Polynomial Association Schemes Research Articles

On few-class Q-polynomial association schemes: feasible parameters and nonexistence results

We present the tables of feasible parameters of primitive 3 -class Q -polynomial association schemes and 4 - and 5 -class Q -bipartite association schemes (on up to 2800 , 10000 , and 50000 vertices, respectively), accompanied by a number of nonexistence results for such schemes obtained by analysing triple intersection numbers of putative open cases.

On some recent progress in the classification of (Phbox { and }Q)-polynomial association schemes

The problem of classification of (Phbox { and }Q)-polynomial association schemes, as a finite analogue of E. Cartan’s classification of compact symmetric spaces, was posed in the monograph “Association schemes” by E. Bannai and T. Ito in the early 1980s. In this expository paper, we report on some recent results towards its solution.

On the Connectivity of Graphs in Association Schemes

Let $(X,\mathcal{R})$ be a commutative association scheme and let $\Gamma=(X,R\cup R^\top)$ be a connected undirected graph where $R\in \mathcal{R}$. Godsil (resp., Brouwer) conjectured that the edge connectivity (resp., vertex connectivity) of $\Gamma$ is equal to its valency. In this paper, we prove that the deletion of the neighborhood of any vertex leaves behind at most one non-singleton component. Two vertices $a,b\in X$ are called "twins" in $\Gamma$ if they have identical neighborhoods: $\Gamma(a)=\Gamma(b)$. We characterize twins in polynomial association schemes and show that, in the absence of twins, the deletion of any vertex and its neighbors in $\Gamma$ results in a connected graph. Using this and other tools, we find lower bounds on the connectivity of $\Gamma$, especially in the case where $\Gamma$ has diameter two.

On Relative $t$-Designs in Polynomial Association Schemes

Motivated by the similarities between the theory of spherical $t$-designs and that of $t$-designs in $Q$-polynomial association schemes, we study two versions of relative $t$-designs, the counterparts of Euclidean $t$-designs for $P$- and/or $Q$-polynomial association schemes. We develop the theory based on the Terwilliger algebra, which is a noncommutative associative semisimple $\mathbb{C}$-algebra associated with each vertex of an association scheme. We compute explicitly the Fisher type lower bounds on the sizes of relative $t$-designs, assuming that certain irreducible modules behave nicely. The two versions of relative $t$-designs turn out to be equivalent in the case of the Hamming schemes. From this point of view, we establish a new algebraic characterization of the Hamming schemes.

Mutually Unbiased Bush-type Hadamard Matrices and Association Schemes

It was shown by LeCompte, Martin, and Owens in 2010 that the existence of mutually unbiased Hadamard matrices and the identity matrix, which coincide with mutually unbiased bases, is equivalent to that of a $Q$-polynomial association scheme of class four which is both $Q$-antipodal and $Q$-bipartite. We prove that the existence of a set of mutually unbiased Bush-type Hadamard matrices is equivalent to that of an association scheme of class five. As an application of this equivalence, we obtain an upper bound of the number of mutually unbiased Bush-type Hadamard matrices of order $4n^2$ to be $2n-1$. This is in contrast to the fact that the best general upper bound for the mutually unbiased Hadamard matrices of order $4n^2$ is $2n^2$. We also discuss a relation of our scheme to some fusion schemes which are $Q$-antipodal and $Q$-bipartite $Q$-polynomial of class $4$.

Characterizations of Regularity for certain $Q$-Polynomial Association Schemes

It was shown that linked systems of symmetric designs with $a_1^*=0$ and mutually unbiased bases (MUBs) are triply regular association schemes. In this paper, we characterize triple regularity of linked systems of symmetric designs by its Krein number. And we prove that a maximal set of MUBs carries a quadruply regular association scheme and characterize the quadruple regularity of MUBs by its parameter.

Deterministic polynomial factoring and association schemes

AbstractThe problem of finding a nontrivial factor of a polynomial$f(x)$over a finite field${\mathbb{F}}_q$has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the generalized Riemann hypothesis (GRH). In this work we improve the state of the art by focusing on prime degree polynomials; let$n$be the degree. If$(n-1)$has a ‘large’$r$-smooth divisor$s$, then we find a nontrivial factor of$f(x)$in deterministic$\mbox{poly}(n^r,\log q)$time, assuming GRH and that$s=\Omega (\sqrt{n/2^r})$. Thus, for$r=O(1)$our algorithm is polynomial time. Further, for$r=\Omega (\log \log n)$there are infinitely many prime degrees$n$for which our algorithm is applicable and better than the best known, assuming GRH. Our methods build on the algebraic-combinatorial framework of$m$-schemes initiated by Ivanyos, Karpinski and Saxena (ISSAC 2009). We show that the$m$-scheme on$n$points, implicitly appearing in our factoring algorithm, has an exceptional structure, leading us to the improved time complexity. Our structure theorem proves the existence of small intersection numbers in any association scheme that has many relations, and roughly equal valencies and indistinguishing numbers.

On $Q$-Polynomial Association Schemes of Small Class

We show an inequality involving the third largest or second smallest dual eigenvalues of $Q$-polynomial association schemes of class at least three. Also we characterize dual-tight $Q$-polynomial association schemes of class three. Our method is based on tridiagonal matrices and can be applied to distance-regular graphs as well.

Bounds on s-Distance Sets with Strength t

A finite set $X$ in the Euclidean unit sphere is called an $s$-distance set if the set of distances between any two distinct elements of $X$ has size $s$. We say that $t$ is the strength of $X$ if $X$ is a spherical $t$-design but not a spherical $(t+1)$-design. Delsarte Goethals, and Seidel gave an absolute bound for the cardinality of an $s$-distance set. The results of Neumaier and Cameron, Goethals, and Seidel imply that if $X$ is a spherical $2$-distance set with strength $2$, then the known absolute bound for $2$-distance sets can be improved. This bound is also regarded as that for a strongly regular graph with a certain condition of the Krein parameters. In this paper, we give two generalizations of this bound to spherical $s$-distance sets with strength $t$ (more generally, to $s$-distance sets with strength $t$ in a two-point-homogeneous space) and to $Q$-polynomial association schemes. First, for any $s$ and $s-1 \leq t \leq 2s-2$, we improve the known absolute bound for the size of a spherical $s$-distance set with strength $t$. Second, for any $s$, we give an absolute bound for the size of a $Q$-polynomial association scheme of class $s$ with some conditions of the Krein parameters.

Nonexistence of exceptional imprimitive [formula omitted]-polynomial association schemes with six classes

Suzuki (1998) [9] showed that an imprimitive Q -polynomial association scheme with first multiplicity at least 3 is Q -bipartite, or is Q -antipodal, or has four or six classes. The exceptional case with four classes has recently been ruled out by Cerzo and Suzuki (2009) [5]. In this paper, we show the nonexistence of the last case with six classes. Hence Suzuki’s theorem now exactly mirrors its well-known counterpart for imprimitive distance-regular graphs.

There are finitely many [formula omitted]-polynomial association schemes with given first multiplicity at least three

In this paper, we will prove a result which is formally dual to the long-standing conjecture of Bannai and Ito which claims that there are only finitely many distance-regular graphs of valency k for each k > 2 . That is, we prove that, for any fixed m 1 > 2 , there are only finitely many cometric association schemes ( X , R ) with the property that the first idempotent in a Q -polynomial ordering has rank m 1 . As a key preliminary result, we show that the splitting field of any such association scheme is at most a degree two extension of the rationals. All of the tools involved in the proof are fairly elementary yet have wide applicability. To indicate this, a more general theorem is proved here with the result alluded to in the title appearing as a corollary to this theorem.

Non-existence of imprimitive [formula omitted]-polynomial schemes of exceptional type with [formula omitted

In [H. Suzuki, Imprimitive Q -polynomial association schemes, J. Algebraic Combin. 7 (2) (1998) 165–180], it was shown that an imprimitive Q -polynomial scheme X = ( X , { R i } 0 ≤ i ≤ d ) is either dual bipartite, dual antipodal or of class 4 or 6. In this paper, it will be shown that the scheme of class 4 does not occur using the integrality conditions of the entries of the first eigenmatrix of X . These integrality conditions arise from the fact that X has exactly one Q -polynomial ordering [H. Suzuki, Association schemes with multiple Q -polynomial structures, J. Algebraic Combin. 7 (2) (1998) 181–196].

Classification of subsets with minimal width and dual width in Grassmann, bilinear forms and dual polar graphs

Brouwer, Godsil, Koolen and Martin [Width and dual width of subsets in polynomial association schemes, J. Combin. Theory Ser. A 102 (2003) 255–271] introduced the width w and the dual width w * of a subset in a distance-regular graph and in a cometric association scheme, respectively, and then derived lower bounds on these new parameters. For instance, subsets with the property w + w * = d in a cometric distance-regular graph with diameter d attain these bounds. In this paper, we classify subsets with this property in Grassmann graphs, bilinear forms graphs and dual polar graphs. We use this information to establish the Erdős–Ko–Rado theorem in full generality for the first two families of graphs.

On the number of irreducible modular representations of a P and Q polynomial scheme

Whereas results about the number of irreducible modular representations of finite groups are known, those of Bose–Mesner algebras of association schemes are not well understood. In this paper, we consider the number of irreducible modular representations of the adjacency algebra of some P and Q polynomial association schemes. These results should be helpful for investigating the structure of association schemes.

Width and dual width of subsets in polynomial association schemes

The width of a subset C of the vertices of a distance-regular graph is the maximum distance which occurs between elements of C. Dually, the dual width of a subset in a cometric association scheme is the index of the “last” eigenspace in the Q-polynomial ordering to which the characteristic vector of C is not orthogonal. Elementary bounds are derived on these two new parameters. We show that any subset of minimal width is a completely regular code and that any subset of minimal dual width induces a cometric association scheme in the original. A variety of examples and applications are considered.

The Multiplicities of a Dual-thin $Q$-polynomial Association Scheme

Let $Y=(X, \{ R_i \}_{1\le i\le D})$ denote a symmetric association scheme, and assume that $Y$ is $Q$-polynomial with respect to an ordering $E_0,...,E_D$ of the primitive idempotents. Bannai and Ito conjectured that the associated sequence of multiplicities $m_i$ $(0 \leq i \leq D)$ of $Y$ is unimodal. Talking to Terwilliger, Stanton made the related conjecture that $m_i \leq m_{i+1}$ and $m_i \leq m_{D-i}$ for $i < D/2$. We prove that if $Y$ is dual-thin in the sense of Terwilliger, then the Stanton conjecture is true.

On designs in compact metric spaces and a universal bound on their size

For finite and compact infinite metric spaces, a concept of a (weighted) τ-design is introduced which depends on a choice of a substitution function. To estimate the minimum size of a τ-design a system of orthogonal polynomials is defined using the average measure of metric balls and the substitution function. A universal lower bound on the size of τ-designs is obtained with the help of the solution of the known extremum problem for systems of orthogonal polynomials. The concept of a τ-design and the bound considered coincide with those in the case of polynomial association schemes of Delsarte and the Euclidean sphere for the proper choices of the substitution functions. This bound is also calculated for some other spaces.

Imprimitive Q-polynomial Association Schemes

It is well known that imprimitive P-polynomial association schemes\(\mathcal{X} = (X,\{ R_i \} _{0 \leqslant i \leqslant d} ) \) with \(\) are either bipartite or antipodal, i.e., intersection numbers satisfy either \(a_i = 0 \) for all\(i,{\text{or }}b_i = c_{d - i} \) for all \(i \ne [d/{\text{2}}] \). In this paper, we show that imprimitive \(Q \)-polynomial association schemes \(\mathcal{X} = (X,\{ R_i \} _{0 \leqslant i \leqslant d} ) \) with \(\) are either dual bipartite or dual antipodal, i.e., dual intersection numbers satisfy either \(a_i^* = 0{\text{ for all }}i, or b_i^* = c_{d - 1}^* {\text{ for all }}i \ne [d/2] \).

Packing and Decomposition Problems for Polynomial Association Schemes

We consider P- and Q-polynomial association schemes and introduce definitions of Delsarte codes and decomposable schemes. Many known combinatorial notions can be defined as Delsarte codes in suitable association schemes, and almost all classical association schemes turn out to be decomposable. For decomposable association schemes we prove some packing bounds, which were proven before only for antipodal schemes. We also prove that any Delsarte code consists of maximal possible numbers of points for its minimal distance. Some statements about the connection between designs in decomposable schemes and designs in their projections are also given. Detailed proofs of some of our results will be published in the longer paper [24], where analogous problems for a wider class of finite and infinite polynomial metric spaces are considered.

P- and Q-Polynomial Association Schemes and Their Antipodal P-Polynomial Covers

We find a condition on the intersection numbers of any P- and Q -polynomial association scheme Y with diameter at least 3, that holds if Y has an antipodal P-polynomial cover with diameter at least 7. If Y is a known example of a P- and Q-polynomial association scheme with diameter at least 3, our condition holds iff Y has an antipodal P-polynomial cover with diameter at least 7.