The problem of finding quantum-error-correcting codes is transformed into the problem of finding additive codes over the field GF(4) which are self-orthogonal 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.

By using totally isotropic subspaces in an orthogonal space Ω + (2i,2), several infinite families of packings of 2 k-dimensional subspaces of real 2 i-dimensional space are constructed, some of which are shown to be optimal packings. A certain Clifford group underlies the construction and links this problem with Barnes-Wall lattices, Kerdock sets and quantumerror-correcting codes.

We derive the Varshamov--Gilbert and Hamming bounds for packings of spheres (codes) in the Grassmann manifolds over $\mathbb R$ and $\mathbb C$. The distance between two $k$-planes is defined as $\rho(p,q)=(\sin^2\theta_1 \dots \sin^2\theta_k)^{1/2}$, where $\theta_i, 1\le i\le k$, are the principal angles between $p$ and $q$.

Quantum theory has found a new field of applications in the realm of information and computation during the recent years. This paper reviews how quantum physics allows information coding in classically unexpected and subtle nonlocal ways, as well as information processing with an efficiency largely surpassing that of the present and foreseeable classical computers. Some outstanding aspects of classical and quantum information theory will be addressed here. Quantum teleportation, dense coding, and quantum cryptography are discussed as a few samples of the impact of quanta in the transmission of information. Quantum logic gates and quantum algorithms are also discussed as instances of the improvement in information processing by a quantum computer. We provide finally some examples of current experimental

The automorphism group of the Barnes-Wall 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 self-dual 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

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 4-spaces in 8-space, 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 \Gamma 1)=2, and leads to a proof that many of our packings are optimal. There are a number of interesting unsolved problems. fTo appear in the Proceedings of the Conference on Algebraic Combinatorics and Related Topics (Yamagata, Japan, Nov. 17--20, 1997).g 1. Introduction In my talk at the Yamagata conference on "Algebraic Combinatorics and Related Topics" (November 1997) I discussed two problems, (a) findin...

This paper considers MIMO transceivers with linear precoders and decision feedback equalizers (DFEs), with bit allocation at the transmitter. Zero-forcing (ZF) is assumed. Considered first is the minimization of transmitted power, for a given total bit rate and a specified set of error probabilities for the symbol streams. The precoder and DFE matrices are optimized jointly with bit allocation. It is shown that the generalized triangular decomposition (GTD) introduced by Jiang, Li, and Hager offers an optimal family of solutions. The optimal linear transceiver (which has a linear equalizer rather than a DFE) with optimal bit allocation is a member of this family. This shows formally that, under optimal bit allocation, linear and DFE transceivers achieve the same minimum power. The DFE transceiver using the geometric mean decomposition (GMD) is another member of this optimal family, and is such that optimal bit allocation yields identical bits for all symbol streams—no bit allocation is necessary—when the specified error probabilities are identical for all streams. The QR-based system used in VBLAST is yet another member of the optimal family and is particularly well-suited when limited feedback is allowed from receiver to transmitter. Two other optimization problems are then considered: a) minimization of power for specified set of bit rates and error probabilities (the QoS problem), and b) maximization of bit rate for fixed set of error probabilities and power. It is shown in both cases that the GTD yields an optimal family of solutions.

Several topics related to packing on a sphere are discussed: packing points, packing lines through the center, and packing k-flats in dimension d. Two frequently asked questions in the graphics community (e.g., on the Usenet newsgroup comp.graphics.algorithms) are: How can points be randomly placed on a sphere?, and, How can points be regularly arranged on a sphere? I will start with these two questions and move to generalizations: packing lines in space and packing affine subspaces in higherdimensional spaces. 1 Random Points on a Sphere Define a zone on a sphere to be the portion between two parallel planes that both intersect the sphere. For a sphere S of radius r, the area of a zone of width h is 2ßrh. This area is independent of where S is sliced, a fact that may seem countertuitive. The implication is that the z-coordinates (say) of random points on a sphere are uniformly distributed. Thus random points (x; y; z) on a sphere of radius 1 can be generated as follows: 1. Choose z ...

The problem of finding quantum-error-correcting codes is transformed into the problem of finding additive codes over the field GF(4) which are self-orthogonal 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.