J.M. Goethals

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.

Line graphs, root systems, and elliptic geometry

Publisher Summary The chapter discusses star-closed sets of lines at 60 ° and 90 ° , leading to a theorem that leaves only a restricted number of possibilities, of a specific structure. These possibilities are realized by the root systems A n , D n , E 8 , E 7 , E 6 defined in terms of lines. The chapter discusses the relations to the official Root Systems, as they occur in geometry and algebra. The chapter also discusses the graphs represented by subsets of the root systems: line graphs of complete bipartite graphs for A n , Hoffmans generalized line graphs for D n , and various exceptional graphs for E 8 . The chapter also presents the application to Hadamard matrices.

Bounds for systems of lines, and Jacobi polynomials

Bounds are obtained for the cardinality of sets of lines having a prescribed number of angles, both in real and in complex Euclidean n-space. Extremal sets provide combinatorial configurations with a particular algebraic structure, such as association schemes and regular two-graphs. The bounds are derived by use of matrix techniques and the addition formula for Jacobi polynomials.

The regular two-graph on 276 vertices

There is a unique regular two-graph on 276 vertices. This provides a characterization of Conways group ... 3. The proof is based on 276 ~ 3 x 11 + 3^5, and uses the ternary Golay code. The paper contains a list of the known strongly regular graphs with the eigenvalue -5.

A skew Hadamard matrix of order 36

Publisher Summary nThis chapter provides an overview of a skew Hadamard matrix of order 36. Hadamard matrices exist for infinitely many orders 4m, m ≥1, m integer, including all 4m < 100. They are conjectured to exist for all such orders. Skew Hadamard matrices have been constructed for all orders 4m < 100, except for 36, 52, 76, and 92. The unit matrix of any order is denoted by me. The square matrices Q and R of order m are defined by their only nonzero elements. Any square matrix H of order 4m is skew Hadamard if its elements are 1 and-1 and HHT = 4ml, H +HT= 2I.

Nearly perfect binary codes

A class of binary codes, satisfying the equality in a specialized version of the Johnson bound, is introduced, which contains perfect codes and the Preparata 2-error correcting codes. These codes are shown to contain t-designs, which can be extended to (t + 1)-designs. It is shown how the weight-distribution of these codes can be uniquely derived.

Unrestricted codes with the golay parameters are unique

The paper contains a proof of the uniqueness of both binary and ternary Golay codes, without assumption of linearity. Similar results are obtained about the extended and expurgated Golay codes. The method consists in proving the linearity, which, according to Pless results, implies the uniqueness.