover instead of the decoding used in earlier publications we apply iterative decoding on both tailbiting and zero tail representations of t to take advantage of the new lower bound and in the meantime to decrease the associated error coefficient caused by the decoding used in earlier publications comparisons among various designs of such a trellis code and some well known coding methods are also provided

multilevel construction of block and trellis group codes the theory of group codes has been shown to be a useful starting point for the construction of good geometrically uniform codes in this paper we study the problem of building multilevel group codes i e codes obtained combining separate coding at different levels in such a way that the resulting code is a group code a construction leading to multilevel group codes for semi direct and direct products is illustrated the codes that can be obtained in this way are identified new geometrically uniform euclidean space codes obtained from multilevel codes over abelian and nonabelian groups are presented

singular values of products of ginibre random matrices multiple orthogonal polynomials and hard edge scaling limits akemann ipsen and kieburg recently showed that the squared singular values of products of m rectangular random matrices with independent complex gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of meijer g functions we show that this point process can be interpreted as a multiple orthogonal polynomial ensemble we give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel which allows us to find its scaling limits at the origin hard edge the limiting kernels generalize the classical bessel kernels for m 2 they coincide with the scaling limits found by bertola gekhtman and szmigielski in the cauchy laguerre two matrix model which indicates that these kernels represent a new universality class in random matrix theory

asymptotic optimality of the gmd and chase decoding algorithms the generalized minimum distance gmd and chase 1972 decoding algorithms are some of the most important suboptimum bounded distance decoding algorithms for binary linear block codes over an additive white gaussian noise awgn channel we compute the limitation of the ratio between the probability of decoding error for the gmd or any one of the chase decoding algorithms and that of the maximum likelihood ml decoding when the signal to noise ratio snr approaches infinity if the minimum hamming distance of the code is greater than 2 the limitation is shown to be equal to 1 and thus the gmd and chase decoding algorithms are asymptotically optimum

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

new multilevel codes over gf q set partitioning is applied to multidimensional signal spaces over gf q i e gf sup n1 q n1 or d are presented these codes use reed solomon codes as component codes longer multilevel block codes are also constructed using q ary block codes with block length longer than q 1 as component codes some quaternary multilevel block codes are presented with the same length and number of information symbols as but larger distance than the best previously known quaternary one level block codes it is proved that if all the component block codes are linear the multilevel block code is also linear low rate q ary convolutional codes word error correcting convolutional codes and binary to q ary convolutional codes can also be used to construct multilevel trellis codes over gf q or binary to q ary trellis codes

large n limit for the product of two coupled random matrices for a pair of coupled rectangular random matrices we consider the squared singular values of their product which form a determinantal point process we show that the limiting mean distribution of these squared singular values is described by the second component of the solution to a vector equilibrium problem this vector equilibrium problem is defined for three measures with an upper constraint on the first measure and an external field on the second measure we carry out the steepest descent analysis for a 4 times 4 matrix valued riemann hilbert problem which characterizes the correlation kernel and is related to mixed type multiple orthogonal polynomials associated with the modified bessel functions a careful study of the vector equilibrium problem combined with this asymptotic analysis ultimately leads to the aforementioned convergence result for the limiting mean distribution an explicit form of the associated spectral curve as well as local sine meijer g and airy universality results for the squared singular values considered

on block coded modulation with interblock memory block coded modulation with interblock memory bcmim is a variation of block coded modulation bcm which is designed for multilevel coding by providing interblock memory between adjacent blocks the coding rate of a bcmim scheme can be increased without decreasing the minimum squared euclidean distance msed as compared to the original bcm in an early version of bcmim interblock coding is provided only between the first two coding levels of adjacent blocks in this paper we design bcmim with a more general form for which interblock coding can be introduced among many coding levels in this way we can further increase the coding rate of bcmim without decreasing the msed we provide many examples to show the advantages of bcmim with the general form most of the examples are designed based on multidimensional signal sets since a multidimensional signal set can provide more coding levels than a two dimensional 2 d signal set

on multilevel block modulation codes the multilevel technique for combining block coding and modulation is investigated a general formulation is presented for multilevel modulation codes in terms of component codes with appropriate distance measures a specific method for constructing multilevel block modulation codes with interdependency among component codes is proposed given a multilevel block modulation code c with no interdependency among the binary component codes the proposed method gives a multilevel block modulation code c that has the same rate as c a minimum squared euclidean distance not less than that of c a trellis diagram with the same number of states as that of c and a smaller number of nearest neighbor codewords than that of c finally a technique is presented for analyzing the error performance of block modulation codes for an additive white gaussian noise awgn channel based on soft decision maximum likelihood decoding error probabilities of some specific codes are evaluated by simulation and upper bounds based on their euclidean weight distributions

statistical analysis of the product high order ambiguity function the high order ambiguity function haf was introduced for the estimation of polynomial phase signals pps embedded in noise since the haf is a nonlinear operator it suffers from noise masking effects and from the appearance of undesired cross terms and possibly spurious harmonics in the presence of multicomponent mc signals the product haf phaf was then proposed as a way to improve the performance of the haf in the presence of noise and to solve the ambiguity problem in this correspondence we derive a statistical analysis of the phaf in the presence of additive white gaussian noise awgn valid for high signal to noise ratio snr and a finite number of data samples the analysis is carried out in detail for single component pps but the multicomponent case is also discussed error propagation phenomena implicit in the recursive structure of the phaf based estimator are explicitly taken into account the analysis is validated by simulation results for both single and multicomponent ppss

a novel weighted multilevel space time trellis coding scheme it has been shown that when perfect or partial channel state information csi is available at the transmitter the performance and capacity of a space time coded system can be further improved multilevel space time trellis codes mlsttc s are capable of simultaneously providing bandwidth efficiency diversity improvement and coding gain with significantly reduced decoding complexity especially for larger constellations and higher throughputs in this paper we present a design of combined multilevel space time trellis codes and beam forming based on feedback from receiver henceforth referred to as weighted multilevel space time trellis codes wmlsttcs the channel profile is used to provide a beam forming scheme that achieves the better performance by properly weighting transmitted signals weights are selected that matches best with a channel profile feedback from the receiver indicative of long term characteristics of the wireless channel wmlsttcs provide improvement in performance of mlsttcs

universality in two dimensional ising models we have used finite size scaling and transfer matrix techniques to calculate accurately the critical exponents of three two dimensional ising like models for which no exact solution is available two spin 12 models with crossing bonds and a spin 1 model the results for the temperature and magnetic exponents are very close to the exact results for exactly solvable models which are assumed to be in the same universality class differences are between 10 5 and a few times 10 4 and within the apparent numerical uncertainties we also present an estimate of the critical point of the spin 1 model and some preliminary results concerning universal properties of critical amplitudes

fast quantizing and decoding and algorithms for lattice quantizers and codes for each of the lattices a n n geq 1 d n n geq 2 e 6 e 7 e 8 and their duals a very fast algorithm is given for finding the closest lattice point to an arbitrary point if these lattices are used for vector quantizing of uniformly distributed data the algorithm finds the minimum distortion lattice point if the lattices are used as codes for a gaussian channel the algorithm performs maximum likelihood decoding

on the spectrum of distances of a multilevel code decoded by a multistage decoder in this correspondence we are concerned with the error rates contributed by component codes in a multilevel code decoded by a multistage decoder these error rates are required for estimating the performance of the structure but more importantly they can serve as a design tool for achieving high coding gain the latter objective can be reached by approaching a balance among these error rates for evaluating the error rate contributed by a component code it is essential to calculate the corresponding spectrum of euclidean distances a method for computing this spectrum is developed in this correspondence the method is general and applicable to any signal constellation and convolutional as well as block component codes the error coefficient available in the literature for some multilevel codes with an infinite size signal constellation is shown to be significantly different from that of a practical finite constellation note that the complete spectrum is essential when codewords of a component code with hamming distance beyond the minimum distance affect the error rate see e g turbo codes

soft decoding techniques for codes and lattices including the golay code and the leech lattice two kinds of algorithms are considered 1 if is a binary code of length n a soft decision decoding algorithm for changes an arbitrary point of r n into a nearest codeword nearest in euclidean distance 2 similarly a decoding algorithm for a lattice lambda in r n changes an arbitrary point of r n into a closest lattice point some general methods are given for constructing such algorithms ami are used to obtain new and faster decoding algorithms for the gosset lattice e 8 the golay code the leech lattice

the supercritical regime in the normal matrix model with cubic potential abstract the normal matrix model with a cubic potential is ill defined and it develops a critical behavior in finite time we follow the approach of bleher and kuijlaars to reformulate the model in terms of orthogonal polynomials with respect to a hermitian form this reformulation was shown to capture the essential features of the normal matrix model in the subcritical regime namely that the zeros of the polynomials tend to a number of segments the motherbody inside a domain the droplet that attracts the eigenvalues in the normal matrix model in the present paper we analyze the supercritical regime and we find that the large n behavior is described by the evolution of a spectral curve satisfying the boutroux condition the boutroux condition determines a system of contours 1 consisting of the motherbody and whiskers sticking out of the domain we find a second critical behavior at which the original motherbody shrinks to a point at the origin and only the whiskers remain in the regime before the second criticality we also give strong asymptotics of the orthogonal polynomials by means of a steepest descent analysis of a 3 3 matrix valued riemann hilbert problem it follows that the zeros of the orthogonal polynomials tend to 1 with the exception of at most three spurious zeros

diversity embedded space time codes rate and diversity impose a fundamental tradeoff in wireless communication high rate space time codes come at a cost of lower reliability diversity and high reliability diversity implies a lower rate however wireless networks need to support applications with very different quality of service qos requirements and it is natural to ask what characteristics should be built into the physical layer link in order to accommodate them in this paper we design high rate space time codes that have a high diversity code embedded within them this allows a form of communication where the high rate code opportunistically takes advantage of good channel realizations while the embedded high diversity code provides guarantees that at least part of the information is received reliably we provide constructions of linear and nonlinear codes for a fixed transmit alphabet constraint the nonlinear constructions are a natural generalization to wireless channels of multilevel codes developed for the additive white gaussian noise awgn channel that are matched to binary partitions of quadrature amplitude modulation qam and phase shift keying psk constellations the importance of set partitioning to code design for the wireless channel is that it provides a mechanism for translating constraints in the binary domain into lower bounds on diversity protection in the complex domain we investigate the systems implications of embedded diversity codes by examining value to unequal error protection rate opportunism and packet delay optimization these applications demonstrate that diversity embedded codes have the potential to outperform traditional single layer codes in moderate signal to noise snr regimes

critical properties of the ashkin teller model from the mean field renormalisation group approach by using small size clusters which carry all the basic symmetries of the ashkin teller model the authors obtain within the mean field renormalisation group approach the complete phase diagram of the model estimates of the thermal critical exponent are also obtained

graph based iterative decoding algorithms for parity concatenated trellis codes we construct parity concatenated trellis codes in which a trellis code is used as the inner code and a simple parity check code is used as the outer code from the tanner wiberg loeliger 1981 1996 graph representation several iterative decoding algorithms can be derived however since the graph of the parity concatenated code contains many short cycles the conventional min sum and sum product algorithms cannot achieve near optimal decoding after some simple modifications we obtain near optimal iterative decoders the modifications include either a introducing a normalization operation in the min sum and sum product algorithms or b cutting the short cycles which arise in the iterative viterbi algorithm iva after modification all three algorithms can achieve near optimal performance but the iva has the least average complexity we also show that asymptotically maximum likelihood ml decoding and a posteriori probability app decoding can be achieved using iterative decoders with only two iterations unfortunately this asymptotic behavior is only exhibited when the bit energy to noise ratio is above the cutoff rate simulation results show that with trellis shaping iterative decoding can perform within 1 2 db of the shannon limit at a bit error rate ber of 4 spl times 10 sup 5 for a block size of 20000 symbols for a block size of 200 symbols iterative decoding can perform within 2 1 db of the shannon limit

renormalization group study of the two dimensional ising model with crossing bonds abstract the simple quadratic ising model with nearest and next nearest neighbor interactions is investigated by means of the mean field renormalization group approach by conveniently choosing the size of the blocks one obtains the expected phase diagram exhibiting ferromagnetic antiferromagnetic paramagnetic and superantiferromagnetic phases the critical lines which separate the paramagnetic phase from the ordered ones have the expected asymptotic behavior for large values of the nearest neighbor coupling

the leech lattice the octacode and decoding algorithms new multilevel constructions of the golay code and the leech lattice are presented these are derived from the turyn construction and the holy construction with the octacode as the glue code further we show that the holy construction of the leech lattice with the octacode as the glue code is essentially different from the permuted turyn construction although both constructions rely on the octacode the turyn construction is based on an odd type of the octacode whereas any type of the octacode can be used in the holy construction moreover the multilevel representation of the holy construction leads to a novel lattice partition chain based on these structures we derive new bounded distance decoders for the golay code and the leech lattice whose effective error coefficient is smaller than that of any previously known bounded distance decoder we provide a general theorem for computing the effective error coefficient of coset decoding with bounded distance decoding for the subcode

large n limit of gaussian random matrices with external source part iii double scaling limit we consider the double scaling limit in the random matrix ensemble with an external source 1 over z n e n hbox tr 1 over 2 m 2 am dm defined on n n hermitian matrices where a is a diagonal matrix with two eigenvalues a of equal multiplicities the value a 1 is critical since the eigenvalues of m accumulate as n on two intervals for a 1 and on one interval for 0 a 1 these two cases were treated in parts i and ii where we showed that the local eigenvalue correlations have the universal limiting behavior known from unitary random matrix ensembles for the critical case a 1 new limiting behavior occurs which is described in terms of pearcey integrals as shown by brezin and hikami and tracy and widom we establish this result by applying the deift zhou steepest descent method to a 3 3 matrix valued riemann hilbert problem which involves the construction of a local parametrix out of pearcey integrals we resolve the main technical issue of matching the local pearcey parametrix with a global outside parametrix by modifying an underlying riemann surface

encoding and decoding binary product lattices a binary product lattice is generated from two binary component lattices of lower dimensions by employing the kronecker product this work focuses on codes carved from binary product lattices defined as such an intriguing problem is that of effectively mapping independent data sequences onto a selected subset of lattice points a novel approach is disclosed yielding an explicit connection between source bits and lattice points decoding methods typically used for binary product codes do not apply for product lattices several alternative decoding approaches are discussed in particular a provably bounded distance decoder is presented it relies on the fact that a product lattice code point may be regarded as a two dimensional array whose rows and columns are points in the component lattices the obtained results are compared with classical lattices known in the art

on the performance evaluation of trellis codes generating function techniques for analyzing the error event and the bit error probabilities for trellis codes are considered the conventional state diagram approach for linear codes where the number of states is equal to the number of trellis states cannot be applied directly to arbitrary trellis codes and instead a state diagram where the number of states is equal to the square of the number of trellis states must be used it is shown that for an interesting class of trellis codes a modified generating function can be defined for which the number of states em is equal to the number of trellis states the class of codes considered includes trellis codes of rate r n 1 n based upon set partitioning whenever the first partition breaks the signal constellation into two subsets which have the same configuration matrix i e the same ordered set of mutual distances the complexity of calculating this modified generating function is the same as for the ordinary generating function of a convolutional code with the same number of trellis states bounds on the performance of some interesting codes are given based upon this method

on the reliability order based decoding algorithms for binary linear block codes in this correspondence we consider the decoding of binary block codes over the additive white gaussian noise awgn channel with binary phase shift keying bpsk signaling by a reliability order based decoding algorithm robda we mean a soft decision decoding algorithm which decodes to the best most likely codeword of the form that is the sum of the hard decision tuple and an error pattern in a set determined only by the order of the reliabilities of the hard decisions examples of robdas include many well known decoding algorithms such as the generalized minimum distance gmd decoding algorithm chase decoding algorithms and the reliability based decoding algorithms proposed by fossorier and lin it is known that the squared error correction radii of robdas can be computed from the minimal squared euclidean distances mseds between the all one sequence and the polyhedra corresponding to the error patterns for the computation of such mseds we give a new method which is more compact than the one proposed by fossorier and lin these results are further used to show that any bounded distance robda is asymptotically optimal the ratio between the probability of decoding error of a bounded distance robda and that of the maximum likelihood ml decoding approaches 1 when the signal to noise ratio snr approaches infinity provided that the minimum hamming distance of the code is greater than 2

practical encoders and decoders for euclidean codes from barnes wall lattices we address the application of barnes wall bw lattice codes for communication over additive white gaussian noise awgn channels we introduce construction a prime of complex bw lattices that makes new connection between linear codes over polynomial rings and lattices we show that construction a prime of bw lattices is equivalent to the multilevel construction from reed muller codes proposed by forney to decode the bw lattice code we adapt the low complexity sequential bw lattice decoder sbwd proposed by micciancio and nicolosi first we study the error performance of sbwd for decoding the infinite lattice and demonstrate that it is powerful in making correct decisions well beyond the packing radius subsequently we use the sbwd to decode lattice codes through a novel noise trimming technique where the received vector is appropriately scaled before applying the sbwd we show that the noise trimming technique is most effective for decoding bw lattice codes in smaller dimensions while the gain diminishes for decoding codes in larger dimensions

large n limit of gaussian random matrices with external source part i we continue the study of the hermitian random matrix ensemble with external source open image in new window where a has two distinct eigenvalues a of equal multiplicity this model exhibits a phase transition for the value a 1 since the eigenvalues of m accumulate on two intervals for a 1 and on one interval for 0 1 was treated in part i where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices that is limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum and in terms of the airy kernel at the edge in this paper we establish the same results for the case 0 a 1 as in part i we apply the deift zhou steepest descent analysis to a 3 3 matrix riemann hilbert problem due to the different structure of an underlying riemann surface the analysis includes an additional step involving a global opening of lenses which is a new phenomenon in the steepest descent analysis of riemann hilbert problems

dropping the independence singular values for products of two coupled random matrices we study the singular values of the product of two coupled rectangular random matrices as a determinantal point process each of the two factors is given by a parameter dependent linear combination of two independent complex gaussian random matrices which is equivalent to a coupling of the two factors via an itzykson zuber term we prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability the parameter dependence allows us to interpolate between the singular value statistics of the laguerre ensemble and that of the product of two independent complex ginibre ensembles which are both known we give exact formulae for the correlation kernel in terms of a complex double contour integral suitable for the subsequent asymptotic analysis in particular we derive a christoffel darboux type formula for the correlation kernel based on a five term recurrence relation for our biorthogonal functions it enables us to find its scaling limit at the origin representing a hard edge the resulting limiting kernel coincides with the universal meijer g kernel found by several authors in different ensembles we show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly

generalized minimum distance decoding of euclidean space codes and lattices it is shown that multistage generalized minimum distance gmd decoding of euclidean space codes and lattices can provide an excellent tradeoff between performance and complexity we introduce a reliability metric for gaussian channels that is easily computed from an inner product and prove that a multistage gmd decoder using this metric is a bounded distance decoder up to the true packing radius the effective error coefficient of multistage gmd decoding is determined two simple modifications in the gmd decoding algorithm that drastically reduce this error coefficient are proposed it is shown that with these modifications gmd decoding achieves the error coefficient of maximum likelihood decoding for block codes and for generalized construction a lattices multistage gmd decoding of the lattices d sub 4 e sub 8 k sub 12 bw sub 16 and spl lambda sub 24 is investigated in detail for k sub 12 bw sub 16 and spl lambda sub 24 the gmd decoders have considerably lower complexity than the best known maximum likelihood or bounded distance decoding algorithms and appear to be the most practically attractive decoders available for high dimensional codes and lattices spl ges 64 dimensions maximum likelihood decoding becomes infeasible while gmd decoding algorithms remain quite practical as an example we devise a multistage gmd decoder for a 128 dimensional sphere packing with a nominal coding gain of 8 98 db that attains an effective error coefficient of 1365760 this decoder requires only about 400 real operations in addition to algebraic errors and erasures decoding of certain bch and hamming codes it therefore appears to be practically feasible to implement algebraic multistage gmd decoders for high dimensional sphere packings and thus achieve high effective coding gains

lattice codes and generalized minimum distance decoding for ofdm systems lattice coding of orthogonal frequency division multiplexing ofdm systems is considered mapping of multilevel construction lattices to ofdm blocks is shown and a methodology for probabilistic analysis of multistage generalized minimum distance gmd decoding of the received ofdm blocks is shown as a case study transmission of points from a 128 dimensional barnes wall lattice is considered tight approximations to the system error rate are obtained and verified by simulation it appears that gmd decoding of lattice encoded ofdm provides high coding gain at low complexity

multilevel codes and multistage decoding h imai and s hirakawa have proposed 1977 a multilevel coding method based on binary block codes that admits a staged decoding procedure the author extends the coding method to coset codes and shows how to calculate minimum squared distance and path multiplicity in terms of the norms and multiplicities of the different cosets the multilevel structure allows the redundancy in the coset selection procedure to be allocated efficiently among the different levels it also allows the use of suboptimal multistage decoding procedures that have performance complexity advantages over maximum likelihood decoding

eigenvalues and singular values of products of rectangular gaussian random matrices we derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions we show that they both have power law behavior at zero and determine the corresponding powers we also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions

reduced lists of error patterns for maximum likelihood soft decoding a method whereby a substantially reduced family of error patterns called survivors may be created for maximum likelihood soft decoding is introduced the family of survivor depends on the received word a decoder based on this approach first forms the survivors then scores them rather than obtaining the survivors by online elimination of error patterns the use of predetermined lists that represent all families of survivors is proposed

renormalisation group recursions by mean field approximations within a renormalisation group strategy different rescaled mean field approximations for the magnetisation are combined good qualitative estimates are obtained for the critical properties of classical and quantum spin systems a particularly simple realisation of the method yields recursions for arbitrary dimensionality d the critical couplings are accurate for all d and have the correct asymptotic behaviour for d approaching infinity

products of rectangular random matrices singular values and progressive scattering we discuss the product of m rectangular random matrices with independent gaussian entries which have several applications including wireless telecommunication and econophysics for complex matrices an explicit expression for the joint probability density function is obtained using the harish chandra itzykson zuber integration formula explicit expressions for all correlation functions and moments for finite matrix sizes are obtained using a two matrix model and the method of biorthogonal polynomials this generalizes the classical result for the so called wishart laguerre gaussian unitary ensemble or chiral unitary ensemble at m 1 and previous results for the product of square matrices the correlation functions are given by a determinantal point process where the kernel can be expressed in terms of meijer g functions we compare the results with numerical simulations and known results for the macroscopic level density in the limit of large matrices the location of the end points of support for the latter are analyzed in detail for general m finally we consider the so called ergodic mutual information which gives an upper bound for the spectral efficiency of a mimo communication channel with multifold scattering

multilevel codes based on partitioning following v v ginzburg 1984 a hierarchy of codes is proposed to match the geometric partitioning of a signal set the authors show that coset codes including ungerboeck lattice and binary codes and indeed any codes which rely on partitioning of the signal set are all subclasses of the proposed coding scheme the combination of such codes in a multilevel scheme often leads to reduced complexity in comparison with previously published schemes a variety of decoder structures is discussed

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

differential systems for biorthogonal polynomials appearing in 2 matrix models and the associated riemann hilbert problem we consider biorthogonal polynomials that arise in the study of a generalization of two matrix hermitian models with two polynomial potentials v1 x v2 y of any degree with arbitrary complex coefficients finite consecutive subsequences of biorthogonal polynomials windows of lengths equal to the degrees of the potentials v1 and v2 satisfy systems of ode s with polynomial coefficients as well as pde s deformation equations with respect to the coefficients of the potentials and recursion relations connecting consecutive windows a compatible sequence of fundamental systems of solutions is constructed for these equations the stokes sectorial asymptotics of these fundamental systems are derived through saddle point integration and the riemann hilbert problem characterizing the differential equations is deduced

a class of optimum block codes in signal space by using convolutional coding coupled with expanded sets of channel signals ungerboeck showed that it is possible to obtain coding gains of several decibels at a given data rate without increasing the transmission bandwidth in this paper we construct a class of optimum signal space codes using short binary block as opposed to convolutional codes our results are compared to those obtained by ungerboeck a simplified decoding procedure that is asymptotically optimum is also presented

efficient bounded distance decoding of the hexacode and associated decoders for the leech lattice and the golay code two soft decision decoding algorithms for the 6 3 4 quaternary code hexacode are presented both algorithms realize half the minimum euclidean distance of the code the proposed algorithms are most practical in using them bounded distance decoding of the golay code and the leech lattice are performed with at most 187 and 519 real number operations respectively compare this to 651 respectively 3595 operations required by the best known maximum likelihood decoders vardy and be ery 1991 1993 and 431 respectively 1007 operations required by the bounded distance decoders amrani et al 1994 we present some simulation results for the proposed leech lattice decoders revealing near optimal performance a comparison to known trellis codes is also provided

sphere bound achieving coset codes and multilevel coset codes a simple sphere bound gives the best possible tradeoff between the volume per point of an infinite array l and its error probability on an additive white gaussian noise awgn channel it is shown that the sphere bound can be approached by a large class of coset codes or multilevel coset codes with multistage decoding including certain binary lattices these codes have structure of the kind that has been found to be useful in practice capacity curves and design guidance for practical codes are given exponential error bounds for coset codes are developed generalizing poltyrev s 1994 bounds for lattices these results are based on the channel coding theorems of information theory rather than the minkowski hlawka theorem of lattice theory

constrained designs for maximum likelihood soft decoding of rm 2 m and the extended golay codes we present algorithms for maximum likelihood soft decoding of the second order reed muller codes rm 2 m and the extended 24 12 8 golay code the decoding procedures are based on a concise representation of appropriately selected cosets of a subcode of the code considered such representation enables application of certain elimination rules a structure called constrained design is established to support the elimination procedures remarkably efficient algorithms are obtainable by this approach the method of elimination is applicable also in combination with existing coset decoding schemes

concatenated multilevel block coded modulation encoding and decoding schemes for concatenated multilevel block codes are presented by one of these structures a real coding gain of 5 6 7 4 db for the bit error range of 10 sup 6 to 10 sup 9 is achieved for transmission through the additive white gaussian noise channel also a rather large asymptotic coding gain is obtained the new coding schemes have very low decoding complexity and increased coding gain in comparison with the conventional block and trellis coded modulation structures a few design rules for concatenated single and multilevel block codes with large coding gain are also provided

the cutoff rate for on off keying it is well known that on off keying ook i e the transmission of a pulse for a data one mark and transmission of no signal for a data zero space exhibits a 3 db inferiority to antipodal signalling in terms of the signal to noise ratio required to achieve a specified bit error probability when the channel is characterized by the cutoff rate r sub 0 this 3 db inferiority persists in the sense that the required e sub b n sub 0 to operate at a given value of r sub 0 is 3 db greater for the ook channel than for the antipodal channel however it is shown that if the definition of cutoff rate is generalized in an appropriate way to reflect the fact that signal energy is expended only in the transmission of marks part of this penalty is recovered in fact it is shown that in the limit of low r sub 0 there is no penalty in signal to noise ratio even at rates of realistic interest the penalty is significantly less than the expected 3 db

trellis coded modulation with multidimensional constellations trellis coded modulation schemes using four eight or 16 dimensional constellations have a number of potential advantages over the usual two dimensional schemes a smaller constituent two dimensional constellation easier tolerance to phase ambiguities and a better trade off between complexity and coding gain a number of such schemes are presented and evaluated starting with a variety of multidimensional lattices we show how to select multidimensional constellations how to partition them into subsets how to construct trellis codes using those subsets and how to map hits to constellation points simplifications of the viterbi decoding algorithm are presented we conclude that there are multidimensional trellis coded modulation schemes that perform better for the same complexity than do two dimensional schemes

non intersecting squared bessel paths and multiple orthogonal polynomials for modified bessel weights we study a model of n non intersecting squared bessel processes in the confluent case all paths start at time t 0 at the same positive value x a remain positive and are conditioned to end at time t t at x 0 in the limit n after appropriate rescaling the paths fill out a region in the tx plane that we describe explicitly in particular the paths initially stay away from the hard edge at x 0 but at a certain critical time t the smallest paths hit the hard edge and from then on are stuck to it for t t we obtain the usual scaling limits from random matrix theory namely the sine airy and bessel kernels a key fact is that the positions of the paths at any time t constitute a multiple orthogonal polynomial ensemble corresponding to a system of two modified bessel type weights as a consequence there is a 3 3 matrix valued riemann hilbert problem characterizing this model that we analyze in the large n limit using the deift zhou steepest descent method there are some novel ingredients in the riemann hilbert analysis that are of independent interest

some exact results from the mean field renormalization group abstract the mean field renormalization group is applied to the study of the directed self avoiding random walk in d dimensions and to the spin ferromagnetic ising model on the bethe lattice in both cases the known exact results are obtained

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

eigenvalue statistics for product complex wishart matrices the eigenvalue statistics for complex wishart matrices where is equal to the product of r complex gaussian matrices and the inverse of s complex gaussian matrices are considered in the case r s the exact form of the global density is computed the averaged characteristic polynomial for the corresponding generalized eigenvalue problem is calculated in terms of a particular generalized hypergeometric function for finite n the eigenvalue probability density function is computed and is shown to be an example of a biorthogonal ensemble a double contour integral form of the corresponding correlation kernel is derived which allows the hard edge scaled limit to be computed the limiting kernel is given in terms of certain meijer g functions and is identical to that found in the recent work of kuijlaars and zhang in the case s 0 properties of the kernel and corresponding correlation functions are discussed

universality conjecture and results for a model of several coupled positive definite matrices the paper contains two main parts in the first part we analyze the general case of p geq 2 matrices coupled in a chain subject to cauchy interaction similarly to the itzykson zuber interaction model the eigenvalues of the cauchy chain form a multi level determinantal point process we first compute all correlations functions in terms of cauchy biorthogonal polynomials and locate them as specific entries of a p 1 times p 1 matrix valued solution of a riemann hilbert problem in the second part we fix the external potentials as classical laguerre weights we then derive strong asymptotics for the cauchy biorthogonal polynomials when the support of the equilibrium measures contains the origin as a result we obtain a new family of universality classes for multi level random determinantal point fields which include the bessel nu universality for 1 level and the meijer g universality for 2 level our analysis uses the deift zhou nonlinear steepest descent method and the explicit construction of a p 1 times p 1 origin parametrix in terms of meijer g functions the solution of the full riemann hilbert problem is derived rigorously only for p 3 but the general framework of the proof can be extended to the cauchy chain of arbitrary length p

cauchy laguerre two matrix model and the meijer g random point field we apply the general theory of cauchy biorthogonal polynomials developed in bertola et al commun math phys 287 3 983 1014 2009 and bertola et al j approx th 162 4 832 867 2010 to the case associated with laguerre measures in particular we obtain explicit formulae in terms of meijer g functions for all key objects relevant to the study of the corresponding biorthogonal polynomials and the cauchy two matrix model associated with them the central theorem we prove is that a scaling limit of the correlation functions for eigenvalues near the origin exists and is given by a new determinantal two level random point field the meijer g random field we conjecture that this random point field leads to a novel universality class of random fields parametrized by exponents of laguerre weights we express the joint distributions of the smallest eigenvalues in terms of suitable fredholm determinants and evaluate them numerically we also show that in a suitable limit the meijer g random field converges to the bessel random field and hence the behavior of the eigenvalues of one of the two matrices converges to the one of the laguerre ensemble

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

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

singular value correlation functions for products of wishart random matrices we consider the product of m quadratic random matrices with complex elements and no further symmetry where all matrix elements of each factor have a gaussian distribution this generalizes the classical wishart laguerre gaussian unitary ensemble with m 1 in this paper we first compute the joint probability distribution for the singular values of the product matrix when the matrix size n and the number m are fixed but arbitrary this leads to a determinantal point process which can be realized in two different ways first it can be written as a one matrix singular value model with a nonstandard jacobian or second for m 2 as a two matrix singular value model with a set of auxiliary singular values and a weight proportional to the meijer g function for both formulations we determine all singular value correlation functions in terms of the kernels of biorthogonal polynomials which we explicitly construct they are given in terms of the hypergeometric and meijer g functions generalizing the laguerre polynomials for m 1 our investigation was motivated from applications in telecommunication of multi layered scattering multiple input and multiple output channels we present the ergodic mutual information for finite n for such a channel model with m 1 layers of scatterers as an example

a bounded distance decoding algorithm for the leech lattice with generalizations an algorithm is given that decodes the leech lattice with not much more than twice the complexity of soft decision decoding of the golay code the algorithm has the same effective minimum distance as maximum likelihood decoding and increases the effective error coefficient by less than a factor or two the algorithm can be recognized as a member of the class of multistage algorithms that are applicable to hierarchical constructions it is readily generalized to lattices that can be expressed in terms of binary code formulas and in particular to construction b lattices

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

the leech lattice and the golay code bounded distance decoding and multilevel constructions multilevel constructions of the binary golay code and the leech lattice are described both constructions are based upon the projection of the golay code and the leech lattice onto the 6 3 4 hexacode over gf 4 however unlike the previously reported constructions the new multilevel constructions make the three levels independent by way of using a different set of coset representatives for one of the quaternary coordinates based upon the multilevel structure of the golay code and the leech lattice efficient bounded distance decoding algorithms are devised the bounded distance decoder for the binary golay code requires at most 431 operations as compared to 651 operations for the best known maximum likelihood decoder efficient bounded distance decoding of the leech lattice is achieved by means of partitioning it into four cosets of q sub 24 beyond the conventional partition into two h sub 24 cosets the complexity of the resulting decoder is only 953 real operations on the average and 1007 operations in the worst case as compared to about 3600 operations for the best known in maximum likelihood decoder it is shown that the proposed algorithms decode correctly at least up to the guaranteed error correction radius of the maximum likelihood decoder thus the loss in coding gain is due primarily to an increase in the effective error coefficient which is calculated exactly for both algorithms furthermore the performance of the leech lattice decoder on the awgn channel is evaluated experimentally by means of a comprehensive computer simulation the results show a loss in coding gain of less than 0 1 db relative to the maximum likelihood decoder for ber ranging from 10 sup 1 to 10 sup 7

maximum likelihood soft decoding of binary block codes and decoders for the golay codes maximum likelihood soft decision decoding of linear block codes is addressed a binary multiple check generalization of the wagner rule is presented and two methods for its implementation one of which resembles the suboptimal forney chase algorithms are described besides efficient soft decoding of small codes the generalized rule enables utilization of subspaces of a wide variety thereby yielding maximum likelihood decoders with substantially reduced computational complexity for some larger binary codes more sophisticated choice and exploitation of the structure of both a subspace and the coset representatives are demonstrated for the 24 12 golay code yielding a computational gain factor of about 2 with respect to previous methods a ternary single check version of the wagner rule is applied for efficient soft decoding of the 12 6 ternary golay code

performance evaluation of coded modulation schemes based on binary lattices we describe computational techniques for the evaluation of error probability of modulation schemes based on binary lattices and multilevel coding these techniques stem from the representation of a binary lattice in the form of a trellis whose structure depends on the block codes on which the binary lattice is based multilevel coded modulation schemes can also be described through a trellis by combining with this the trellis structure of the lattice constellation we obtain a single trellis that can be used for maximum likelihood decoding using the viterbi algorithm and for computing the error probability an upper bound to error probability can be obtained from the transfer function of the trellis with suitable labels and is applicable to both maximum likelihood and staged decoding

a code decomposition approach for decoding cyclic and algebraic geometric codes an approach based on code decomposition and partial transform for constructing algorithms for decoding cyclic codes and algebraic geometric ag codes is introduced a general decoding procedure applicable to arbitrary linear codes and their subfield subcodes is then developed in particular we have developed algorithms for error and erasure decoding of cyclic codes up to their actual minimum distance error and erasure decoding of ag codes up to their designed minimum distance and decoding of subfield subcodes of ag codes geometric bch codes moreover a generalization of geometric bch codes is introduced it is shown by an example that for a simple class of ag codes the so called one point codes this generalization brings a better estimate of their minimum distance it is also shown that the algorithms developed in this paper can be applied to decode these codes up to their estimated minimum distance

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

multilevel coded modulation for unequal error protection and multistage decoding ii asymmetric constellations in this paper theoretical upper bounds and computer simulation results on the error performance of multilevel block coded modulations for unequal error protection uep and multistage decoding are presented it is shown that nonstandard signal set partitionings and multistage decoding provide excellent uep capabilities beyond those achievable with conventional coded modulation the coding scheme is designed in such a way that the most important information bits have a lower error rate than other information bits the large effective error coefficients normally associated with standard mapping by set partitioning are reduced by considering nonstandard partitionings of the underlying signal set the bits to signal mappings induced by these partitionings allow the use of soft decision decoding of binary block codes moreover parallel operation of some of the staged decoders is possible to achieve high data rate transmission so that there is no error propagation between these decoders hybrid partitionings are also considered that trade off increased intraset distances in the last partition levels with larger effective error coefficients in the middle partition levels the error performance of specific examples of multilevel codes over 8 psk and 64 qam signal sets are simulated and compared with theoretical upper bounds on the error performance

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

the error probability of m ary psk block coded modulation schemes a tight upper bound on the decoding error probability is derived for block coded modulation structures where an m ary phase shift keying m psk signal constellation is employed this bound called a tangential sphere bound is tight for very low as well as for high signal to noise ratios snrs berlekamp s tangential union bound previously derived for binary codes can be derived for an m psk block coded modulation structure as well however it is proven that our tangential sphere bound is tighter than berlekamp s 1980 tangential bound for particular schemes it is shown that for low snrs our bound is considerably tighter than the tangential bound as one of the examples a multistage decoder is considered

some ternary and quaternary codes and associated sphere packings tables of good ternary and quaternary codes are presented and they are used in the construction of dense sphere packings results include 1 tables of the best ternary and quaternary constacyclic codes including cyclic codes up to block length 50 2 a class of optimal n 2 codes over gf q 3 the u v w mod 2u v mod u construction a new ternary code construction technique that can be used to construct the ternary reed muller codes and others and 4 tables of linear ternary and quaternary codes obtained by modifying and combining various codes here and in the literature packings are generated in even dimensions up to 100 using these codes and a sphere packing construction in dimensions 36 and 60 new record densities appear to have been achieved

the simple quadratic ising model with crossing bonds abstract we investigate the ferromagnetic critical line of the simple quadratic ising model with nearest neighbour and next nearest neighbour interactions by means of monte carlo renormalisation group and analytic methods a comparison of the monte carlo results with the renormalisation group trajectories yields the asymptotic behaviour of the critical line in the strong coupling limit a number of figures of spin configurations is included to illustrate the physical behaviour of the model

multilevel coded cpfsk systems for awgn and fading channels in this paper we introduce multilevel coded continuous phase frequency shift keying cpfsk systems for both additive white gaussian noise and fading channels based on multilevel coding and multistage decoding techniques these schemes are designed under the constraint m spl ges 2p for m ary cpfsk with modulation index j p in order to maintain the phase continuity property after multilevel coding we use some specific set partitioning rules we construct examples to show that the proposed systems outperform the corresponding conventional one level schemes for the fading case we consider an ideal fading channel where adjacent fading symbols are assumed to be independent and a correlated fading channel where bit interleavers are needed at each coding level the computer simulation results show that the proposed systems have bit error performance and decoder complexity advantages over the corresponding reference codes taken from the literature it is also shown that there is only a slight degradation on the bit error performance for correlated fading compared to the ideal fading case

coded phase frequency modulation channel coding combined with expanded signal sets has been shown 1 to improve error performance over uncoded modulation without expanding the bandwidth of the transmitted signals in this paper new coded modulation formats defined over an expanded set of signals varying both in phase and frequency are presented the new schemes combine fsk and psk modulation and make use of trellis coding and viterbi decoding to improve error performance over uncoded modulation the free euclidean distance is calculated for several classes of codes and upper bounds and simulation results are also presented for some simple codes the spectral characteristics of the proposed coded modulation formats are evaluated and compared to conventional two dimensional modulation formats differential encoding and various extensions of the basic scheme are also discussed

two level coding based on trellis coded modulation and reed solomon codes a number of two level codes based on trellis coded modulation tcm schemes and their matched single or double error correcting reed solomon codes using qam constellations are presented and evaluated compared to the well known 4d 16 state tcm scheme with a comparable bandwidth efficiency these two level codes provide greater immunity from both gaussian and impulse noise at the cost of a reasonably higher complexity and a longer decoding delay on the order of a few hundred symbols the authors also show how to cope with phase ambiguities of the constellation in these two level codes both rotationally invariant coding and rotation detecting techniques are considered

universality in the two matrix model a riemann hilbert steepest descent analysis the eigenvalue statistics of a pair m 1 m 2 of n x n hermitian matrices taken randomly with respect to the measure 1 z n exp n tr v m 1 w m 2 tau m 1 m 2 dm 1 dm 2 can be descri

maximum likelihood decoding of the leech lattice an algorithm for maximum likelihood decoding of the leech lattice is presented the algorithm involves projecting the points of the leech lattice directly onto the codewords of the 6 3 4 quaternary code the hexacode projection on the hexacode induces a partition of the leech lattice into four cosets of a certain sublattice 24 such a partition into cosets enables maximum likelihood decoding of the leech lattice with 3595 real operations in the worst case and only 2955 operations on the average this is about half the worst case and the average complexity of the best previously known algorithm

a bounded distance decoding algorithm for binary linear block codes achieving the minimum effective error coefficient a new bounded distance bd decoding algorithm is presented for binary linear n k d block codes on additive white gaussian noise channels the algorithm is based on the generalized minimum distance gmd decoding algorithm of forney 1989 using the acceptance criterion of taipale and pursley gmd tp proposed in 1991 it is shown that the gmd tp decoding algorithm is a bd decoding algorithm with effective error coefficient sup n sub d it is also shown that the decision regions of gmd tp are good inner approximations of those of full gmd decoding and therefore full gmd decoding is bd and has an effective error coefficient that is well approximated by sup n sub d moreover by adding a d erasure correction step to gmd decoding the effective error coefficient can be reduced to a sub d the number of minimum weight codewords which is the same as the effective error coefficient of maximum likelihood decoding the decoding algorithm is mainly based on algebraic errors and erasures decoding and therefore has polynomial rather than exponential complexity

an approach to ungerboeck coding for rectangular signal sets corresp a method of finding good ungerboeck codes for large rectangular quadrature amplitude modulation qam signal sets is described using the concept of euclidean weights due to ungerboeck we prove that a 2 n point basic constellation may be employed to determine exactly the free distance for an ungerboeck coded rectangular 2 m point set when m n 1 bits are uncoded and the remaining bits pass through a rate n 1 n convolutionai encoder it is shown that rate 2 3 encoders may be used to achieve most of the theoretically possible coding gain in the proposed scheme where the effect of the error coefficient on the coding gain has been considered

trellis codes for the optical direct detection channel it is observed that the reduced complexity minimum distance search method of zehavi and wolf 1987 can be extended from codes designed using the squared euclidean distance measure to any measure which is additive over the code branches c n georghiades 1989 has derived such a metric for codes for the direct detection optical channel the search method is applied to find codes for four and eight point overlapping pulse position modulation oppm with large minimum distance and a small number of codewords at that distance

block coded psk modulation using two level group codes over dihedral groups a length n group code over a group g is a subgroup of g sup n under component wise group operation group codes over dihedral groups d sub m with 2m elements that are two level constructible using a binary code and a code over z sub m residue class integer ring modulo m as component codes are studied for arbitrary m a set of necessary and sufficient conditions on the component codes for the two level construction to result in a group code over d sub m are obtained the conditions differ for m odd and even using two level group codes over d sub m as label codes the performance of a block coded modulation scheme is discussed under all possible matched labelings of 2m apsk and 2m spsk asymmetric and symmetric psk signal sets in terms of the minimum squared euclidean distance matched labelings that lead to automorphic euclidean distance equivalent codes are identified it is shown that depending upon the ratio of hamming distances of the component codes some labelings perform better than others the best labeling is identified under a set of restrictive conditions finally conditions on the component codes for phase rotational invariance properties of the signal space codes are discussed

semiclassical orthogonal polynomials matrix models and isomonodromic tau functions the differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived as well as the compatible systems of deformation equations obtained from varying such measures these are shown to preserve the generalized monodromy of the associated rank 2 rational covariant derivative operators the corresponding matrix models consisting of unitarily diagonalizable matrices with spectra supported on these contours are analyzed and it is shown that all coefficients of the associated spectral curves are given by logarithmic derivatives of the partition function or more generally the gap probabilities the associated isomonodromic tau functions are shown to coincide within an explicitly computed factor with these partition functions