on the construction of equiangular frames from graphs abstract we give details of the 1 1 correspondence between equiangular frames of n vectors for r d and graphs with n vertices this has been studied recently for tight equiangular frames because of applications to signal processing and quantum information theory the nontight examples given here which correspond to graphs with more than 2 eigenvalues have the potential for similar applications e g the frame corresponding to the 5 cycle graph is the unique grassmannian frame of 5 vectors in openr 3 further the associated canonical tight frames have a small number of angles in many cases

on decoding unequal error protection product codes the decoding of unequal error protection product codes which are a combination of linear unequal error protection uep codes and product codes is addressed a nonconstructive proof of the existence of a good error erasure decoding algorithm is presented however obtaining the decoding procedure is still an open research problem a particular subclass of uep product codes is considered including a decoding algorithm that is an extension of the blokh zyablov decoding algorithm for product codes for this particular subclass the decoding problem is solved

subelliptic spin c dirac operators ii basic estimates we assume that the manifold with boundary x has a spinc structure with spinor bundle s along the boundary this structure agrees with the structure defined by an infinite order integrable almost complex structure and the metric is kahler in this case the spinc dirac operator agrees with along the boundary the induced cr structure on bx is integrable and either strictly pseudoconvex or strictly pseudoconcave we assume that e x is a complex vector bundle which has an infinite order integrable complex structure along bx compatible with that defined along bx in this paper we use boundary layer methods to prove subelliptic estimates for the twisted spinc dirac operator acting on sections on s e we use boundary conditions that are modifications of the classical neumann condition these results are proved by using the extended heisenberg calculus

optical orthogonal codes with unequal auto and cross correlation constraints an optical orthogonal code ooc is a collection of binary sequences with good auto and cross correlation properties they were defined by salehi and others as a means of obtaining code division multiple access on optical networks up to now all work on oocs have assumed that the constraint placed on the autocorrelation and that placed on the cross correlation are the same we consider codes for which the two constraints are not equal specifically we develop bounds on the size of such oocs and demonstrate constriction techniques for building them the results demonstrate that a significant increase in the code size is possible by letting the autocorrelation constraint exceed the cross correlation constraint these results suggest that for a given performance requirement the optimal ooc may be one with unequal constraints this paper also views oocs with unequal auto and cross correlation constraints as constant weight unequal error protection uep codes with two levels of protection the bounds derived are interpreted from this viewpoint

an entropy maximization problem related to optical communication corresp motivated by a problem in optical communication we consider the general problem of maximizing the entropy of a stationary random process that is subject to an average transition cost constraint using a recent result of justesen and hoholdt we present an exact solution to the problem and suggest a class of finite state encoders that give a good approximation to the exact solution

operator valued frames we develop a natural generalization of vector valued frame theory we term operator valued frame theory using operator algebraic methods this extends work of the second author and d han which can be viewed as the mul tiplicity one case and extends to higher multiplicity their dilation approach we prove several results for operator valued frames concerning duality dis jointeness complementarity and composition of operator valued frames and the relationship between the two types of similarity left and right of such frames a key technical tool is the parametrization of parseval operator val ued frames in terms of a class of partial isometries in the hilbert space of the analysis operator we apply these notions to an analysis of multiframe gener ators for the action of a discrete group g on a hilbert space one of the main results of the han larson work was the parametrization of the parseval frame generators in terms of the unitary operators in the von neumann algebra gen erated by the group representation and the resulting norm path connectedness of the set of frame generators due to the connectedness of the group of unitary operators of an arbitrary von neumann algebra in this paper we general ize this multiplicity one result to operator valued frames however both the parameterization and the proof of norm path connectedness turn out to be necessarily more complicated and this is at least in part the rationale for this paper our parameterization involves a class of partial isometries of a different von neumann algebra these partial isometries are not path connected in the norm topology but only in the strong operator topology we prove that the set of operator frame generators is norm pathwise connected precisely when the von neumann algebra generated by the right representation of the group has no minimal projections as in the multiplicity one theory there are analogous results for general non parseval frames

the symmetry group of a finite frame abstract we define the symmetry group of a finite frame as a group of permutations on its index set this group is closely related to the symmetry group of vale and waldron 2005 12 for tight frames they are isomorphic when the frame is tight and has distinct vectors the symmetry group is the same for all similar frames in particular for a frame its dual and canonical tight frames it can easily be calculated from the gramian matrix of the canonical tight frame further a frame and its complementary frame have the same symmetry group we exploit this last property to construct and classify some classes of highly symmetric tight frames

equiangular tight frames and signature sets in groups abstract quiangular frames are an important class of finite dimensional frames because of their superior performance and numerous applications the objective of this paper is to present a new tool to construct equiangular tight frames from groups we prove that many equiangular tight frames arise from subsets of groups which we call signature sets subsequently we define quasi signature sets and examine real equiangular tight frames associated with these subsets of groups this approach yields further results and establishes new correspondences we extend these results to complex equiangular tight frames where the inner product between any pair of vectors is a common multiple of a cube root of unity and exhibit equiangular tight frames that arise from groups in this manner

morse inequalities for fourier components of kohn rossi cohomology of cr manifolds with s 1 action let x be a compact connected cr manifold of dimension 2n 1 n ge 2 with a transversal cr s 1 action on x we study the fourier components of the kohn rossi cohomology with respect to the s 1 action by studying the szego kernel of the fourier components we establish the morse inequalities on x using the morse inequalities we have established on x we prove that there are abundant cr functions on x when x is weakly pseudoconvex and strongly pseudoconvex at a point

construction of robust frames in erasure recovery abstract erasures occur often in signal transmission in 10 the authors introduced robust frames with which one can recover erased frame coefficients even if the locations are unknown in this paper we give some explicit construction of robust frames

burst erasures and the mean square error for cyclic parseval frames this paper investigates the performance of frames for the linear redundant encoding of vectors when consecutive frame coefficients are lost due to the occurrence of random burst errors we assume that the distribution of bursts is invariant under cyclic shifts and that the burst length statistics are known in analogy with rate distortion theory we wish to find frames of a given size which minimize the mean square reconstruction error for the encoding of vectors in a complex finite dimensional hilbert space we obtain an upper bound for the mean square reconstruction error for a given parseval frame and in the case of cyclic parseval frames we find a family of frames which minimizes this upper bound under certain conditions these minimizers are identical to complex bose chaudhuri hocquenghem codes discussed in the literature the accuracy of our upper bounds for the mean square error is substantiated by complementary lower bounds all estimates are based on convexity arguments and a discrete rearrangement inequality

improved upper bounds for codes with unequal error protection asymptotic nonexistence bounds for unequal error protecting codes with two protection levels are considered we show that the improved estimates on the possible distance distributions for codes may sometimes yield sharper upper bounds than the previously known ones on the higher significance protection level of both nonlinear and linear codes having two protection levels

strongly pseudoconvex cr structures over small balls part i an a priori estimate the purpose of the present series of papers is to develop the theory of harmonic integrals on strongly pseudoconvex cr structures over small balls along the line developed by d c spencer c b morrey j j kohn and l nirenberg 5 6 7 8 for strongly pseudoconvex complex manifolds we consider a strongly pseudoconvex cr structure on a manifold of real dimension 2n 1 in part i we establish the a priori estimate for the neumann boundary value problem on the complex associated with the structure in the case the structure is induced by an embedding in c and restricted to a small ball of special type provided i c q c n 3 where q is the degree of differential forms we consider in part ii we derive the regularity theorem of solutions of the neumann boundary value problem based on the a priori estimate of part i as an application we prove in part il that when n 5 the structure is realized on a neighborhood of a reference point by an embedding in cn we note here that l nirenberg gave an example of a strongly pseudoconvex cr structure which cannot be induced by such an embedding when n 2 9 l boutet de monvel proved that in the case of such structures over a compact manifold we can find such an embedding when n 3 1 there are two major differences between the cases of strongly pseudoconvex integrable almost complex structures and cr structures firstly the principal parts of the associated laplacians in cr structures are no longer elliptic secondly we have characteristic points on the boundary as a result when we follow over the predecessor s calculation in our case we end up with terms which may not be absorbed in the main non negative terms fortunately these leftover terms depend mostly on the levi form of the cr structure and the hessian form with respect to the cr structure of the equation of the boundary of our

optimal reconstruction systems for erasures and for the q potential abstract in this paper we introduce the q potential as an extension of the benedetto fickus frame potential defined on general reconstruction systems and show that protocols are the minimizers of this potential under certain restrictions we extend recent results of b g bodmann on the structure of optimal protocols with respect to 1 and 2 lost packets where the worst normalized reconstruction error is computed with respect to a compatible unitarily invariant norm we finally describe necessary and sufficient spectral conditions that we call q fundamental inequalities for the existence of protocols with prescribed properties by relating this problem to klyachko s and fulton s theory on sums of hermitian operators

robustness and surgery of frames abstract we characterize frames in r n that are robust to k erasures the characterization is given in terms of the support of the null space of the synthesis operator of the frame a necessary and sufficient condition is given for when an r k surgery on unit norm tight frames in r 2 are possible also a generalization of a known characterization of the existence of tight frames with prescribed norms is given

new binary linear block codes corresp using some known techniques several new binary linear codes are constructed i e codes with a greater minimum distance than any previously known binary code of the same length and dimension 8 a detailed description of one of the constructions is given

coding for the binary symmetric broadcast channel with two receivers block coding for the binary symmetric broadcast channel with two receivers is investigated a graph theoretic approach to the construction of a class of block codes with unequal error protection for two different sets of messages is presented a code in this class is a direct sum of two component codes each set of messages is encoded based on one component code the codes in this class are easy to implement decoding of these codes is presented and lower bounds on the achievable rates of these codes are derived the bounds are tighter than the katsman s bounds

estimates for derivatives of holomorphic functions in pseudoconvex domains in this paper we will prove several sobolev type inequalities for holomorphic functions in strictly pseudoconvex domains to a large extent this work was motivated by the following problem let d be a strictly pseudoconvex domain and let m be the intersection with d of a submanifold of a neighborhood o f which intersects 0d transversally let f be a holomorphic function on m lying in some l p class e g a hardy or bergman class we want to find a holomorphic extension of f to d satisfying an optimal l p estimate on d the case p c was solved by henkin 14 who showed that a bounded holomorphic function on m is necessarily the restriction of a bounded holomorphic function on d for certain weighted bergman spaces on m optimal l p estimates for extensions were obtained by cumenge 9 in the case l p o o and by beatrous 4 in the case 0 p c more precisely letting 6 z denote the distance from z to 0d the following result was proved let f be a holomorphic function on m satisfying

frame theoretic analysis of robust filter bank frames to quantization and erasures this paper presents the theoretic analysis for the robustness of filter bank fb frames in infinite dimensional hilbert space l2 bbz to quantization and erasures as well as studies the design of such robust frames from the perspective of both frame and fb theory first a characterization of the eigenstructure for the frame operator and the induced gram matrix of general fb frames is presented the robustness of fb frames to erasures is investigated in detail especially on the necessary and sufficient condition maximally robust frames are further analyzed by explicitly constructive methods moreover the stability and even the possible fir reconstruction of the subframe by the pseudoinverse are studied thoroughly next we examine the optimal quantized fb frames introducing a novel notion named as frame energy the universal optimality of tight fb frames to quantization is established in contrast to the optimality of only equal norm tight frames shown in previous works the added design freedom obtained by removing the equal norm constraint is explained and illustrated with examples finally the effect of erasures on the quantized fb frames is studied by incorporating the probability of erasures which shows the universal optimality of uniform tight fb frames for one erasure the optimal fb frame is usually not of equal norm and actually its norm distribution follows the reverse waterfilling principle an example of two state erasure channel model is shown to further explain our result followed by an analysis of the successive reconstruction which shows its potential application to frame based multiple description coding

on frames for krein spaces abstract a definition of frames for krein spaces is proposed which extends the notion of j orthonormal bases of krein spaces a j frame for a krein space h is in particular a frame for h in the hilbert space sense but it is also compatible with the indefinite inner product meaning that it determines a pair of maximal uniformly j definite subspaces an analogue to the maximal dual pair associated to a j orthonormal basis also each j frame induces an indefinite reconstruction formula for the vectors in h which resembles the one given by a j orthonormal basis

numerically and spectrally optimal dual frames in hilbert spaces abstract finding a best dual frame that minimizes the reconstruction errors when erasures occur is a deep rooted problem in frame theory the primary purpose of this paper is to introduce a new measurement for constructing optimal duals we consider the numerical radius of the error operator which has some advantages over the previous measures then we give several equivalent conditions for which the canonical dual and alternate duals are numerically optimal dual frames moreover we obtain some relations between optimal duals spectrally optimal duals and numerically optimal duals for erasures finally we survey the effect of operator perturbation on spectrally optimal duals this leads to finding some equivalent conditions under which the canonical dual is a spectrally optimal dual for any r erasures

non realizable cr structures the examples given by nirenberg in 5 and 6 show that not all three dimensional cr manifolds may be realized as real hypersurfaces in 1172 these examples are perturbations of the cr structure on the sphere in this paper we extend his construction and show that any strictly pseudo convex realizable cr structure of dimension 3 may be perturbed so as to obtain a non realizable one this is done in sect 1 in sect 2 we show non realizability for higher dimensional structures provided the levi form is non degenerate and has only one positive eigenvalue in our last section we summarize cartan s approach to cr geometry from this viewpoint it was important to decide if nirenberg s example was really limited to perturbations of the sphere we have made this paper self contained and have provided proofs of several known results for instance lemma 1 l and the first part of lemma 1 5 were proved by lewy 2 and lemma 1 2 is the well known observation that a strictly pseudo convex hypersurface is convex in some coordinate system

on the stability of equivariant embedding of compact cr manifolds with circle action we prove the stability of the equivariant embedding of compact strictly pseudoconvex cr manifolds with transversal cr circle action under circle invariant deformations of the cr structures

stability of the embeddability under perturbations of the cr structure for compact cr manifolds we study the stability of the embeddability of compact 2 concave cr manifolds in complex manifolds under small horizontal perturbations of the cr structure

cr structures with group action and extendability of cr functions part i presents results on local embedding of cr structures we consider an abstract cr manifold whose structure is invariant under a transversal lie group action we show that such a manifold can always be locally embedded in complex space as a generic submanifold the proof is based on selection of canonical coordinates and repeated use of the newlander nirenberg theorem 13 when the lie group is abelian the embedding can be given a particularly simple form let l 1 be the codimension of our submanifold called m throughout the paper it is then convenient to denote by n i the dimension of the ambient complex space and by zl z w 1 w t the complex coordinates we shall systematically write

subelliptic boundary conditions for spinc dirac operators gluing relative indices and tame fredholm pairs abstract let x be a spin manifold with boundary such that the spin structure is defined near the boundary by an almost complex structure which is either strictly pseudoconvex or pseudoconcave and hence contact using generalized szeg projectors we define modified neumann boundary conditions eo for spinors which lead to subelliptic fredholm boundary value problems for the spin dirac operator eo to study the index of these boundary value problems we introduce a generalization of fredholm pairs to the tame category in this context we show that the index of the graph closure of eo eo equals the tame relative index on the boundary between eo and the calderon projector let x0 and x1 be strictly pseudoconvex spin manifolds as above let bx1 bx0 be a contact diffeomorphism s0 s1 denote generalized szeg projectors on bx0 bx1 respectively and 0eo 1eo the subelliptic boundary conditions they define if x1 is the manifold x1 with its orientation reversed then the glued manifold x x0 x1 has a canonical spin structure and dirac operator xeo applying these results we obtain a formula for the relative index r ind s0 s1 as a special case this formula verifies a conjecture of atiyah and weinstein 1997 rims kokyuroku 1014 1 14 for the index of the quantization of a contact transformation between cosphere bundles

frames graphs and erasures two uniform frames and their use for the coding of vectors are the main subject of this paper these frames are known to be optimal for handling up to two erasures in the sense that they minimize the largest possible error when up to two frame coefficients are set to zero here we consider various numerical measures for the reconstruction error associated with a frame when an arbitrary number of the frame coefficients of a vector are lost we derive general error bounds for two uniform frames when more than two erasures occur and apply these to concrete examples we show that among the 227 known equivalence classes of two uniform 36 15 frames arising from hadamard matrices there are 5 that give smallest error bounds for up to 8 erasures

a large automorphism group decreases the number of computations in the construction of an optimal encoder decoder pair for linear block code for a linear block code it is shown how the number of computations needed for the determination of an optimal encoder decoder pair can be reduced by using the code s automorphism group furthermore it is shown that the use of an unequal error protection optical generator matrix and a minimum weight coset leader decoder is suboptimal for a q ary symmetry channel but their determination needs less computational effort

probability modelled optimal frames for erasures abstract we establish a probability model for constructing optimal parseval frames when erasures occur during the transmission process of the frame coefficient data set such frames are called probability modelled pm optimal frames for erasures while pm optimal frames exist for all erasures it is usually difficult to construct them we characterize all the pm optimal frames for one and two erasures and propose an algorithm to construct these frames examples are given to demonstrate the construction and to compare the decoding reconstruction effects when both pm optimal parseval frames and uniform length parseval frames are used for encoding

optimal frames for erasures we study frames from the viewpoint of coding theory we introduce a numerical measure of how well a frame reconstructs vectors when some of the frame coefficients of a vector are lost and then attempt to find and classify the frames that are optimal in this setting

duality in reconstruction systems we consider the notion of finite dimensional reconstructions systems rs s which includes the fusion frames as projective rs s we study erasures some geometrical properties of these spaces the spectral picture of the set of all dual systems of a fixed rs the spectral picture of the set of rs operators for the projective systems with fixed weights and the structure of the minimizers of the joint potential in this setting we give several examples

binary parseval frames from group orbits abstract binary parseval frames share many structural properties with real and complex ones on the other hand there are subtle differences for example that the gramian of a binary parseval frame is characterized as a symmetric idempotent whose range contains at least one odd vector here we study binary parseval frames obtained from the orbit of a vector under a group representation in short binary parseval group frames in this case the gramian of the frame is in the algebra generated by the right regular representation we identify equivalence classes of such parseval frames with binary functions on the group that satisfy a convolution identity this allows us to find structural constraints for such frames we use these constraints to catalogue equivalence classes of binary parseval frames obtained from group representations as an application we study the performance of binary parseval frames generated with abelian groups for purposes of error correction we show that if p is an odd prime then the group z p q is always preferable to z p q when searching for best performing codes associated with binary parseval group frames

computer search for binary cyclic uep codes of odd length up to 65 exhaustive computation by a computer was used to find the unequal error protection capabilities of all binary cyclic codes of odd length up to 65 that have minimum distances of at least 3 for those codes for which upper bounds can only be computed on their unequal error protection capabilities an analytic method developed by v n dynkin and v a togonidze 1976 is used to show that the upper bounds meet the exact unequal error protection capabilities

cyclic unequal error protection codes constructed from cyclic codes of composite length the unequal error correction capabilities of binary cyclic codes of composite length are investigated under certain conditions direct sums of concatenated codes have unequal error correction capabilities by a modified hartmann and tzeng 1973 algorithm it is shown that a binary cyclic code of composite length is equivalent to the direct sum of concatenated codes with this some binary cyclic unequal error protection uep codes are constructed finally the authors present a class of two level uep cyclic direct sum codes which provide error correction capabilities higher than those guaranteed by the blokh zyablov 1974 constructions

subelliptic mathrm spin _ mathbb c dirac operators i let x be a compact kahler manifold with strictly pseudoconvex bound ary y in this setting the spinc dirac operator is canonically identified with c x 0 e c x 0 o we consider modifications of the classi cal neumann conditions that define fredholm problems for the spinc dirac operator in part 2 7 we use boundary layer methods to obtain subelliptic estimates for these boundary value problems using these results we obtain an expression for the finite part of the holomorphic euler characteristic of a strictly pseudoconvex manifold as the index of a spinc dirac operator with a subellip tic boundary condition we also prove an analogue of the agranovich dynin formula expressing the change in the index in terms of a relative index on the boundary if x is a complex manifold partitioned by a strictly pseudoconvex hypersurface then we obtain formulae for the holomorphic euler characteristic of x as sums of indices of spinc dirac operators on the components this is a subelliptic analogue of bojarski s formula in the elliptic case

causal compensation for erasures in frame representations in a variety of signal processing and communications contexts erasures occur inadvertently or can be intentionally introduced as part of a data reduction strategy this paper discusses causal compensation for erasures in frame representations of signals the approach described assumes linear synthesis of the signal using a prespecified frame but no specific generation mechanism for the coefficients under this assumption it is demonstrated that erasures can be compensated for using low complexity causal systems if the transmitter is aware of the occurrence of the erasure an optimal compensation is to project the erasure error to the remaining coefficients it is demonstrated that the same compensation can be executed using a transmitter receiver combination in which the transmitter is not aware of the erasure occurrence the transmitter precompensates using projections as if assuming erasures will occur the receiver undoes the compensation for the coefficients that have not been erased thus maintaining the compensation only of the erased coefficients the stability of the resulting systems is explored and stability conditions are derived it is shown that stability for any erasure pattern can be enforced by optimizing a constrained quadratic program at the system design stage the paper concludes with examples and simulations that verify the theoretical results and illustrate key issues in the algorithms

signal reconstruction from the magnitude of subspace components we consider signal reconstruction from the norms of subspace components generalizing standard phase retrieval problems in the deterministic setting a closed reconstruction formula is derived when the subspaces satisfy certain cubature conditions that require at least a quadratic number of subspaces moreover we address reconstruction under the erasure of a subset of the norms using the concepts of p fusion frames and list decoding we propose an algorithm that outputs a finite list of candidate signals one of which is the correct one in the random setting we show that a set of subspaces chosen at random and of cardinality scaling linearly in the ambient dimension allows for exact reconstruction with high probability by solving the feasibility problem of a semidefinite program

linearly connected sequences and spectrally optimal dual frames for erasures abstract in the case that a frame is prescribed for applications and erasures occur in the process of data transmissions we examine optimal dual frames for the recovery from single erasures in contrast to earlier papers we consider the spectral radius of the error operator instead of its operator norm as a measure of optimality this notion of optimality is natural when the neumann series is used to recover the original data in an iterative manner we obtain a complete characterization of spectrally one erasure optimal dual frames in terms of the redundancy distribution of the prescribed frame our characterization relies on the connection between erasure optimal frames and the linear connectivity property of the frame we prove that the linear connectivity property is equivalent to the intersection dependent property and is also closely related to the well known concept of a k independent set additionally we also establish several necessary and sufficient conditions for the existence of an alternate dual frame to make the iterative reconstruction work

on primitive bch codes with unequal error protection capabilities presents a class of binary primitive bch codes that have unequal error protection uep capabilities the authors use a previous result on the span of their minimum weight vectors to show that binary primitive bch codes containing second order punctured reed muller rm codes of the same minimum distance are binary cyclic uep codes the values of the error correction levels for this class of binary luep codes are estimated

constructing linear unequal error protection codes from algebraic curves we show that the concept of generalized algebraic geometry codes which was introduced by xing niederreiter and lam see ibid vol 45 p 2498 2501 nov 1999 gives a natural framework for constructing linear unequal error protection codes

decoding binary two error correcting cyclic codes with zech logarithms a decoding method for binary two error correcting cyclic codes whose generator polynomials have at most two irreducible factors is presented this class includes binary narrow sense bch codes with designed distance 5 the decoding algorithm uses the zech logarithm for the finite field in which the roots of the code lie

qpsk block modulation codes for unequal error protection unequal error protection uep codes find applications in broadcast channels as well as in other digital communication systems where messages have different degrees of importance binary linear uep luep codes combined with a gray mapped qpsk signal set are used to obtain new efficient qpsk block modulation codes for unequal error protection several examples of qpsk modulation codes that have the same minimum squared euclidean distance as the best qpsk modulation codes of the same rate and length are given in the new constructions of qpsk block modulation codes even length binary luep codes are used good even length binary luep codes are obtained when shorter binary linear codes are combined using either the well known u u v construction or the so called construction x both constructions have the advantage of resulting in optimal or near optimal binary luep codes of short to moderate lengths using very simple linear codes and may be used as constituent codes in the new constructions luep codes lend themselves quite naturally to multistage decoding up to their minimum distance using the decoding of component subcodes a new suboptimal two stage soft decision decoding of luep codes is presented and its application to qpsk block modulation codes for uep illustrated

nonlinear unequal error protection codes are sometimes better than linear ones the parameters of some binary nonlinear two level unequal error protection uep codes are evaluated and compared with nonexistence bounds for linear uep codes the major points is that for uep codes linearity is an essential restriction the results demonstrate that at least in certain cases nonlinear codes perform better than linear ones this is in contrast to the situation for ordinary codes when in most cases the best possible result can be obtained even with the restriction of linearity

achieving the orthoplex bound and constructing weighted complex projective 2 designs with singer sets abstract equiangular tight frames are examples of grassmannian line packings for a hilbert space more specifically according to a bound by welch they are minimizers for the maximal magnitude occurring among the inner products of all pairs of vectors in a unit norm frame this paper is dedicated to packings in the regime in which the number of frame vectors precludes the existence of equiangular frames the orthoplex bound then serves as an alternative to infer a geometric structure of optimal designs we construct frames of unit norm vectors in k dimensional complex hilbert spaces that achieve the orthoplex bound when k 1 is a prime power we obtain a tight frame with k 2 1 vectors and when k is a prime power with k 2 k 1 vectors in addition we show that these frames form weighted complex projective 2 designs that are useful additions to maximal equiangular tight frames and maximal sets of mutually unbiased bases in quantum state tomography our construction is based on singer s family of difference sets and the related concept of relative difference sets

optimal symbol error rate encoding corresp it is shown that in a digital communication system that uses an n k linear code the symbol error rate measured on the information symbols is a function of the encoder g used for the code an algorithm is proposed identical in nature to that encountered in unequal error protection codes for finding an optimal linear encoder g 0 that minimizes the symbol error rate for a fixed complete standard array decoder

the pythagorean theorem i the finite case the pythagorean theorem and variants of it are studied the variations evolve to a formulation in terms of noncommutative conditional expectations on von neumann algebras that displays the theorem as the basic result of noncommutative metric euclidean geometry the emphasis in the present article is finite dimensionality both discrete and continuous

statistical mechanics of low density parity check codes we review recent theoretical progress on the statistical mechanics of error correcting codes focusing on low density parity check ldpc codes in general and on gallager and mackay neal codes in particular by exploiting the relation between ldpc codes and ising spin systems with multispin interactions one can carry out a statistical mechanics based analysis that determines the practical and theoretical limitations of various code constructions corresponding to dynamical and thermodynamical transitions respectively as well as the behaviour of error exponents averaged over the corresponding code ensemble as a function of channel noise we also contrast the results obtained using methods of statistical mechanics with those derived in the information theory literature and show how these methods can be generalized to include other channel types and related communication problems

extreme points and identification of optimal alternate dual frames abstract the main purpose of this paper is to identify the set of optimal dual frames under erasures to this end we characterize extreme points in the set of all optimal duals for 1 erasure also we give some conditions under which an alternate dual frame is either not optimal or is a non unique optimal dual moreover we obtain a new characterization of frames when the canonical dual is the unique optimal dual frame for erasures

reconstruction of signals from frame coefficients with erasures at unknown locations we propose new approaches to the problems of recovering signals from the rearranged frame coefficients or frame coefficients with erasures at either known or unknown locations these problems naturally arise from applications where the encoded information needs to be transmitted for example in signal image processing information and coding theory and communications we show that with the appropriate choices of the frames that are used for encoding the signal with erasures occurring at known locations can be easily recovered without inverting the sub frame operators each time our new easy to implement and cost efficient algorithm provides perfect reconstruction of the original signal to address the problem of recovering erased coefficients from unknown locations we propose to use a class of frames that are almost robust with respect to m erasures we prove that every frame with uniform excess can be rescaled to an almost robust frame and the locations of erased data can be perfectly recovered for almost all the signals similar results are obtained for recovering the original order of a disordered rearranged set of frame coefficients numerical examples are presented to test the main results whenever the received data are noise free we can recover the original signal exactly from frame coefficients with erasures at unknown locations or from a disordered set of frame coefficients

binary coding by integration of polynomials based on the idea of integration and differentiation of polynomials a large class of linear unequal error protection luep codes is constructed many of these codes are optimal a codeword is generated by binary discrete integration of an all zero vector using the information bits as integration constants decoding is performed by discrete differentiation of the received word for special designs all information bits are equally protected and in this case all classes of reed muller codes are obtained thus a new and very comprehensive description of these codes is given the described codes are decoded by majority decisions over corresponding derivatives based on the structure of pascal s triangle reduced modulo 2

on the existence of equiangular tight frames abstract an equiangular tight frame etf is a d n matrix that has unit norm columns and orthogonal rows of norm n d its key property is that the absolute inner products between pairs of columns are i identical and ii as small as possible etfs have applications in communications coding theory and sparse approximation numerical experiments indicate that etfs arise for very few pairs d n and it is an important challenge to develop restrictions on the pairs for which they can exist this article uses field theory to provide detailed conditions on real and complex etfs in particular it describes restrictions on harmonic etfs a specific type of complex etf that appears in applications finally the article offers empirical evidence that these conditions are sharp or nearly sharp especially in the real case

complex equiangular tight frames and erasures in this paper we demonstrate that there are distinct dierences between real and complex equiangular tight frames etfs with regards to erasures for example we prove that there exist arbitrarily large non trivial complex equiangular tight frames which are robust against three erasures and that such frames come from a unique class of complex etfs in addition we extend certain results in 1 to complex vector spaces as well as show that other results regarding real etfs are not valid for complex etfs

constructive codes with unequal error protection an asymptotic lower bound on the rates of two level binary codes with unequal error protection is presented the bound is based on a class of codes that are constructive in the sense that the description complexity is polynomial in the codeword length in some cases the bound exceeds an upper bound for linear codes proving the existence of constructive nonlinear uep codes that are better than any linear codes

extension of the bch decoding algorithm to decode binary cyclic codes up to their maximum error correction capacities the bch algorithm is extended to correct more errors than indicated by the bch bound in the first step of the decoding procedure a number of errors corresponding to a particular case of the hartmann tzeng bound are corrected in the second step full error correction is the goal a measure for the worst case number of field elements of an extension field gf 2 sup m that must be tested for this purpose is given for binary cyclic linear unequal error protection codes as well as for conventional binary cyclic codes

moduli for pointed convex domains a moduli space for the class of pointed strictly linearly convex domains in n is obtained it is shown that the space of pointed smoothly bounded strictly linearly convex domains with a fixed indicatrix is parameterized by a class of deformations of the cr structure of the boundary of the indicatrix these deformations are constructed by using the circular representation of a domain to pull back its complex structure tensor to the indicatrix a careful study of the pull back structure shows that the allowable deformations are parameterized by a class of complex hamiltonian vector fields the proof of this fact is based on the folland stein estimates for the bar partial b complex of the boundary of the indicatrix

steiner equiangular tight frames abstract we provide a new method for constructing equiangular tight frames etfs the construction is valid in both the real and complex settings and shows that many of the few previously known examples of etfs are but the first representatives of infinite families of such frames it provides great freedom in terms of the frame s size and redundancy this method also explicitly constructs the frame vectors in their native domain as opposed to implicitly defining them via their gram matrix moreover in this domain the frame vectors are very sparse the construction is extremely simple a tensor like combination of a steiner system and a regular simplex this simplicity permits us to resolve an open question regarding etfs and the restricted isometry property rip we show that the rip behavior of some etfs is unfortunately no better than their coherence indicates

the road to equal norm parseval frames abstract the construction of equal norm parseval frames is fundamental for many applications of frame theory we present a construction method based on a system of ordinary differential equations which generates a flow on the set of parseval frames that converges to equal norm parseval frames we developed this method to address a question posed by vern paulsen how close is a nearly equal norm nearly parseval frame to an equal norm parseval frame the distance estimate derived here can be used to substantiate numerically found approximate constructions of equal norm parseval frames the estimate is valid for a fairly general class of frames requiring that the dimension of the hilbert space and the number of frame vectors is relatively prime in addition we re phrase our distance estimate to show that certain projection matrices which are nearly constant on the diagonal are close in hilbert schmidt norm to ones which have a constant diagonal

complex equiangular cyclic frames and erasures we derive various interesting properties of complex equiangular cyclic frames for many pairs n k using gauss sums and number theory we further use these results to study the random and burst errors of some special classes of complex equiangular cyclic n k frames

on an infinite series of 4n 2n binary codes corresp this correspondence deals with an infinite series of binary reversible 4n 2n 4 n geq 2 unequal error protection codes which are majority logic decodable the weight enumerators and automorphism groups are determined completely for n even the codes are self dual by pasting together copies of a 4n 2n 4 code binary codes with parameters 8n 2n 8 for n geq 2 8n 2n 12 for n geq 4 and 12n 2n 16 and 16n 2n 24 for n geq 3 are obtained

embeddability for three dimensional cr manifolds a question of principal interest in the theory of compact three dimensional cr manifolds is to understand when a given strictly pseudoconvex cr structure can be realized by an embedding in c we call a structure for which this is possible embeddable and nonembeddable otherwise this question is only of interest in three dimensions as a theorem of boutet de monvel states that any strictly pseudoconvex cr structure on a compact 2d 1 manifold is realizable as an embedding in some c provided d 1 on the other hand there is a classical example of rossi which shows that an arbitrarily small real analytic perturbation of the standard structure on the three sphere may fail to be embeddable ros later nirenberg constructed w perturbations for which the associated ob equation fails to have even local solutions n in fact this situation is generic in the f topology it is now understood principally from the work of eliashberg that cr structures with

encoding and decoding for the minimization of message symbol error rates in linear block codes given any fixed linear block code the error rates for the message symbols depend both on the encoding function and on the decoding map this research shows how to optimize the choice of a generator matrix and decoding map simultaneously to minimize the error rates for all message symbols the model used assumes that the distribution of messages is flat and that the distribution of error vectors defining the channel is independent of the message transmitted in addition it is shown that with proper choice of coset leaders standard array decoding is optimal in this circumstance the results generalize previously known results on unequal error protection and are sufficiently general to apply when a code is used for error detection only

equiangular tight frames and fourth root seidel matrices in this paper we construct complex equiangular tight frames etfs in particular we study the grammian associated with an etf whose off diagonal entries consist entirely of fourth roots of unity these etfs are classified and we also provide some computational techniques which give rise to previously undiscovered etfs

on unequal error protection of convolutional codes from an algebraic perspective in this paper convolutional codes are studied for unequal error protection uep from an algebraic theoretical viewpoint we first show that for every convolutional code there exists at least one optimal generator matrix with respect to uep the uep optimality of convolutional encoders is then combined with several algebraic properties e g systematic basic canonical and minimal to establish the fundamentals of convolutional codes for uep in addition a generic lower bound on the length of a uep convolutional code is proposed good uep codes with their lengths equal to the derived lower bound are obtained by computer search

on three dimensional cauchy riemann manifolds of the almost complex tensor of c2 multiplication by i the collection of these subspaces hp m forms a bundle hm c tm called the horizontal or contact bundle indeed strict pseudoconvexity of m implies that the plane field hpm is nondegenerate i e defines a contact structure the almost complex tensor of c2 then restricts to a bundle endomorphism j hm hm such that j2 id the bundle hm c tm together with this endomorphism j defines the cr structure of m slightly more generally let d d u m be a four dimensional compact manifold with boundary m interior d endowed with a smooth almost complex tensor which is integrable on d if d is strictly pseudoconvex i e in a neighborhood of any p e m there is a smooth strictly plurisubharmonic function u negative on d 0 on m but du 0 on m then m inherits a strictly pseudoconvex cr structure from d in this case we shall say that m bounds a strictly pseudoconvex surface this leads us to the abstract definition a strictly pseudoconvex threedimensional cr manifold is a compact manifold m without boundary dim m 3 endowed with a contact structure hm hpm p e m c tm and an endomorphism j of hm such that j2 id throughout this paper unless otherwise stated we shall be working with infinitely differentiable objects thus m hm j are assumed to be smooth in this sense furthermore a differentiable function f m c is a cr function if for any p e m the restriction df ihm hpm tf p c

two topics on linear unequal error protection codes bounds on their length and cyclic code classes it is possible for a linear block code to provide more protection for selected positions in the input message words than is guaranteed by the minimum distance of the code linear codes having this property are called linear unequal error protection luep codes bounds on the length of a luep code that ensures a given unequal error protection are derived a majority decoding method for certain classes of cyclic binary uep codes is treated a list of short i e of length less than 16 binary luep codes of optimal i e minimal length and a list of all cyclic binary uep codes of length less than 40 are included

optimal dual frames for erasures for any given frame for the purpose of encoding in a finite dimensional hilbert space we investigate its dual frames that are optimal for erasures for the purpose of decoding we show that in general the canonical dual is not necessarily optimal moreover optimal dual frames are not necessarily unique we present some sufficient conditions under which the canonical dual is the unique optimal dual frame for the erasure problem as an application we get that the canonical dual is the only optimal dual when either the frame is induced by a group representation or the frame is uniform tight

frames of uniform subframe bounds with applications to erasures abstract we introduce a class of frames called usb frames frames of uniform subframe bounds which mean that any subsequence of the frame is a frame for the subspace it spans and the frame bounds depend only on the cardinality of the subsequence we give several necessary and sufficient conditions for a frame to be a usb frame as an application we show that a tight usb frame is optimal for erasures in the sense that the reconstruction error is independent of the position of erasures we give several explicit construction of usb frames

unequal error protection an information theoretic perspective an information theoretic framework for unequal error protection is developed in terms of the exponential error bounds the fundamental difference between the bit wise and message wise unequal error protection uep is demonstrated for fixed length block codes on discrete memoryless channels dmcs without feedback effect of feedback is investigated via variable length block codes it is shown that feedback results in a significant improvement in both bit wise and message wise ueps except the single message case for missed detection the distinction between false alarm and missed detection formalizations for message wise uep is also considered all results presented are at rates close to capacity

on the singularities of the szegő projections on lower energy forms let x be an abstract not necessarily compact orientable cr manifold of dimension 2n 1 n geqslant 2 let box q b be the gaffney extension of kohn laplacian on 0 q forms we show that the spectral function of box q b admits a full asymptotic expansion on the non degenerate part of the levi form as a corollary we deduce that if x is compact and the levi form is non degenerate of constant signature on x then the spectrum of box q b in 0 infty consists of point eigenvalues of finite multiplicity moreover we show that a certain microlocal conjugation of the associated szeg kernel admits an asymptotic expansion under a local closed range condition as applications we establish the szeg kernel asymptotic expansions on some weakly pseudoconvex cr manifolds and on cr manifolds with transversal cr s 1 actions by using these asymptotics we establish some local embedding theorems on cr manifolds and we give an analytic proof of a theorem of lempert asserting that a compact strictly pseudoconvex cr manifold of dimension three with a transversal cr s 1 action can be cr embedded into mathbb c n for some n in mathbb n

optimal dual frames for communication coding with probabilistic erasures assume that a frame is preselected for encoding in a communication system we investigate the optimal dual frames for signal reconstruction decoding which minimize the maximal error when the probabilistic erasures occur in the transmission process from the perspective of mathematical theory of frames we set up a probability model under which we define the probability optimal po dual frames for a given frame when the frame based coding involves probabilistic erasures we obtain a sufficient and necessary condition under which the canonical dual frame is the unique po dual frame additionally we derive some general sufficient conditions for which the canonical dual frame is either not optimal or it is optimal but not the unique optimal one we present two simulation examples to compare the reconstruction effects when both the po dual frames and the general optimal go dual frames are used for reconstruction