Recently A. Schrijver derived new upper bounds for binary codes using semidefinite programming. In this paper we adapt this approach to codes on the unit sphere and we compute new upper bounds for the kissing number in several dimensions. In particular our computations give the (known) values for the cases
n
=
3
,
4
,
8
,
24
n = 3, 4, 8, 24
.
We study self-dual codes over certain finite rings which are quotients of quadratic imaginary fields or of totally definite quaternion fields over Q. A natural weight taking two different nonzero values is defined over these rings; using invariant theory, we give a basis for the space of invariants to which belongs the three variables weight enumerator of a self-dual code. A general bound for the weight of such codes is derived. We construct a number of extremal self-dual codes, which are the codes reaching this bound, and derive some extremal lattices of level l=2, 3, 7 and minimum 4, 6, 8. 1997 Academic Press
Summary. This chapter provides the reader with the necessary background for dealing with semidefinite programs which have symmetry. The basic theory is given and it is illustrated in applications from coding theory, combinatorics, geometry, and polynomial optimization.
Abstract. We develop the linear programming method in order to obtain bounds for the cardinality of Grassmannian codes endowed with the chordal distance. We obtain a bound and its asymptotic version that generalize the well-known bound for codes in the real projective space obtained by Kabatyanskiy and Levenshtein, and improve the Hamming bound for sufficiently large minimal distances.
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