86 research outputs found
High accuracy semidefinite programming bounds for kissing numbers
The kissing number in n-dimensional Euclidean space is the maximal number of
non-overlapping unit spheres which simultaneously can touch a central unit
sphere. Bachoc and Vallentin developed a method to find upper bounds for the
kissing number based on semidefinite programming. This paper is a report on
high accuracy calculations of these upper bounds for n <= 24. The bound for n =
16 implies a conjecture of Conway and Sloane: There is no 16-dimensional
periodic point set with average theta series 1 + 7680q^3 + 4320q^4 + 276480q^5
+ 61440q^6 + ...Comment: 7 pages (v3) new numerical result in Section 4, to appear in
Experiment. Mat
Statistical Analysis of a Posteriori Channel and Noise Distribution Based on HARQ Feedback
In response to a comment on one of our manuscript, this work studies the
posterior channel and noise distributions conditioned on the NACKs and ACKs of
all previous transmissions in HARQ system with statistical approaches. Our main
result is that, unless the coherence interval (time or frequency) is large as
in block-fading assumption, the posterior distribution of the channel and noise
either remains almost identical to the prior distribution, or it mostly follows
the same class of distribution as the prior one. In the latter case, the
difference between the posterior and prior distribution can be modeled as some
parameter mismatch, which has little impact on certain type of applications.Comment: 15 pages, 2 figures, 4 table
Semidefinite code bounds based on quadruple distances
Let be the maximum number of words of length , any two
having Hamming distance at least . We prove , which implies
that the quadruply shortened Golay code is optimal. Moreover, we show
, , , ,
, , , ,
, , , ,
, , and .
The method is based on the positive semidefiniteness of matrices derived from
quadruples of words. This can be put as constraint in a semidefinite program,
whose optimum value is an upper bound for . The order of the matrices
involved is huge. However, the semidefinite program is highly symmetric, by
which its feasible region can be restricted to the algebra of matrices
invariant under this symmetry. By block diagonalizing this algebra, the order
of the matrices will be reduced so as to make the program solvable with
semidefinite programming software in the above range of values of and .Comment: 15 page
Optimization-based search for nordsieck methods of high order with quadratic stability polynomials
We describe the search for explicit general linear methods in Nordsieck form for which the stability function has only two nonzero roots. This search is based on state-of-the-art optimization software. Examples of methods found in this way are given for order p = 5, p = 6, and p = 7
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