N. J. A. Sloane

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We all recognize 0, 1, 1, 2, 3, 5, 8, 13, . . . but what about 1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, . . .? If you come across a number sequence and want to know if it has been studied before, there is only one place to look, the On-Line Encyclopedia of Integer Sequences (or OEIS). Now in its 49th year, the OEIS contains over 220,000 sequences and 20,000 new(More)
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. Manuscript received ;(More)
Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson , Kerdock, Preparata, Goethals, and Delsarte-Goethals . It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z4, the integers mod 4(More)
This paper addresses the question: how shouldN n-dimensional subspaces ofm-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(More)
A quantum error-correcting code is a way of encoding quantum states into qubits (two-state quantum systems) so that error or decoherence in a small number of individual qubits has little or no effect on the encoded data. The existence of quantum error-correcting codes was discovered only recently [1]. Although the subject is relatively new, a large number(More)
The problem of designing a multiple description vector quantizer with lattice codebook Λ 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 A2 and , i = 1, 2, 4, 8, that make use of this labeling(More)
For each of the lattices A,(n 2 I), D,,(n 2 2), EC, E,, E,, and their duals a very fast algorithm is given for finding the closest lattice point to an arbitrary point. If these lattices are used for vector quantizing of uniformly distributed data, the algorithm finds the min imum distortion lattice point. If the lattices are used as codes for a Gaussian(More)
A table of binary constant weight codes of length n <; 28 is presented. Explicit constructions are given for most of the 600 codes in the table; the majority of these codes are new. The known techniques for constructing constant weight codes are surveyed, and also a table is given of (unrestricted) binary codes of length J1 <; 28.
Ah&act-If a point is picked at random inside a regular simplex, octahedron, 600-cell, or other polytope, what is its average squared distance from the centroid? In n-dimensional space, what is the average squared distance of a random point from the closest point of the lattice A, (or D,, , En, A: or D,*)? The answers are given here, together with a(More)