J.J. Seidel

SPHERICAL CODES AND DESIGNS

Publisher Summary This chapter provides an overview of spherical codes and designs. A finite non-empty set X of unit vectors in Euclidean space R d has several characteristics, such as the dimension d ( X ) of the space spanned by X , its cardinality n = | X |, its degree s( X ), and its strength t ( X ).The chapter presents derivation of bounds for the cardinality of spherical A -codes in terms of the Gegenbauer coefficients of polynomials compatible with A . It also discusses spherical ( d , n , s , i)- configurations X . These are sets X of cardinality n on the unit sphere Ω d , which are spherical t -designs and spherical A -codes with I A I = s ; in other words, the strength t ( X ) is at least t and the degree s ( X ) is at most s . A condition is given for a spherical A -code to be a spherical t -design, in terms of the Gegenbauer coefficients of an annihilator of the set A . The chapter presents many examples of spherical ( d , n , s , t )-configurations; there exist tight spherical t-designs with t = 2, 3, 4, 5, 7, 11, and non-tight spherical ( 2s − 1)-designs. The constructions of these examples use sets of lines with few angles and association schemes, respectively.

Spherical codes and designs

Publisher Summary This chapter provides an overview of spherical codes and designs. A finite non-empty set X of unit vectors in Euclidean space R d has several characteristics, such as the dimension d ( X ) of the space spanned by X , its cardinality n = | X |, its degree s( X ), and its strength t ( X ).The chapter presents derivation of bounds for the cardinality of spherical A -codes in terms of the Gegenbauer coefficients of polynomials compatible with A . It also discusses spherical ( d , n , s , i)- configurations X . These are sets X of cardinality n on the unit sphere Ω d , which are spherical t -designs and spherical A -codes with I A I = s ; in other words, the strength t ( X ) is at least t and the degree s ( X ) is at most s . A condition is given for a spherical A -code to be a spherical t -design, in terms of the Gegenbauer coefficients of an annihilator of the set A . The chapter presents many examples of spherical ( d , n , s , t )-configurations; there exist tight spherical t-designs with t = 2, 3, 4, 5, 7, 11, and non-tight spherical ( 2s − 1)-designs. The constructions of these examples use sets of lines with few angles and association schemes, respectively.

Z4-kerdock codes, orthogonal spreads, and extremal euclidean line-sets

When

Equilateral point sets in elliptic geometry

m

Line graphs, root systems, and elliptic geometry

is odd, spreads in an orthogonal vector space of type

Bounds for systems of lines, and Jacobi polynomials

\Omega^+ (2m+2,2)

A SURVEY OF TWO-GRAPHS

are related to binary Kerdock codes and extremal line-sets in

Quadratic forms over GF(2)

\RR^{2^{m+1}}

The regular two-graph on 276 vertices

with prescribed angles. Spreads in a

MATHEMATICSEquilateral point sets in elliptic geometry

2m