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40
Quantum Error Correction Via Codes Over GF(4)
, 1997
"... The problem of finding quantumerrorcorrecting codes is transformed into the problem of finding additive codes over the field GF(4) which are selforthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on s ..."
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Cited by 304 (18 self)
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The problem of finding quantumerrorcorrecting codes is transformed into the problem of finding additive codes over the field GF(4) which are selforthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.
Multimode precoding for MIMO wireless systems
 IEEE TRANS. SIGNAL PROCESSING
, 2005
"... Multipleinput multipleoutput (MIMO) wireless systems obtain large diversity and capacity gains by employing multielement antenna arrays at both the transmitter and receiver. The theoretical performance benefits of MIMO systems, however, are irrelevant unless low error rate, spectrally efficient si ..."
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Cited by 35 (5 self)
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Multipleinput multipleoutput (MIMO) wireless systems obtain large diversity and capacity gains by employing multielement antenna arrays at both the transmitter and receiver. The theoretical performance benefits of MIMO systems, however, are irrelevant unless low error rate, spectrally efficient signaling techniques are found. This paper proposes a new method for designing high datarate spatial signals with low error rates. The basic idea is to use transmitter channel information to adaptively vary the transmission scheme for a fixed data rate. This adaptation is done by varying the number of substreams and the rate of each substream in a precoded spatial multiplexing system. We show that these substreams can be designed to obtain full diversity and full rate gain using feedback from the receiver to transmitter. We model the feedback using a limited feedback scenario where only finite sets, or codebooks, of possible precoding configurations are known to both the transmitter and receiver. Monte Carlo simulations show substantial performance gains over beamforming and spatial multiplexing.
A New Proof for the Existence of Mutually Unbiased Bases
, 2001
"... We develop a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. A necessary condition of the existence of mutually unbiased bases for any finite dimension is obtained. Then a constructive proof of the existence of mutually unbiased bases f ..."
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Cited by 30 (1 self)
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We develop a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. A necessary condition of the existence of mutually unbiased bases for any finite dimension is obtained. Then a constructive proof of the existence of mutually unbiased bases for dimensions that are powers of primes is presented. It is also proved that in any dimension d the number of mutually unbiased bases is at most d + 1. An explicit representation of mutually unbiased observables in terms of Pauli matrices are provided for d = 2 . 1
Linear programming bounds for codes in Grassmannian spaces
 IEEE Trans. Inf. Th
, 2006
"... ABSTRACT. We develop the linear programming method to obtain bounds for the cardinality of Grassmannian codes endowed with the chordal distance. We obtain a bound and its asymptotic version that generalize the wellknown bound for codes in the real projective space obtained by Kabatyanskiy and Leven ..."
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Cited by 27 (10 self)
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ABSTRACT. We develop the linear programming method to obtain bounds for the cardinality of Grassmannian codes endowed with the chordal distance. We obtain a bound and its asymptotic version that generalize the wellknown bound for codes in the real projective space obtained by Kabatyanskiy and Levenshtein. 1.
Robust dimension reduction, fusion frames, and Grassmannian packings,
 Appl. Comput. Harmon. Anal.
, 2009
"... Abstract We consider estimating a random vector from its measurements in a fusion frame, in presence of noise and subspace erasures. A fusion frame is a collection of subspaces, for which the sum of the projection operators onto the subspaces is bounded below and above by constant multiples of the ..."
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Cited by 24 (5 self)
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Abstract We consider estimating a random vector from its measurements in a fusion frame, in presence of noise and subspace erasures. A fusion frame is a collection of subspaces, for which the sum of the projection operators onto the subspaces is bounded below and above by constant multiples of the identity operator. We first consider the linear minimum meansquared error (LMSSE) estimation of the random vector of interest from its fusion frame measurements in the presence of additive white noise. Each fusion frame measurement is a vector whose elements are inner products of an orthogonal basis for a fusion frame subspace and the random vector of interest. We derive bounds on the meansquared error (MSE) and show that the MSE will achieve its lower bound if the fusion frame is tight. We then analyze the robustness of the constructed LMMSE estimator to erasures of the fusion frame subspaces. We limit our erasure analysis to the class of tight fusion frames and assume that all erasures are equally important. Under these assumptions, we prove that tight fusion frames consisting of equidimensional subspaces have maximum robustness (in the MSE sense) with respect to erasures of one subspace among all tight fusion frames, and that the optimal subspace dimension depends on signaltonoise ratio (SNR). We also prove that tight fusion frames consisting of equidimensional subspaces with equal pairwise chordal distances are most robust with respect to two and more subspace erasures, among the class of equidimensional tight fusion frames. We call such fusion frames equidistance tight fusion frames. We prove that the squared chordal distance between the subspaces in such fusion frames meets the socalled simplex bound, and thereby establish connections between equidistance tight fusion frames and optimal Grassmannian packings. Finally, we present several examples for the construction of equidistance tight fusion frames.
The Invariants of the Clifford Groups
, 2000
"... The automorphism group of the BarnesWall lattice Lm in dimension 2m (m ̸ = 3) is a subgroup of index 2 in a certain “Clifford group ” of structure 2 1+2m +.O+(2m, 2). This group and its complex analogue CIm of structure (2 1+2m + YZ8).Sp(2m, 2) have arisen in recent years in connection with the con ..."
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Cited by 23 (7 self)
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The automorphism group of the BarnesWall lattice Lm in dimension 2m (m ̸ = 3) is a subgroup of index 2 in a certain “Clifford group ” of structure 2 1+2m +.O+(2m, 2). This group and its complex analogue CIm of structure (2 1+2m + YZ8).Sp(2m, 2) have arisen in recent years in connection with the construction of orthogonal spreads, Kerdock sets, packings in Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs. In this paper we give a simpler proof of Runge’s 1996 result that the space of invariants for Cm of degree 2k is spanned by the complete weight enumerators of the codes C ⊗ F2m, where C ranges over all binary selfdual codes of length 2k; these are a basis if m ≥ k − 1. We also give new constructions for Lm and Cm: let M be the Z [ √ [ 2]lattice with Gram matrix
On QuasiOrthogonal Signatures for CDMA Systems
 IEEE Trans. Inform. Theory. [Online]. Available: http://www.ece.utexas.edu/˜rheath/papers/2002/quasicdma
, 2002
"... Codebooks constructed from Welch bound equality sequences are optimal in terms of sum capacity in synchronous CDMA systems. These optimal codebooks depend on both the sequence length as well as the number of active sequences. Thus to maintain optimality a typically different codebook is needed fo ..."
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Cited by 22 (6 self)
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Codebooks constructed from Welch bound equality sequences are optimal in terms of sum capacity in synchronous CDMA systems. These optimal codebooks depend on both the sequence length as well as the number of active sequences. Thus to maintain optimality a typically different codebook is needed for every possible number of active users. Further, all the sequences need to be reassigned as the number of active users changes. This paper describes and analyzes two promising subclasses of Welch bound equality sequences that have good properties when only a subset of sequences is active. One subclass, based on maximum Welch bound equality sequence sets, has equiangular sequences thus each user experiences the same amount of interference. The interference power depends only on the total number of active users. Another subclass, constructed by concatenating multiple orthonormal basis, comes closer to the Welch bound when not all signatures are active. Optimal unions of orthonormal basis are derived. Performance when subsets of sequences are active is characterized in terms of sumsquared correlation, average interference, and condition number of the Gram matrix.
Packing planes in four dimensions and other mysteries
 in Proceedings of the Conference on Algebraic Combinatorics and Related Topics
, 1997
"... How should you choose a good set of (say) 48 planes in four dimensions? More generally, how do you find packings in Grassmannian spaces? In this article I give a brief introduction to the work that I have been doing on this problem in collaboration with A. R. Calderbank, J. H. Conway, R. H. Hardin, ..."
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Cited by 10 (0 self)
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How should you choose a good set of (say) 48 planes in four dimensions? More generally, how do you find packings in Grassmannian spaces? In this article I give a brief introduction to the work that I have been doing on this problem in collaboration with A. R. Calderbank, J. H. Conway, R. H. Hardin, E. M. Rains and P. W. Shor. We have found many nice examples of specific packings (70 4spaces in 8space, for instance), several general constructions, and an embedding theorem which shows that a packing in Grassmannian space G(m,n) is a subset of a sphere in R D, D = (m + 2)(m − 1)/2, and leads to a proof that many of our packings are optimal. There are a number of interesting unsolved problems. 1.
Tight pfusion frames
"... Fusion frames enable signal decompositions into weighted linear subspace components. For positive integers p, we introduce pfusion frames, a sharpening of the notion of fusion frames. Tight pfusion frames are closely related to the classical notions of designs and cubature formulas in Grassmann s ..."
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Cited by 8 (4 self)
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Fusion frames enable signal decompositions into weighted linear subspace components. For positive integers p, we introduce pfusion frames, a sharpening of the notion of fusion frames. Tight pfusion frames are closely related to the classical notions of designs and cubature formulas in Grassmann spaces and are analyzed with methods from harmonic analysis in the Grassmannians. We define the pfusion frame potential, derive bounds for its value, and discuss the connections to tight pfusion frames.
A Simple Construction for the BarnesWall Lattices
, 2002
"... A certain family of orthogonal groups (called “Clifford groups” by G. E. Wall) has arisen in a variety of different contexts in recent years. These groups have a simple definition as the automorphism groups of certain generalized BarnesWall lattices. This leads to an especially simple construction ..."
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Cited by 8 (3 self)
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A certain family of orthogonal groups (called “Clifford groups” by G. E. Wall) has arisen in a variety of different contexts in recent years. These groups have a simple definition as the automorphism groups of certain generalized BarnesWall lattices. This leads to an especially simple construction for the usual BarnesWall lattices. {This is based on the third author’s talk at the ForneyFest, M.I.T., March 2000, which in turn is based on our paper “The Invariants of the Clifford Groups” Designs, Codes, Crypt., 24 (2001), 99–121, to which the reader is referred for further details and proofs.}