value distribution of meromorphic solutions of certain difference painleve equations abstract the borel exceptional value and the exponents of convergence of poles zeros and fixed points of finite order transcendental meromorphic solutions for difference painleve i and ii equations are estimated and the forms of rational solutions of the difference painleve ii equation and the autonomous difference painleve i equation are also given it is also proved that the non autonomous difference painleve i equation has no rational solution

nearest neighbor algorithm for spherical codes from the leech lattice the leech lattice is a regular arrangement of points in 24 dimensional euclidean space that yields an extremely dense packing when equal spheres are centered at these points a subset of the leech lattice can be used as a signal set for the gaussian channel or as representative vectors for a vector quantizer of particular interest are the spherical codes or code books that consist of the points of the leech lattice which lie on a sphere centered at the origin the code points do not have to be stored because they can be obtained from a very small set of basic vectors using permutations of the components in a manner dictated by the words of the extended golay code a nearest neighbor algorithm that works on this is developed to determine the point in the code closet to some arbitrary vector in r sup 24 the performance of this approach when quantizing independent identically distributed gaussian samples is reported

growth and zeros of meromorphic solution of some linear difference equations abstract in this paper we study growth and zeros of linear difference equations p n z f z n p 1 z f z 1 p 0 z f z f z where f z p n z p 0 z are polynomials with f p n p 0 0 and satisfy deg p n p 0 max deg p j j 0 n 1 the corresponding homogeneous equation of the above equation is also investigated

growth and integrability in the dynamics of mappings the growth of some numerical characteristics of the mappings under their iterations in the context of the general problem of integrability is discussed in the general case such characteristics as complexity by arnold or the number of the different images for the multiple valued mappings are growing exponentially it is shown that the integrability is closely related with thepolynomial growth the analogies with quantum integrable systems are discussed

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

relationships between characteristic functions of δnf and f abstract in this paper we obtain the accurate relationships between the characteristic functions of a finite order meromorphic function f z and its forward differences n f z where f z has two borel exceptional values

on the error performance of the a_ n lattices we consider the root lattice a n and derive explicit recursive formulas for the moments of its voronoi cell these formulas enable accurate prediction of the error probability of lattice codes constructed from a n

on properties of meromorphic solutions for difference equations concerning gamma function abstract in this paper we mainly consider the properties of differences of meromorphic solutions for the difference equation a 1 z f z 1 a 0 z f z 0 concerning a gamma function where a 1 z and a 0 z are nonzero polynomials by these properties we deduce that a gamma function satisfies that for every n n n z z 0 n z has only n zeros z z z j z 1 j 0 1 n 1 z 1 z 1 n 1 z and 1 z have same zeros at most except n exceptional zeros where g denotes the order of growth of a meromorphic function g and g and g denote the exponents of convergence of fixed points and zeros of g respectively

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 probability density of intersymbol interference for trellis coded modulation knowledge of the probability density function pdf of intersymbol interference isi can be used in the computation of system error probabilities and to help identify the cause of the errors exact computation of the probability density is only practicable in a few special cases several algorithms exist for computing approximate pdfs of isi but these only work with certain signal constellations and transmitted symbols that are mutually independent or coded by partial response this paper considers an algorithm for the computation of isi pdfs for trellis coded modulation with general 1d and 2d constellations the results suggest that symbol dependencies introduced into the transmitted signal by symmetric trellis codes have negligible effect on the pdf of the isi although the ideas presented are general in nature examples are given for isi pdfs on a line of sight radio channel

bilinear trilinear forms and exact solution of certain fourth order integrable difference equations a systematic investigation of finding bilinear or trilinear representations of fourth order autonomous ordinary difference equation x n 4 f x n x n 1 x n 2 x n 3 or xn 4 f xn xn 1 xn 2 xn 3 is made as an illustration we consider fourth order symplectic integrable difference equations reported by capel and sahadevan physica a 289 86 2001 and derived their bilinear or trilinear forms also it is shown that the obtained bilinear representations admit exact solution of rational form

decoding the golay codes we introduce exceptionally simple decoding algorithms for the two extended golay codes the algorithms are based on recent methods of conway and curtis of finding the unique blocks containing five points in either the 5 8 24 steiner system or the 5 6 12 steiner system these decoding methods are simple enough to enable decoding extended golay codes by hand both of the methods involve relations between the extended golay codes and other self dual codes proofs are given explaining these relationships and why the decoding methods work the decoding algorithms are explained and illustrated with many examples 3 chap 12 has facts about the mathieu group and some details about decoding the golay codes

tables of sphere packings and spherical codes the theta function of a sphere packing gives the number of centers at each distance from the origin the theta functions of a number of important packings a n d n e n the leech lattice and others and tables of the first fifty or so of their coefficients are given in this paper

the second main theorem for small functions and related problems we shall establish the following three results in more general forms 1 the second main theorem for small functions let f be a meromorphic function on the complex plane c let a1 aq be distinct meromorphic functions on c assume that ai are small with respect to f i e t r a 0 corollary 1 here as usual in nevanlinna theory the terms t r f and n r ai f denote for the characteristic function and the truncated counting function respectively 2 application to functional equations let kc be the field of meromorphic functions on c for a function r 0 r put k c a kc t r a o r which is a subfield of kc then the following holds let f x y k c x y be a polynomial in two variables over k c assume that the curve f x y 0 over k c has genus greater than one if 1 2 kc satisfy the functional equation f 1 2 0 then both 1 and 2 are contained in k c corollary 2 3 height inequality for curves over function fields let k be a function field of one variable over c let x be a smooth projective curve over k let d x be a reduced divisor let l be an ample line bundle on x and let e 0 then we have hk kx d p n 1 k s d p dk p ehk l p oe 1 for all p x k d theorem 5 here the notations are introduced in v1 v3 see also section 9 our proof uses ahlfors theory of covering surfaces and the geometry of the moduli space of q pointed stable curves of genus 0

meromorphic solutions of difference equations integrability and the discrete painleve equations the painlev e property is closely connected to differential equations that are integrable via related iso monodromy problems many apparently integrable discrete analogues of the painlev e equations have appeared in the literature the existence of sufficiently many finite order meromorphic solutions appears to be a good analogue of the painlev e property for discrete equations in which the independent variable is taken to be complex a general introduction to nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q difference operators new results are presented concerning equations of the form w z 1 w z 1 r z w where r is rational in w with meromorphic coefficients

existence of finite order meromorphic solutions as a detector of integrability in difference equations abstract the existence of sufficiently many finite order in the sense of nevanlinna meromorphic solutions of a difference equation appears to be a good indicator of integrability it is shown that out of a large class of second order difference equations the only equation that can admit a sufficiently general finite order meromorphic solution is the difference painleve ii equation the proof given relies on estimates obtained by arguments related to singularity confinement the existence of meromorphic solutions of a general class of first order difference equations is also proven by a simple method based on banach s fixed point theorem

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

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

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

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 nordstrom robinson code representation over gf 4 and efficient decoding we show that the nordstrom robinson 1968 code may be represented as the union of binary images of two isomorphic linear 4 2 3 codes over gf 4 certain symmetries of the unique quaternary 4 2 3 quadracode are discussed it is shown how the properties of the nordstrom robinson code itself such as weight distribution distance invariance and the well known representation as the union of eight cosets of the 16 5 8 reed muller code may be easily rederived from these symmetries in addition we introduce a decoding algorithm for the nordstrom robinson code which involves projecting its codewords onto the codewords of the quadracode the algorithm is simple enough to enable hard decision decoding of the nordstrom robinson code by hand furthermore we present an algorithm for maximum likelihood soft decision decoding of the nordstrom robinson code based on its representation over gf 4 the complexity of the proposed algorithm is at most 205 real operations this is to the best of our knowledge less than the complexity of any existing decoder and twice as efficient as the fast maximum likelihood decoder of adoul 1987 we also investigate several related topics such as bounded distance decoding sphere packings and the 20 2048 6 code

the art of signaling fifty years of coding theory in 1948 shannon developed fundamental limits on the efficiency of communication over noisy channels the coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero fifty years later codes for the gaussian channel have been discovered that come close to these fundamental limits there is now a substantial algebraic theory of error correcting codes with as many connections to mathematics as to engineering practice and the last 20 years have seen the construction of algebraic geometry codes that can be encoded and decoded in polynomial time and that beat the gilbert varshamov bound given the size of coding theory as a subject this review is of necessity a personal perspective and the focus is reliable communication and not source coding or cryptography the emphasis is on connecting coding theories for hamming and euclidean space and on future challenges specifically in data networking wireless communication and quantum information theory

on the value distribution of some difference polynomials abstract in this paper we continue to investigate the value distribution of some difference polynomials which can be considered as difference counterparts of some classic results of hayman

closed form expressions for integrals of mkdv and sine gordon maps we present closed form expressions for approximately n integrals of 2n dimensional maps the maps are obtained by travelling wave reductions of the modified korteweg de vries equation and of the sine gordon equation respectively we provide the integrating factors corresponding to the integrals moreover we show how the integrals and the integrating factors relate to the staircase method

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

elementary exact calculations of degree growth and entropy for discrete equations second order discrete equations are studied over the field of rational functions c z where z is a variable not appearing in the equation the exact degree of each iterate as a function of z can b

singularity complexity and quasi integrability of rational mappings we investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models by exploiting their nature of birational transformations of projective spaces we give an algorithmic analysis of the structure of invariants of such mappings we discuss some characteristic conditions for their quasi integrability and in particular its links with their singularities in the 2 plane finally we describe some of their propertiesqua dynamical systems making contact with arnol d s notion of complexity and exemplify remarkable behaviours

optimal sectionalization of a trellis while the complexity of trellis decoding for a given block code is essentially a function of the number of states and branches in its trellis the decoding complexity may be often reduced by means of an appropriate sectionalization of the trellis notwithstanding the many examples of such sectionalizations for particular codes that appeared in the literature no systematic method for finding the best sectionalization of a given trellis is presently known we present a polynomial time algorithm which produces the optimal sectionalization of a given trellis t in time o n sup 2 where n is the length of the code generated by t the algorithm is developed in a general setting of certain operations and functions defined on the set of trellises it therefore applies to both linear and nonlinear codes and easily accommodates a broad range of optimality criteria the particular optimality criterion based on minimizing the total number of additions and comparisons required for maximum likelihood trellis decoding is investigated in detail several different methods for decoding a given trellis are discussed and compared in a number of examples finally analysis of the dynamical properties of certain optimal sectionalizations is presented

closest point search in lattices in this semitutorial paper a comprehensive survey of closest point search methods for lattices without a regular structure is presented the existing search strategies are described in a unified framework and differences between them are elucidated an efficient closest point search algorithm based on the schnorr euchner 1995 variation of the pohst 1981 method is implemented given an arbitrary point x spl isin spl ropf sup m and a generator matrix for a lattice spl lambda the algorithm computes the point of spl lambda that is closest to x the algorithm is shown to be substantially faster than other known methods by means of a theoretical comparison with the kannan 1983 1987 algorithm and an experimental comparison with the pohst 1981 algorithm and its variants such as the viterbo boutros see ibid vol 45 p 1639 42 1999 decoder modifications of the algorithm are developed to solve a number of related search problems for lattices such as finding a shortest vector determining the kissing number computing the voronoi 1908 relevant vectors and finding a korkine zolotareff 1873 reduced basis

voronoi regions of lattices second moments of polytopes and quantization if a point is picked at random inside a regular simplex octahedron 600 cell or other polytope what is its average squared distance from the centroid in n dimensional space what is the average squared distance of a random point from the closest point of the lattice a n or d n e n a n ast or d n ast the answers are given here together with a description of the voronoi or nearest neighbor regions of these lattices the results have applications to quantization and to the design of signals for the gaussian channel for example a quantizer based on the eight dimensional lattice e8 has a mean squared error per symbol of 0 0717 cdots when applied to uniformly distributed data compared with 0 08333 cdots for the best one dimensional quantizer

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

mathematical problems in error calculations for interference several treatments of interference problems are examined with a view to determine whether doubtful mathematical assumptions about the existence of probability density functions affect the conclusions most derivations can be recast so that the existence of a density function is not necessary for the conclusions to be valid

some properties of meromorphic solutions of q difference equations abstract we investigate the growth of transcendental meromorphic solutions of some complex q difference equations and find lower bounds for nevanlinna lower order for meromorphic solutions of such equations we also obtain a q difference version of tumura clunie theorem

multiple description vector quantization with lattice codebooks design and analysis the problem of designing a multiple description vector quantizer with lattice codebook spl lambda is considered a general solution is given to a labeling problem which plays a crucial role in the design of such quantizers numerical performance results are obtained for quantizers based on the lattices a sub 2 and z sup i i 1 2 4 8 that make use of this labeling algorithm the high rate squared error distortions for this family of l dimensional vector quantizers are then analyzed for a memoryless source with probability density function pdf p and differential entropy h p spl infin for any a spl epsiv 0 1 and rate pair r r it is shown that the two channel distortion d sub o and the channel 1 or channel 2 distortion d sub s satisfy lim sub r spl rarr spl infin d sub o 2 sup 2r 1 a 1 4 g spl lambda 2 sup 2h p and lim sub r spl rarr spl infin d sub s 2 sup 2r 1 a g s sub l 2 sup 2h p where g spl lambda is the normalized second moment of a voronoi cell of the lattice spl lambda and g s sub l is the normalized second moment of a sphere in l dimensions

hkz and minkowski reduction algorithms for lattice reduction aided mimo detection recently lattice reduction has been widely used for signal detection in multiinput multioutput mimo communications in this paper we present three novel lattice reduction algorithms first using a unimodular transformation a significant improvement on an existing hermite korkine zolotareff reduction algorithm is proposed then we present two practical algorithms for constructing minkowski reduced bases to assess the output quality we compare the orthogonality defect of the reduced bases produced by lll algorithm and our new algorithms and find that in practice minkowski reduced basis vectors are the closest to being orthogonal an error rate analysis of suboptimal decoding algorithms aided by different reduction notions is also presented to this aim the proximity factor is employed as a measurement we improve some existing results and derive upper bounds for the proximity factors of minkowski reduction aided decoding mrad to show that mrad can achieve the same diversity order with infinite lattice decoding ild

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

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

baseband trellis codes with a spectral null at zero a method is described for modifying classical n dimensional trellis codes to provide baseband codes that combine a spectral null at dc with significant coding gain the information rate of the classical code is decreased by one bit and this extra redundancy is used to keep the running digital sum bounded equivalently if the rate is held constant then twice as many signal points are needed causing a power penalty of 6 n db baseband trellis codes are presented for several information rates together with complete spectral plots and performance comparisons a method of constructing baseband codes with multiple spectral nulls is also described

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

on the growth of logarithmic differences difference quotients and logarithmic derivatives of meromorphic functions a crucial ingredient in the recent discovery by ablowitz halburd herbst and korhonen 2000 2007 that a connection exists between discrete painleve equations and finite order nevanlinna theory is an estimate of the integrated average of log f z 1 f z on z r we obtained essentially the same estimate in our previous paper 2008 independent of halburd et al 2006 we continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences difference quotients and logarithmic derivatives for finite order meromorphic functions in addition to the potential applications of our new estimates in integrable systems they are also of independent interest in particular our findings show that there are marked differences between the growth of meromorphic functions with nevanlinna order less than and greater than one we have established a difference analogue of the classical wiman valiron type estimates for meromorphic functions with order less than one which allow us to prove that all entire solutions of linear difference equations with polynomial coefficients of order less than one must have positive rational order of growth we have also established that any entire solution to a first order algebraic difference equation with polynomial coefficients must have a positive order of growth which is a difference analogue of a classical result of polya

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

gaussian source coding with spherical codes a fixed rate shape gain quantizer for the memoryless gaussian source is proposed the shape quantizer is constructed from wrapped spherical codes that map a sphere packing in spl ropf sup k 1 onto a sphere in spl ropf sup k and the gain codebook is a globally optimal scalar quantizer a wrapped leech lattice shape quantizer is used to demonstrate a signal to quantization noise ratio within 1 db of the distortion rate function for rates above 1 bit per sample and an improvement over existing techniques of similar complexity an asymptotic analysis of the tradeoff between gain quantization and shape quantization is also given

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

meromorphic solutions of difference painleve equations we show that under certain conditions some difference painleve equations have nontrivial meromorphic solutions in the whole complex plane these meromorphic solutions are obtained by analytic continuation of asymptotic solutions given in sectors of zero opening angle the existence of these asymptotic solutions is shown by using fundamental matrix solutions of the associated linear systems with double unit characteristic roots and with coefficients expressed by asymptotic series in fractional powers

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

nevanlinna theory for the q difference operator and meromorphic solutions of q difference equations it is shown that if f is a meromorphic function of order zero and q in mathbb c then begin equation label abstid m bigg r frac f qz f z bigg o t r f tag ddag end equation for all r on a set of logarithmic density 1 the remainder of the paper consists of applications of identity eqref abstid to the study of value distribution of zero order meromorphic functions and in particular zero order meromorphic solutions of q difference equations the results obtained include q shift analogues of the second main theorem of nevanlinna theory picard s theorem and clunie and mohon ko lemmas

hexacode based quantization of the gaussian source at 1 2 bit per sample we present the performance of several suboptimal algebraic quantizers in 24 dimensions the gaussian source is encoded at 1 2 bit per sample using the binary extended golay code c sub 24 and the hexacode h sub 6 we also propose two new suboptimal decoding algorithms for the hexacode h sub 6

performance bounds for trellis coded modulation on time dispersive channels the performance of trellis coded modulation tcm on additive white gaussian noise channels is well understood and tight analytical bounds exist on the probability of the viterbi decoder making a decision error when a channel is also time dispersive the performance of tcm systems has been studied mainly by simulation however simulation is limited to symbol error probabilities greater than 10 sup 6 and is not a particularly useful tool for designing codes tight analytical bounds on the error probability of tcm on time dispersive channels are required for a more thorough study of performance moreover the design of good codes and optimum metrics for time dispersive channels requires tight analytical bounds in this paper we derive analytical upper bounds which although requiring numerical techniques for tractable evaluation are tight for a wide range of time dispersive channel conditions the bounds are based on a union bound of error events that leads to a summation of pairwise error probabilities which are themselves upper bounded