This paper addresses the question: how should N n-dimensional subspaces of m-dimensional Euclidean space be arranged so that they are as far apart as possible? The results of extensive computations for modest values of N, n, m are described, as well as a reformulation of the problem that was suggested by these computations. The reformulation gives a way to describe n-dimensional subspaces of m-space as points on a sphere in dimension (m − 1)(m + 2)/2, which provides a (usually) lower-dimensional representation than the Plücker embedding, and leads to a proof that many of the new packings are optimal. The results have applications to the graphical display of multi-dimensional data via Asimov's "Grand Tour" method.

Evidence is presented to suggest that, in three dimensions, spherical 6-designs with N points exist for N = 24, 26, ≥ 28; 7-designs for N = 24, 30, 32, 34, ≥ 36; 8-

Every good quantum error-correcting code discovered thus far, such as those known as "stabilizer" or "additive" codes, has had the structure of an eigenspace of an Abelian group generated by tensor products of Pauli matrices. In this Letter we present the first example of a code that is better than any code of this type. It encodes six states in five qubits and can correct the erasure of any single qubit. [S0031-9007 (97)03628-4] PACS numbers: 89.70. + c, 03.65.Bz Most present models for quantum computers require quantum error-correcting codes [1-4] for their operations.

What is the tightest packing of N equal nonoverlapping spheres, in the sense of having minimal energy, i.e., smallest second moment about the centroid? The putatively optimal arrangements are described for N < 32. A number of new and interesting polyhedra arise.

By combining a modified version of Hooke and Jeeves' pattern search with exact or Monte Carlo moment calculations, it is possible to find I-, D-and A-optimal (or nearly optimal) designs for a wide range of response-surface problems. The algorithm routinely handles problems involving the minimization of functions of 1000 variables, and so for example can construct designs for a full quadratic response-surface depending on 12 continuous process variables. The algorithm handles continuous or discrete variables, linear equality or inequality constraints, and a response surface that is any low degree polynomial. The design may be required to include a specified set of points, so a sequence of designs can be obtained, each optimal given that the earlier runs have been made. The modeling region need not coincide with the measurement region. The algorithm has been implemented in a program called gosset, which has been used to compute extensive tables of designs. Many of these are more efficient than the best designs previously known.