17 research outputs found
New Upper Bounds for Some Spherical Codes
The maximal cardinality of a code W on the unit sphere in n dimensions
with (x, y) ≤ s whenever x, y ∈ W, x 6= y, is denoted by A(n, s). We use two
methods for obtaining new upper bounds on A(n, s) for some values of n and s.
We find new linear programming bounds by suitable polynomials of degrees which
are higher than the degrees of the previously known good polynomials due to
Levenshtein [11, 12]. Also we investigate the possibilities for attaining the Levenshtein
bounds [11, 12]. In such cases we find the distance distributions of the corresponding
feasible maximal spherical codes. Usually this leads to a contradiction showing
that such codes do not exist
Energy bounds for codes and designs in Hamming spaces
We obtain universal bounds on the energy of codes and for designs in Hamming
spaces. Our bounds hold for a large class of potential functions, allow unified
treatment, and can be viewed as a generalization of the Levenshtein bounds for
maximal codes.Comment: 25 page