n channel symmetric multiple descriptions part ii an achievable rate distortion region in this part ii of a two part paper we present a new achievable rate distortion region for the symmetric n channel multiple descriptions coding problem n 2 where the rate of every description is the same and the reconstruction distortion depends only on the number of descriptions received using a new approach for the random coding constructions along with a generalization of the technique used in the two channel el gamal and cover region to any n the rate region presented here achieves points that have not been known in the literature previously this rate region is derived from a concatenation of source channel erasure codes developed in part i of this work by deploying the framework of source coding with side information random binning the key idea is that by using the framework of source coding with side information multiple statistically identical realizations representing the coarse version of a source can be simultaneously refined by a single encoding we point out that there is an important conceptual difference in random coding construction for the multiple descriptions coding problem between the case of n 2 and n 2 to illustrate the framework we also present the important case of the gaussian source in detail

asymptotical analysis of several multiple description scenarios with l 3 descriptions this work compares theoretically the performance of several representative practical multiple description md frameworks with l 3 symmetric descriptions the first scenario is the classic unequal erasure protection uep scheme using a successively refinable code src and reed solomon codes the second scenario is an improvement upon uep by applying domain partitioning and permuted reed solomon codes the third scenario uses a finer partitioning and erasure correction via repetition codes additionally the md lattice vector quantizer and another recent md scheme are considered in the comparison the aforementioned md schemes are compared in terms of the expected squared error asymptotically achievable as the rate r of a description approaches assuming independent description losses our analysis reveals that the improvement of the second scenario upon the first one when r can reach up to 1 68 db but it approaches 0 as the description loss rate p goes to 0 and l approaches additionally we find that the first two schemes outperform the third one with an unbounded gain when r as p 0 or l further we show that the first three scenarios achieve unbounded improvements over the other two as r and p 0 on the other hand we point out that some of the results of our asymptotic analysis rely on strong assumptions and therefore an experimental validation is needed before applying them to practical situations

association schemes based on maximal totally isotropic subspaces in singular pseudo symplectic spaces this paper provides some new families of symmetric association schemes based on maximal totally isotropic subspaces in singular pseudo symplectic spaces all intersection numbers of these schemes are computed

n channel asymmetric entropy constrained multiple description lattice vector quantization this paper is about the design and analysis of an index assignment ia based multiple description coding scheme for the n channel asymmetric case we use entropy constrained lattice vector quantization and restrict attention to simple reconstruction functions which are given by the inverse ia function when all descriptions are received or otherwise by a weighted average of the received descriptions we consider smooth sources with finite differential entropy rate and mse fidelity criterion as in previous designs our construction is based on nested lattices which are combined through a single ia function the results are exact under high resolution conditions and asymptotically as the nesting ratios of the lattices approach infinity for any n the design is asymptotically optimal within the class of ia based schemes moreover in the case of two descriptions and finite lattice vector dimensions greater than one the performance is strictly better than that of existing designs in the case of three descriptions we show that in the limit of large lattice vector dimensions points on the inner bound of pradhan can be achieved furthermore for three descriptions and finite lattice vector dimensions we show that the ia based approach yields in the symmetric case a smaller rate loss than the recently proposed source splitting approach

flexible symmetric multiple description lattice vector quantizer with l geq 3 descriptions in the previous work on multiple description lattice vector quantizers mdlvqs with l geq 3 descriptions once the central and side lattice codebooks are fixed the decoding quality is determined for all numbers k of received descriptions therefore it is not possible to achieve tradeoffs between the quality of reconstruction for different values of k 1 leq k leq l 1 this paper proposes a flexible mdlvq capable of overcoming the above drawback for this a different reconstruction method is employed and a heuristic index assignment ia algorithm which uses l 2 parameters to control the distortions for 2 leq k leq l 1 is developed experimental results show that the proposed mdlvq in addition to achieving the desired tradeoffs significantly outperforms the classic md scheme based on unequal erasure protection the second contribution of this paper is a structured ia for the case of l 3 and the derivation of the corresponding expressions of the distortions at high resolution the proposed ia has a simple mechanism for controlling the tradeoff between the reconstruction quality for k 1 2 the ia is able to achieve a wide range of distortion values while keeping the product of the distortions for k 1 2 the same as in the prior work

bit error resilient index assignment for multiple description scalar quantizers this paper addresses the problem of increasing the robustness to bit errors for two description scalar quantizers our approach is to start with an m diagonal index assignment and further apply a permutation to the indexes of each description to increase the minimum hamming distance d rm min of the set of valid index pairs in particular we show how to construct linear permutation pairs achieving d rm min geq 3 and establish a lower bound in terms of the description rate r for the highest value of m for which such permutations exist

information content for quantum states a method for representing probabilistic aspects of quantum systems by means of a density function on the space of pure quantum states is introduced in particular a maximum entropy argument allows us to obtain a natural density function that only reflects the information provided by the density matrix this result is applied to derive the shannon entropy of a quantum state the information theoretic quantum entropy thereby obtained is shown to have the desired concavity property and to differ from the conventional von neumann entropy this is illustrated explicitly for a two state system

asymmetric multiple description lattice vector quantizers we consider the design of asymmetric multiple description lattice quantizers that cover the entire spectrum of the distortion profile ranging from symmetric or balanced to successively refinable we present a solution to a labeling problem which is an important part of the construction along with a general design procedure the high rate asymptotic performance of the quantizer is also studied we evaluate the rate distortion performance of the quantizer and compare it to known information theoretic bounds the high rate asymptotic analysis is compared to the performance of the quantizer

n channel entropy constrained multiple description lattice vector quantization in this paper we derive analytical expressions for the central and side quantizers which under high resolution assumptions minimize the expected distortion of a symmetric multiple description lattice vector quantization md lvq system subject to entropy constraints on the side descriptions for given packet loss probabilities we consider a special case of the general n channel symmetric multiple description problem where only a single parameter controls the redundancy tradeoffs between the central and the side distortions previous work on two channel md lvq showed that the distortions of the side quantizers can be expressed through the normalized second moment of a sphere we show here that this is also the case for three channel md lvq furthermore we conjecture that this is true for the general n channel md lvq for given source target rate and packet loss probabilities we find the optimal number of descriptions and construct the md lvq system that minimizes the expected distortion we verify theoretical expressions by numerical simulations and show in a practical setup that significant performance improvements can be achieved over state of the art two channel md lvq by using three channel md lvq

association schemes based on the subspaces of type 2 1 0 in singular symplectic space over finite fields abstract in this paper we provide a construction of association scheme based on the subspaces of type 2 1 0 in singular symplectic space over finite fields moreover all parameters of the association scheme are computed

optimal index assignment for multiple description scalar quantization with translated lattice codebooks we design a k description scalar quantizer whose construction is based on a structure of translated scalar lattices and a lattice in k 1 dimensional space the use of translated lattices provides a performance advantage by exploiting a so called staggering gain the use of the k 1 dimensional lattice facilitates analytic insight into the performance and significantly speeds up the computation of the index assignment compared to state of the art methods using a common decoding method the proposed index assignment is proven to be optimal for the k description case it is shown that the optimal index assignment is not unique this is illustrated for the two description case where a periodic index assignment is selected from possible optimal assignments and described in detail the performance of the proposed quantizer accurately matches theoretic analysis over the full range of operational redundancies moreover the quantizer outperforms the state of the art md scheme as the redundancy among the description increases

suborbits of m dimensional totally isotropic subspaces under finite singular classical groups it is well known that the set of all the totally isotropic subspaces with the same dimension in a singular classical space forms an orbit under the action of the corresponding classical group in this paper we determine all the orbitals and the rank of this action and calculate the length of each suborbit

geometry of stochastic state vector reduction the state space of a quantum mechanical system is a complex projective space the space of rays in the associated hilbert space the state space comes equipped with a natural riemannian metric the fubini study metric and a compatible symplectic structure the operations of ordinary quantum mechanics can thus be reinterpreted in the language of differential geometry it is interesting in this spirit to scrutinize the probabilistic assumptions that are brought in at various stages in the analy sis of quantum dynamics particularly in connection with state vector reduction a promising approach to understanding reduction studied recently by a number of authors involves the use of nonlinear stochastic dynamics to modify the ordinary linear schrodinger evolution here we use stochastic differential geometry to give a systematic geometric formulation for such stochastic models of state vector collapse in this picture the conventional schrodinger evolution which corresponds to the unitary flow associated with a killing vector of the fubini study metric is replaced by a more general stochastic flow on the state manifold in the simplest example of such a flow the volatility term in the stochastic differential equation for the state trajectory is proportional to the gradient of the expectation of the hamiltonian the conservation of energy is represented by the requirement that the actual process followed by the expectation of the hamiltonian as the state evolves should be a martingale this requirement implies the existence of a nonlinear term in the drift vector of the state process which is always oriented opposite to the direction of in creasing energy uncertainty as a consequence the state vector necessarily collapses to an energy eigenstate and a martingale argument can be used to show that the probability of collapse to a given eigenstate from any particular initial state is in fact given by precisely the usual quantum mechanical probability

association schemes based on singular symplectic geometry over finite fields and its application abstract the paper provides the construction of association scheme on the subspaces of type k 0 k in singular symplectic geometry over finite fields all intersection numbers of the scheme are computed at last an authentication code with perfect secrecy from the association scheme is construction

multiple description coding by dithered delta sigma quantization we address the connection between the multiple description md problem and delta sigma quantization the inherent redundancy due to oversampling in delta sigma quantization and the simple linear additive noise model resulting from dithered lattice quantization allow us to construct a symmetric and time invariant md coding scheme we show that the use of a noise shaping filter makes it possible to trade off central distortion for side distortion asymptotically as the dimension of the lattice vector quantizer and order of the noise shaping filter approach infinity the entropy rate of the dithered delta sigma quantization scheme approaches the symmetric two channel md rate distortion function for a memoryless gaussian source and mean square error mse fidelity criterion at any side to central distortion ratio and any resolution in the optimal scheme the infinite order noise shaping filter must be minimum phase and have a piecewise flat power spectrum with a single jump discontinuity an important advantage of the proposed design is that it is symmetric in rate and distortion by construction so the coding rates of the descriptions are identical and there is therefore no need for source splitting

multiple description vector quantization with a coarse lattice a multiple description md lattice vector quantization technique for two descriptions was previously introduced in which fine and coarse codebooks are both lattices the encoding begins with quantization to the nearest point in the fine lattice this encoding is an inherent optimization for the decoder that receives both descriptions performance can be improved with little increase in complexity by considering all decoders in the initial encoding step the altered encoding relies only on the symmetries of the coarse lattice this allows us to further improve performance without a significant increase in complexity by replacing the fine lattice codebook with a nonlattice codebook that respects many of the symmetries of the coarse lattice examples constructed with the two dimensional 2 d hexagonal lattice demonstrate large improvement over time sharing between previously known quantizers

approximating the gaussian multiple description rate region under symmetric distortion constraints we consider multiple description md coding for the gaussian source with k descriptions under the symmetric mean squared error mse distortion constraints and provide an approximate characterization of the rate region we show that the rate region can be sandwiched between two polytopes between which the gap can be upper bounded by constants dependent on the number of descriptions but independent of the distortion constraints underlying this result is an exact characterization of the lossless multilevel diversity source coding problem a lossless counterpart of the md problem this connection provides a polytopic template for the inner and outer bounds to the rate region in order to establish the outer bound we generalize ozarow s technique to introduce a strategic expansion of the original probability space by more than one random variable for the symmetric rate case with any number of descriptions we show that the gap between the upper bound and the lower bound for the individual description rate distortion function is no larger than 0 92 bit the results developed in this work also suggest that the ldquoseparationrdquo approach of combining successive refinement quantization and lossless multilevel diversity coding is a competitive one since its performance is only a constant away from the optimum the results are further extended to general sources under the mse distortion measure where a similar but looser bound on the gap holds

suborbits of subspaces of type m k under finite singular general linear groups abstract suppose f q n l denotes the n l dimensional vector space over a finite field f q and gl n l n f q denotes the corresponding singular general linear group all the subspaces of type m k form an orbit under gl n l n f q denoted by m m k n l n let be the set of all the orbitals of gl n l n f q m m k n l n then m m k n l n is a symmetric association scheme in this paper we determine all the orbitals and the rank of gl n l n f q m m k n l n calculate the length of each suborbit finally we compute all the intersection numbers of the symmetric association scheme m m k n l n where k 1 or k l 1

multiple description quantization via gram schmidt orthogonalization the multiple description md problem has received considerable attention as a model of information transmission over unreliable channels a general framework for designing efficient md quantization schemes is proposed in this paper we provide a systematic treatment of the el gamal cover egc achievable md rate distortion region and show it can be decomposed into a simplified egc segc region and a superimposed refinement operation furthermore any point in the segc region can be achieved via a successive quantization scheme along with quantization splitting for the quadratic gaussian case the proposed scheme has an intrinsic connection with the gram schmidt orthogonalization which implies that the whole gaussian md rate distortion region is achievable with a sequential dithered lattice based quantization scheme as the dimension of the optimal lattice quantizers becomes large moreover this scheme is shown to be universal for all independent and identically distributed i i d smooth sources with performance no worse than that for an i i d gaussian source with the same variance and asymptotically optimal at high resolution a class of md scalar quantizers in the proposed general framework is also constructed and is illustrated geometrically the performance is analyzed in the high resolution regime which exhibits a noticeable improvement over the existing md scalar quantization schemes

the quantum canonical ensemble the phase space of quantum mechanics can be viewed as the complex projective space cpn endowed with a kahlerian structure given by the fubini study metric and an associated symplectic form we can then interpret the schrodinger equation as generating a hamiltonian dynamics on based upon the geometric structure of the quantum phase space we introduce the corresponding natural microcanonical and canonical ensembles the resulting density matrix for the canonical ensemble differs from the density matrix of the conventional approach as an illustration the results are applied to the case of a spin one half particle in a heat bath with an applied magnetic field

m description lattice vector quantization index assignment and analysis in this paper we investigate the design of symmetric entropy constrained multiple description lattice vector quantization mdlvq more specifically mdlvq index assignment we consider a fine lattice containing clean similar sublattices with s similarity due to the s similarity of the sublattices an m fraction lattice can be used to regularly partition the fine lattice with smaller voronoi cells than a sublattice does with the partition the mdlvq index assignment design can be translated into a transportation problem in operations research both greedy and general algorithms are developed to pursue optimality of the index assignment under high resolution assumption we compare the proposed schemes with other relevant techniques in terms of optimality and complexity following our index assignment design we also obtain an asymptotical close form expression of k description side distortion simulation results on coding different sources of gaussian speech and image are presented to validate the effectiveness of the proposed schemes

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

new coding schemes for the symmetric k description problem we propose novel coding schemes for the k description problem with symmetric rates and symmetric distortion constraints there are two main new ingredients in these schemes the first one is akin to the method seen in the well known butterfly network of network coding literature and systematic erasure channel codes are applied on certain carefully chosen source coding component the second approach is built on the quantization splitting technique which was previously proven useful in the gaussian ceo problem we first focus on a special case of the three description problem where any two descriptions are rate distortion optimal jointly referred to as the no two description excess rate case for this special case and the quadratic gaussian source we show that the two aforementioned approaches lead to rate distortion points outside the achievable region based on the source channel erasure codes previously proposed by pradhan puri and ramchandran interestingly though only the symmetric problem is considered in our work the proposed schemes in fact benefit from time sharing several asymmetric rate distortion points the insights gained through the no two description excess rate case lead to strategic combination of the new ingredients with the existing coding scheme yielding new coding schemes for the symmetric k description problem

a generalization of the formulas for intersection numbers of dual polar association schemes and their applications dual polar association schemes form an important family of association schemes whose intersection numbers were computed in wan et al studies in finite geometry and the construction of incomplete block designs science press beijing 1966 in chinese in this paper we generalize the formulas for the intersection numbers and introduce their applications to pooling designs cartesian authentication codes and vertex transitive graphs

association schemes based on maximal isotropic subspaces in singular classical spaces the paper provides the construction of association schemes on the sets of the maximal totally isotropic subspaces in singular classical spaces all intersection numbers of these schemes are computed

n channel symmetric multiple descriptions part i n k source channel erasure codes in this two part paper we present a new achievable rate region for the general n channel symmetric multiple descriptions problem in part i inspired by the concept of maximum distance separable mds erasure channel codes we consider a special case of this rate region where the source is encoded into n descriptions each with rate r these descriptions are transmitted over n bandwidth constrained and errorless channels during transmission a subset of these channels can break down thus erasing the corresponding descriptions the decoder is interested in recovering the source with the reception of at least k descriptions thus the encoder is allowed to sample only one realization of this breakdown process during the entire transmission for gaussian sources we have the following interesting result when any k descriptions arrive the achievable distortion exactly matches the optimal distortion rate performance corresponding to a source rate of kr bits with the reception of any m k descriptions the source reconstruction quality is strictly better the improvement being nearly linear in the number of descriptions received

high rate analysis of symmetric l channel multiple description coding this paper studies the tight rate distortion bound for l channel symmetric multiple description coding of a scalar gaussian source with two levels of receivers each of the first level receivers obtains of the l descriptions l the second level receiver obtains all l descriptions we find that if the central distortion corresponding to the second level receiver is much smaller than the side distortion corresponding to the first level receivers the product of a function of the side distortions and the central distortion is asymptotically independent of the redundancy between the descriptions using this property we analyze the asymptotic behavior of a practical multiple description lattice vector quantizer mdlvq our analysis includes the treatment of the mdlvq system from a new geometric viewpoint which results in an expression for the side distortions using the normalized second moment of a sphere of higher dimensionality than the quantization space the expression of the distortion product derived from the lower bound is then applied as a criterion to assess the performance loss of the considered mdlvq system in principle the efficiency of other practical md systems can also be evaluated using the derived distortion product