2,382 research outputs found
McLaren's Improved Snub Cube and Other New Spherical Designs in Three Dimensions
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-designs for N=36, 40, 42, >= 44; 9-designs for N=48, 50, 52, >=
54; 10-designs for N=60, 62, >= 64; 11-designs for N=70, 72, >= 74; and
12-designs for N=84, >= 86. The existence of some of these designs is
established analytically, while others are given by very accurate numerical
coordinates. The 24-point 7-design was first found by McLaren in 1963, and --
although not identified as such by McLaren -- consists of the vertices of an
"improved" snub cube, obtained from Archimedes' regular snub cube (which is
only a 3-design) by slightly shrinking each square face and expanding each
triangular face. 5-designs with 23 and 25 points are presented which, taken
together with earlier work of Reznick, show that 5-designs exist for N=12, 16,
18, 20, >= 22. It is conjectured, albeit with decreasing confidence for t >= 9,
that these lists of t-designs are complete and that no others exist. One of the
constructions gives a sequence of putative spherical t-designs with N= 12m
points (m >= 2) where N = t^2/2 (1+o(1)) as t -> infinity.Comment: 16 pages, 1 figur
The Primary Pretenders
We call a composite number q such that there exists a positive integer b with
b^p == b (mod q) a prime pretender to base b. The least prime pretender to base
b is the primary pretender q_b. It is shown that there are only 132 distinct
primary pretenders, and that q_b is a periodic function of b whose period is
the 122-digit number
19568584333460072587245340037736278982017213829337604336734362-
294738647777395483196097971852999259921329236506842360439300.Comment: 7 page
Nonintersecting Subspaces Based on Finite Alphabets
Two subspaces of a vector space are here called ``nonintersecting'' if they
meet only in the zero vector. The following problem arises in the design of
noncoherent multiple-antenna communications systems. How many pairwise
nonintersecting M_t-dimensional subspaces of an m-dimensional vector space V
over a field F can be found, if the generator matrices for the subspaces may
contain only symbols from a given finite alphabet A subseteq F? The most
important case is when F is the field of complex numbers C; then M_t is the
number of antennas. If A = F = GF(q) it is shown that the number of
nonintersecting subspaces is at most (q^m-1)/(q^{M_t}-1), and that this bound
can be attained if and only if m is divisible by M_t. Furthermore these
subspaces remain nonintersecting when ``lifted'' to the complex field. Thus the
finite field case is essentially completely solved. In the case when F = C only
the case M_t=2 is considered. It is shown that if A is a PSK-configuration,
consisting of the 2^r complex roots of unity, the number of nonintersecting
planes is at least 2^{r(m-2)} and at most 2^{r(m-1)-1} (the lower bound may in
fact be the best that can be achieved).Comment: 14 page
Quantum Error Correction and Orthogonal Geometry
A group theoretic framework is introduced that simplifies the description of
known quantum error-correcting codes and greatly facilitates the construction
of new examples. Codes are given which map 3 qubits to 8 qubits correcting 1
error, 4 to 10 qubits correcting 1 error, 1 to 13 qubits correcting 2 errors,
and 1 to 29 qubits correcting 5 errors.Comment: RevTex, 4 pages, no figures, submitted to Phys. Rev. Letters. We have
changed the statement of Theorem 2 to correct it -- we now get worse rates
than we previously claimed for our quantum codes. Minor changes have been
made to the rest of the pape
Quantum Error Correction via Codes over GF(4)
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.Comment: Latex, 46 pages. To appear in IEEE Transactions on Information
Theory. Replaced Sept. 24, 1996, to correct a number of minor errors.
Replaced Sept. 10, 1997. The second section has been completely rewritten,
and should hopefully be much clearer. We have also added a new section
discussing the developments of the past year. Finally, we again corrected a
number of minor error
A linear construction for certain Kerdock and Preparata codes
The Nordstrom-Robinson, Kerdock, and (slightly modified) Pre\- parata codes
are shown to be linear over \ZZ_4, the integers . The Kerdock and
Preparata codes are duals over \ZZ_4, and the Nordstrom-Robinson code is
self-dual. All these codes are just extended cyclic codes over \ZZ_4. This
provides a simple definition for these codes and explains why their Hamming
weight distributions are dual to each other. First- and second-order
Reed-Muller codes are also linear codes over \ZZ_4, but Hamming codes in
general are not, nor is the Golay code.Comment: 5 page
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