Eiichi Bannai

Subschemes of some association schemes

Abstract By using the character tables of known commutative association schemes, we can, in many instances, construct some new commutative association schemes as their subschemes. In particular, we obtain many new non-group-case association schemes from the group association schemes of finite simple groups. Although examples worked out in this paper are limited to some special cases, it might be possible that this phenomenon occurs universally for all the group association schemes of finite Chevalley groups. The subschemes obtained in this way could be considered as an association scheme version of Weyl groups for Chevalley groups.

Regular graphs with excess one

Abstract It is shown that there exists no regular graph with excess e =1 and girth 2 r +1⩾5.

Tables of the association schemes of finite orthogonal groups acting on the nonisotropic points

Abstract We study the character tables of the association schemes obtained from the following actions of finite (simple) orthogonal groups: 1. (i) O 2 m + (2 n ) and O 2 m − (2 n ) acting on the set of nonisotropic points. 2. (ii) O 2 m + ( p n ) and O 2 m − ( p n ) ( p an odd prime) acting on each half of the set of nonisotropic points. 3. (iii) O 2 m + 1 ( p n ) ( p an odd prime) acting on the set of square-type and the set of non-square-type nonisotropic points. It is shown that the character tables of the association schemes in the above (i) and (ii) are controlled by the character table of the group PSL (2, q ), while that the character tables of the association schemes in (iii) are controlled by the character tables of the association schemes obtained from the action of the group PGL (2, q ) acting on the cosets by dihedral subgroups D 2( q − 1) and D 2( q + 1) , respectively.

On distance-regular graphs with fixed valency

We prove the following result. LetΓ be a finite distance-regular graph. Letci,ai,bi be the intersection numbers ofΓ. IfΓ is not an ordinaryn-gon, then the number of (ci,ai,bi) such thatci =bi is bounded by a certain function of the valencyk, say 10k2k.

Current research on algebraic combinatorics: Supplements to our Book, Algebraic Combinatorics I

This paper gives an account of recent activity in the field of algebraic combinatorics and thus updates our book, Algebraic Combinatorics I (Benjamin/Cummings, 1984).

On distance-regular graphs with fixed valency, IV

Distance-regular graphs with certain specific intersection arrays are investigated. In particular, it is shown that if the columns of the intersection array are all of the form t (*, 0, k ), t (1, a , k - a - 1), t ( k - a - 1, a , 1) and t ( k - e , e , *) except for t intermediate columns, then the diameter d of the graph is bounded by a function depending only on t and the valency k if k ⩾ 3.

Tight t -designs and squarefree integers

The authors prove, using a variety of number-theoretical methods, that tight t-designs in the projective spaces FPn of ‘lines’ through the origin in Fn+1 (F = ℂ, or the quarternions H) satisfy t ⩽ 5. Such a design is a generalisation of a combinatorial t-design. It is known that t ⩽ 5 in the cases F = ℝ , O (the octonions) and that t ⩽ 11 for tight spherical t-designs; hence the authors result essentially completes the classification of tight t-designs in compact connected symmetric spaces of rank 1.

How many P-polynomial Structures can an Association Scheme have?

As the Johnson association scheme J (2 k - 1, k - 1) shows, an association scheme sometimes has more than one P -polynomial structure. In this paper we prove that any one association scheme can have no more than two P -polynomial structures (except for the association scheme of an n -gon) and that an association scheme can have two P -polynomial structures in only a few different ways.

On perfect codes in the hamming schemes H(n, q) with q arbitrary

Abstract For each e ⩾ 3, there are at most finitely many nontrivial perfect e-codes in the Hamming schemes H(n, q) where n and q are arbitrary.

Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics

This paper surveys the role of orthogonal polynomials in Algebraic Combinatorics, an area which includes association schemes, coding theory, design theory, various theories of group representation, and so on. The main topics discussed in this paper include the following: The connection between orthogonal polynomials and P -polynomial (or Q -polynomial) association schemes. The classification problem for P - and Q -polynomial association schemes and its connection with Askey-Wilson orthogonal polynomials. Delsarte theory of codes and designs in association schemes. The nonexistence of perfect e-codes and tight t-designs through the study of the zeros of orthogonal polynomials. The possible importance of multi-variable versions of Askey-Wilson polynomials in the future study of general commutative association schemes.