on the laplacian eigenvalues of a graph abstract in the note we present an upper bound for the spectral radius of laplacian matrix of a graph in terms of a 2 degree of a vertex

a fast encoding method for lattice codes and quantizers in an earlier paper the authors described a very fast method which for the root lattices a n d n e n their duals and certain other lattices finds the closest lattice point to an arbitrary point of the underlying space if the lattices are used as codes for a gaussian channel the algorithm provides a fast decoding procedure or if they are used as vector quantizers the algorithm performs the analog to digital conversion efficiently the present paper offers a solution to the inverse problem for the same lattices the encoding problem for channel codes or the digital to analog part of quantizing namely given an integer k to find the kth code vector and to the closely related problem of finding the index k of a given code vector

shaping multidimensional signal spaces i optimum shaping shell mapping the structure of the regions which provide the optimum tradeoff between gamma sub s shaping gain and cer constellation expansion ratio and between gamma sub s and par peak to average power ratio in a finite dimensional space is introduced analytical expressions are derived for the corresponding tradeoff curves in general the initial parts of the curves have a steep slope this means that an appreciable portion of the maximum shaping gain corresponding to a spherical region can be achieved with a small value of cer sub s par the technique of shell mapping is introduced this is a change of variable which maps the optimum shaping region to a hypercube truncated within a simplex this mapping is a useful tool in computing the performance and also in facilitating the addressing of the optimum shaping region using the shell mapping a practical addressing scheme is presented that achieves a point on the optimum tradeoff curves for dimensionalities around 12 the point achieved is located near the knee of the corresponding tradeoff curve for larger dimensionalities a general shaping region with two degrees of freedom is used this region provides more flexibility in selecting the tradeoff point

nonvanishing of siegel poincare series we prove that under suitable conditions certain siegel poincare series of exponential type of even integer weight and degree 2 do not vanish identically we also find estimates for twisted kloosterman sums and salie sums in all generality

on some forests determined by their laplacian or signless laplacian spectrum we consider the class of graphs whose each component is either a proper subgraph of some smith graphs or belongs to a precized subset of smith graphs we classify the graphs from the considered class into those which are determined or not determined by laplacian or signless laplacian spectrum

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

rotational invariance of trellis codes i encoders and precoders we present a theoretical framework for rotational invariance of trellis codes the distinction between codes and encoders plays a pivotal role necessary and sufficient conditions for rotational invariance are derived under general assumptions and a construction is presented that obtains a rotationally invariant encoder for almost any rotationally invariant code independent of the code s algebraic structure encoders that use a differential precoder are considered as a separate case where a system theoretic characterization of precoding is used to find two alternative and slightly less general encoder constructions

channel coding with quadrature quadrature phase shift keying q sup 2 psk signals channel coding due to trellis modulation has been proved to be useful for bandlimited channels however these modulations mostly designed with n use of 2d signals are primarily aimed at coding gain only it is pointed out that utilization of all available signal dimensions which is limited by the time bandwidth product may improve the bandwidth efficiency and simultaneously bring an additional coding gain trellis coding with a spectrally efficient 4d signal set based on q sup 2 psk is addressed without any expansion of the q sup 2 psk signal set a simple hand designed 16 state trellis code provides a coding gain of 6 02 db with the same number of states in g ungergoeck s 1982 8 psk trellis the gain is 4 1 db in low intersymbol interference situations the bandwidth efficiency of this coded q sup 2 psk is twice that of coded 8 psk if both operate at 2 b s hz and p sub b e 10 sup 5 the coded q sup 2 psk provides a saving of about 4 db over the coded 8 psk some fully connected trellises with an expanded signal due to 2 use of q sup 2 psk signals are also presented two such codes ar rate 7 8 achieve a gain of 5 45 db with only eight states

multilevel codes theoretical concepts and practical design rules this paper deals with 2 sup l ary transmission using multilevel coding mlc and multistage decoding msd the known result that mlc and msd suffice to approach capacity if the rates at each level are appropriately chosen is reviewed using multiuser information theory it is shown that there is a large space of rate combinations such that mlc and full maximum likelihood decoding mld can approach capacity it is noted that multilevel codes designed according to the traditional balanced distance rule tend to fall in the latter category and therefore require the huge complexity of mld the capacity rule the balanced distances rules and two other rules based on the random coding exponent and cutoff rate are compared and contrasted for practical design simulation results using multilevel binary turbo codes show that capacity can in fact be closely approached at high bandwidth efficiencies moreover topics relevant in practical applications such as signal set labeling dimensionality of the constituent constellation and hard decision decoding are emphasized bit interleaved coded modulation proposed by caire et al see ibid vol 44 p 927 46 1998 is reviewed in the context of mlc finally the combination of signal shaping and coding is discussed significant shaping gains are achievable in practice only if these design rules are taken into account

new rotationally invariant four dimensional trellis codes two new classes of rotationally invariant trellis codes are constructed a simple method is used to check the rotational invariance of a given code in the process of searching for optimum trellis codes a class of linear trellis codes with constraint lengths 2 9 using four dimensional constellations is presented simulation results show that a 180 spl deg rotationally invariant constraint length 8 linear trellis code achieves about 0 4 db real coding gain compared to the best constraint length 6 code while the trellis complexity is only four times that of the constraint length 6 code on the other hand the constraint length 6 code that has been adopted for use in the v 34 28 8 kbit s modem standard has the same 0 4 db real coding gain compared to a constraint length 4 code which has also been adopted for the standard but it requires sixteen times the trellis complexity a class of fully rotationally invariant nonlinear trellis codes with constraint lengths 6 11 is also presented simulation results show that a 90 spl deg rotationally invariant constraint length 8 nonlinear trellis code performs almost as well as the best linear code

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

a note on the spectral characterization of θ graphs we consider graphs that is graphs obtained by subdividing the edges of the multigraph consisting of 3 parallel edges it is shown that any graph g is determined by the spectrum the multiset of eigenvalues except possibly when it contains a unique 4 cycle

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

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

multiple trellis coded modulation mtcm the authors demonstrate a trellis coded modulation technique referred to as multiple trellis coded modulation mtcm wherein more than one channel symbol per trellis branch is transmitted they have found simple two state trellis codes for symmetric mpsk multiple phase shift keying and am modulations that can achieve 3 db gain over uncoded modulation at very high signal to noise ratios without bandwidth expansion or reduction in information bit rate the gain of these codes with respect to previously reported two state trellis codes is between 1 and 2 db at very high signal to noise ratios depending on the number of bits per hertz transmitted these gains are achieved for those of the equivalent conventional trellis codes with the same number of states in the trellis diagram the authors note that additional computations per branch are needed for the multiple trellis coding scheme the concept can be extended to a higher number of states and other types of modulations

line star sets for laplacian eigenvalues abstract let g be a mixed graph with a nonzero laplacian eigenvalue of multiplicity k a line star set for in g is a set y of k edges of g such that is not a laplacian eigenvalue of g y it is shown that line star set exists for any nonzero laplacian eigenvalue of any mixed graph some basic properties for line star sets are given we also obtain some results on line star sets in undirected graphs

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

trellis coding with multidimensional qam signal sets trellis coding using multidimensional quadrature amplitude modulation qam signal sets is investigated finite size 2d signal sets are presented that have minimum average energy are 90 degrees rotationally symmetric and have from 16 to 1024 points the best trellis codes using the finite 16 qam signal set with two four six and eight dimensions are found by computer search the multidimensional multi d signal set is constructed from the 2 d signal set the best moderate complexity trellis codes for infinite lattices with two four six and eight dimensions are also found the minimum free squared euclidean distance and number of nearest neighbors for these codes were used as the selection criteria many of the multi d codes are fully rotationally invariant and give asymptotic coding gains up to 6 0 db from the infinite lattice codes the best codes for transmitting j j 1 4 j 1 3 j 1 2 j 2 3 and j 3 4 b sym j an integer are presented

use of spl sigma trees as constellations in trellis coded modulation sigma trees spl sigma trees are a class of geometric structures that include lattices as a constrained special case these structures allow for signal sets in spaces of arbitrary dimension that are more spherical in shape than signal sets based on lattices in this correspondence it is shown that spl sigma trees can be used in the construction of non lattice trellis coded modulation schemes tcm schemes a low complexity spl sigma tree encoder is presented for multidimensional tcm codes an optimal tcm decoder that uses the efficient sequential search property of spl sigma trees is then described simulation results based on one and two dimensional examples are used to show that the performance of spl sigma tree based codes is comparable to that provided by conventional lattice based codes

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

note the t shape tree is determined by its laplacian spectrum a t shape is a tree with exactly one of its vertices having maximal degree 3 it is proved in this paper that the t shape tree is determined by its laplacian spectrum

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

an efficient block based addressing scheme for the nearly optimum shaping of multidimensional signal spaces we introduce an efficient addressing scheme for the nearly optimum shaping of a multidimensional signal constellation the 2 d two dimensional subspaces are partitioned into k energy shells of equal cardinality the average energy of a 2 d shell can be closely approximated by a linear function of its index in an n 2n d space we obtain k sup n shaping clusters of equal cardinality shaping is achieved by selecting t spl les k sup n of the n d clusters with the least sum of the 2 d indices this results in a set of t integer n tuples with the components in the range 0 k 1 and the sum of the components being at most a given number l the problem of addressing is to find a one to one mapping between the set of such n tuples and the set of integers 0 t 1 such that the mapping and its inverse can be easily implemented in the proposed scheme the n d clusters are grouped into blocks of identical binary weight vectors this results in a simple rule for the addressing of points within the blocks the addressing of the blocks is based on some recursive relationship which allows us to decompose the problem into simpler parts the overall scheme requires a modest amount of memory and has a small computational complexity

wei type trellis coded modulation with 2n dimensional rectangular constellation for n not a power of two the wei method of constructing trellis codes with 2n dimensional rectangular signal constellations is limited to n being an integer power of two in the present correspondence this limitation is removed and examples of six dimensional 6 d and twelve dimensional 12 d trellis codes are given

geometrically uniform codes a signal space code c is defined as geometrically uniform if for any two code sequences in c there exists an isometry that maps one sequence into the other while leaving the code c invariant geometrical uniformity a strong kind of symmetry implies such properties as a the distance profiles from code sequences in c to all other code sequences are all the same and b all voronoi regions of code sequences in c have the same shape it is stronger than ungerboeck zehavi wolf symmetry or calderbank sloane regularity nonetheless most known good classes of signal space codes are shown to be generalized coset codes and therefore geometrically uniform including a lattice type trellis codes based on lattice partitions lambda lambda such that z sup n lambda lambda 4z sup n is a lattice partition chain and b phase shift keying psk type trellis codes based on up to four way partitions of a 2 sup n psk signal set

nonvanishing of koecher maass series attached to siegel cusp forms we prove a nonvanishing result for koecher maass series attached to siegel cusp forms of weight k and degree n in certain strips on the complex plane when n 2 we prove such a result for forms orthogonal to the space of the saito kurokawa lifts up to finitely many exceptions in bounded regions c 2015 elsevier inc all rights reserved

trellis shaping the author discusses trellis shaping a method of selecting a minimum weight sequence from an equivalence class of possible transmitted sequences by a search through the trellis diagram of a shaping convolutional code c sub s shaping gains on the order of 1 db may be obtained with simple four state shaping codes and with moderate constellation expansion the shaping gains obtained with more complicated codes approach the ultimate shaping gain of 1 53 db with a feedback free syndrome former for c sub s transmitted data can be recovered without catastrophic error propagation constellation expansion and peak to average energy ratio may be effectively limited by peak constraints with lattice theoretic constellations the shaping operation may be characterized as a decoding of an initial sequence in a channel trellis code by a minimum distance decoder for a shaping trellis code based on the shaping convolutional code and the set of possible transmitted sequences is then the set of code sequences in the channel trellis code that lie in the voronoi region of the trellis shaping code

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

estimates for fourier coefficients of siegel cusp forms the proof of 1 was based on appropriate estimates both for the fourier coefficients of cusp forms on the jacobi group f1 2 more precisely let

a systematic approach to the construction of bandwidth efficient multidimensional trellis codes we present a unified approach to the construction of multidimensional trellis codes that have small constellation expansion ratio cer and small peak to average power ratio par and hence are of high bandwidth efficiency in the approach we extend wei s 1987 construction to dimensionalities other than a power of two and propose a low complexity encoder we also present a general partition technique that gives geometrically uniform partitions with a good distance profile the proposed method can generate many good codes including previously reported ungerboeck codes gallager calderbank sloane codes and wei codes with quadrature amplitude modulation signals some newly discovered six dimensional trellis codes are compared with these known codes on their cer par effective coding gain and encoder decoder complexities

on a laplacian spectral characterization of graphs of index less than 2 a graph is said to be determined by the adjacency respectively laplacian spectrum if there is no other non isomorphic graph with the same adjacency respectively laplacian spectrum the maximum eigenvalue of a g is called the index of g the connected graphs with index less than 2 are known and each is determined by its adjacency spectrum in this paper we show that graphs of index less than 2 are determined by their laplacian spectrum

trellis coded quantization of memoryless and gauss markov sources trellis coded quantization tcq is developed and applied to the encoding of memoryless and gauss markov sources the theoretical justification for the approach is alphabet constrained rate distortion theory which is a dual to the channel capacity argument that motivates trellis coded modulation tcm the authors adopt the notions of signal set expansion set partitioning and branch labeling of tcm but modify the techniques to account for the source distribution to design tcq coders of low complexity with excellent mean squared error mse performance for a memoryless uniform source tcq provides an mse within 0 21 db of the distortion rate bound at all positive integral rates the performance is superior to that promised by the coefficient of quantization for all of the best lattices known in dimensions 24 or less for a memoryless gaussian source the tcq performance at rates of 0 5 1 and 2 b sample is superior to all previous results the authors found in the literature the encoding complexity of tcq is very modest tcq is incorporated into a predictive coding structure for the encoding of gauss markov sources simulation results for first second and third order gauss markov sources are presented

six dimensional trellis coding with qam signal sets a family of 6 d trellis coded modulation tcm schemes which involve a 2 step partitioning of the constituent qam signal alphabet is presented with infinite constellations without shaping the asymptotic coding gain is 3 db for the 2 state code 4 db for the 4 and 8 state codes and 5 db for the 16 and 32 state codes which involve a smaller alphabet expansion the authors also describe a rotationally invariant 16 state code that achieves the same asymptotic gain as its linear counterpart practical signal constellations are described for 6 d tcm with the spectral efficiency of uncoded 64 qam and the performance of these schemes is studied by means of computer simulations it was found that they achieve an additional coding gain of 0 2 0 3 db over infinite hypercube type constellations the performance of the presented schemes at practical signal to noise ratio values is evaluated using transfer function techniques

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

trellis coded multidimensional phase modulation a 2l dimensional multiple phase shift keyed l mpsk signal set is obtained by forming the cartesian product of l two dimensional mpsk signal sets a systematic approach to partitioning l mpsk signal sets that is based on block coding is used an encoder system approach is developed it incorporates the design of a differential precoder a systematic convolutional encoder and a signal set mapper trellis coded l 4psk l 8psk and l 16psk modulation schemes are found for 1

on a signless laplacian spectral characterization of t shape trees let m be an associated matrix of a graph g the adjacency laplacian and signless laplacian matrix two graphs are said to be cospectral with respect to m if they have the same m spectrum a graph is said to be determined by m spectrum if there is no other non isomorphic graph with the same spectrum with respect to m it is shown that t shape trees are determined by their laplacian spectra moreover among them those are determined by their adjacency spectra are characterized in this paper we identify graphs which are cospectral to a given t shape tree with respect to the signless laplacian matrix subsequently t shape trees which are determined by their signless laplacian spectra are identified

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

the spectral characterization of graphs of index less than 2 with no path as a component a graph is said to be determined by the adjacency and laplacian spectrum or to be a ds graph for short if there is no other non isomorphic graph with the same adjacency and laplacian spectrum respectively it is known that connected graphs of index less than 2 are determined by their adjacency spectrum in this paper we focus on the problem of characterization of ds graphs of index less than 2 first we give various infinite families of cospectral graphs with respect to the adjacency matrix subsequently the results will be used to characterize all ds graphs with respect to the adjacency matrix of index less than 2 with no path as a component moreover we show that most of these graphs are ds with respect to the laplacian matrix

multidimensional mappings for iteratively decoded bicm on multiple antenna channels multidimensional binary mappings for bit interleaved coded modulations bicms on ergodic multiple antenna channels with iterative decoding are presented after derivation of a closed form expression for the pairwise error probability under ideal maximum likelihood ml decoding the design criterion for mapping optimization is established from the ml performance of the ideally interleaved channel it coincides with the figure of merit derived from the genie condition when the iterative receiver converges to perfect a priori information multidimensional mapping constructions that exhibit high signal to noise ratio snr gains without increasing the complexity of the a posteriori probability app detection are proposed they allow for a reduced decoding complexity as they achieve near turbo code performance with a single convolutional code