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There is a unique regular two-graph on 276 vertices. This provides a characterization of Conway's group · 3. The proof is based on 276 = 3 × 11 + 35, and uses the ternary Golay code. The paper contains a list of the known strongly regular graphs with the eigenvalue −5.

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 An, Dn, E8, E7, E6 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 An, Hoffman's generalized line graphs for Dn, and various exceptional graphs for E8. The chapter also presents the application to Hadamard matrices.

The chapter reviews the Bose–Mesner algebra of a strongly regular graph Γ , and presents in the i th eigenspace V i , i ∈ {1, 2} a special basis relative to some vertex of the graph. The Krein parameter is related to the components, with respect to the special basis, of a symmetric 3-tensor. The chapter also presents a definition of spherical i -designs in terms of tensors. The theorems explained in the chapter show that q i ii = 0 if and only if Γ yields a spherical 3-design in V i , and if and only if the sub-constituents of Γ yield spherical 2-designs in a hyperplane of V i . The chapter presents a comparison of the restricted spectra of the sub-constituents of Γ . The chapter focuses on Smith graphs having (q + l) (q 3 + 1) vertices, and on those having 16, 27, 100, 112, 162, and 275 vertices. Any graph of the first family is the point graph of a generalized quadrangle ( q , q 2 ).

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 , Hoffman's generalized line graphs for D n , and various exceptional graphs for E 8 . The chapter also presents the application to Hadamard matrices.

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.

Several concepts in discrete mathematics such as block designs, Latin squares, Hadamard matrices, tactical configurations, errorcorrecting codes, geometric configurations, finite groups, and graphs are by no means independent. Combinations of these notions may serve the development of any one of them, and sometimes reveal hidden interrelations. In the present paper a central role in this respect is played by the notion of strongly regular graph, the definition of which is recalled below.In § 2, a fibre-type construction for graphs is given which, applied to block designs withλ= 1 and Hadamard matrices, yields strongly regular graphs. The method, although still limited in its applications, may serve further developments. In § 3 we deal with block designs, first considered by Shrikhande[22],in which the number of points in the intersection of any pair of blocks attains only two values.

1. Introduction. C-matrices appear in the literature at various places; for a survey, see [11]. Important for the construction of Hadamard matrices are the symmetric C-matrices, of order v = 2 (mod 4), and the skew C-matrices, of order v = 0 (mod 4). In § 2 of the present paper it is shown that there are essentially no other C-matrices. A more general class of matrices with zero diagonal is investigated, which contains the C-matrices and the matrices of (v, k, X)-systems on k and k + 1 in the sense of Bridges and Ryser [6]. Skew C-matrices are interpreted in § 3 as the adjacency matrices of a special class of tournaments, which we call strong tournaments. They generalize the tourna­ ments introduced by Szekeres [24] and by Reid and Brown [21]. In § 4 we introduce the notion of negacyclic C-matrices, analogous to the similar notion introduced by Berlekamp in the setting of coding theory (cf. [4, p. 211]). Eigenvalues of negacyclic matrices are characterized and standard forms are obtained. Negacyclic C-matrices are interpreted in § 5 as the matrices of a special class of the relative difference sets introduced by Butson [7]. Exploiting some results of Elliott and Butson [10], we obtain a "multiplier theorem" for negacyclic C-matrices, and adapting a result of [2], we show that any negacyclic C-matrix has a nontrivial multiplier. Necessary conditions for the existence of a negacyclic C-matrix of order v are obtained in § 6. The nonexistence of nega­ cyclic C-matrices of all orders v ^ 226, v ^ 1 + Ph, with p prime, has been verified. This leads to the conjecture that they do not exist, unless v = 1 + PkPaley [19] constructed C-matrices of all orders v = 1 + pk, p prime. In § 7 it is shown that every Paley matrix is equivalent to a negacyclic C-matrix, a fact

Publisher Summary This chapter discusses the combinatorial aspects of coding theory. Codes are used for the correction of errors occurring during transmission over a noisy channel. Many aspects of coding theory are of a combinatorial rather than of an algebraic nature. Typical problems, in this respect, are the construction of maximal codes and of majority decodable codes. The problem of constructing majority decodable linear block codes has been extensively studied in the recent literature. Almost all known majority decoding methods use the properties of block designs, mainly those arising from finite geometries over finite fields. The chapter presents a survey of the literature devoted to this problem, with emphasis on geometric codes. It also reviews connections between geometric codes and polynomial codes. Polynomial codes have the property that any vector of the code can be described by the set of values assumed by a polynomial at a set of points.

A description is given of two new infinite families of nonlinear binary codes. For all block lengths n = 22m, m ≥ 3, two nonlinear binary codes are defined, which are called dual as their distance distributions satisfy the MacWilliams identities. The first code has minimum distance 8 and has four times as many codewords as the corresponding extended BCH code. In the other code, there are only six distinct nonzero values for the distances between codewords.