High accuracy semidefinite programming bounds for kissing numbers

Bachoc and Vallentin [1] develop a method (Section 2 recalls it) to find upper bounds for the kissing number based on semidefinite programming. Table 1 in Section 3, the main contribution of this paper, gives the values, i.e. the first ten significant digits, of these upper bounds for all dimensions 3, . . . , 24. In all cases they are the best known upper bounds. Dimension 5 is the first dimension in which the kissing number is not known. With our computation we could limit the range of possible values from {40, . . . , 45} to {40, . . . , 44}. In Section 4 we show that the high accuracy computations for the upper bounds in dimension 4 result into a question about a possible approach to prove the uniqueness of the kissing configuration in 4 dimensions.

Although acquiring the data for the table is a purely computational task we think that providing this table is valuable for several reasons: The kissing number is an important constant in geometry and results can depend on good upper bounds for it. For instance in Section 5 we show that there is no periodic point set in dimension 16 with average theta series

This proves a conjecture of Conway and Sloane [6,Chapter 7,page 190]. Furthermore, the actual computation of the table was very challenging. Bachoc and Vallentin [1] gave results for dimension 3, . . . , 10. However, they report on numerical difficulties which prevented them from extending their results. Now using new, more sophisticated high accuracy software and faster computers and more computation time we could overcome some of the numerical difficulties. Section 3 contains details about the computations.

In this section we set up the notation which is needed for our computation. For more information we refer to [1]. For natural numbers d and n ≥ 3 let s d (n) be the optimal value of the minimization problem

:

is positive semidefinite,

Here P n k is the normalized Jacobi polynomial of degree k with P n k (1) = 1 and parameters ((n -3)/2, (n -3)/2). In general, Jacobi polynomials with parameters (α, β) are orthogonal polynomials for the measure (1-u) α (1+u) β du on the interval [-1, 1]. Before we can define the matrices S n k we first define the entry (i, j) with i, j ≥ 0 of the (infinite) matrix Y n k containing polynomials in the variables u, v, w by

Then we get S n k by symmetrization:

, where σ runs through all permutations of the variables u, v, t which acts on the matrix coefficients in the obvious way. The polynomials p, p 1 , . . . , p 4 are given by

By A, B we denote the inner product between symmetric matrices trace(AB).

In [1] it is shown that this minimization problem is a semidefinite program and that every upper bound on s d (n) provides an upper bound for the kissing number in dimension n. Clearly, the numbers s d (n) form a monotonic decreasing sequence in d. Finding the solution of the semidefinite program defined in Section 2 is a computational challenge. It turns out that the major obstacle is numerical instability and not the problem size. When d is fixed, then the size of the input matrices stays constant with n; when n is fixed, then it grows rather moderately with d.

There is a number of available software packages for solving semidefinite programs. Mittelmann compares many existing packages in [10]. For our purpose first order, gradient-based methods such as SDPLR are far too inaccurate, and second order, primal-dual interior point methods are more suitable. Here increasingly ill-conditioned linear systems have to be solved even if the underlying problem is well-conditioned. This happens in the final phase of the algorithm when one approaches an optimal solution. Our problems are not well-conditioned and even the most robust solver SeDuMi which uses partial quadruple arithmetic in the final phase does not produce reliable results for d > 10.

We thus had to fall back on the implementation SDPA-GMP [8] which is much slower but much more accurate than other software packages because it uses the GNU Multiple Precision Arithmetic Library. We worked with 200-300 binary digits and relative stopping criteria of 10 -30 . The ten significant digits listed in the table are thus guaranteed to be correct. One problem was the convergence. Even with the control parameter settings recommended by the authors of SDPA-GMP for “slow but stable” computations, the algorithm failed to converge in several instances. However, we found parameter choices which worked for all cases: We varied the parameter la

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