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Inspired by some intriguing examples, we study uniform association schemes and uniform coherent configurations, including cometric Q-antipodal association schemes. After a review of imprimitivity, we show that an imprimitive association…

Coherent configurations are a generalization of association schemes. In this paper, we introduce the concept of $Q$-polynomial coherent configurations and study the relationship among intersection numbers, Krein numbers, and eigenmatrices.…

Penttila and Williford constructed a $4-$class association scheme from a generalized quadrangle with a doubly subtended subquadrangle. We show that an association scheme with appropriate parameters and satisfying some assumption about…

We characterize that the image of the embedding of the $Q$-polynomial association scheme into eigenspace by primitive idempotent $E_1$ is a spherical $t$-design in terms of the Krein numbers. And we show that the strengths of $P$- and…

We complete the classification of tight $4$-designs in Hamming association schemes $H(n,q)$, i.e., that of tight orthogonal arrays of strength $4$, which had been open since a result by Noda (1979). To do so, we construct an association…

In this paper, the association scheme defined on the flags of a finite generalized quadrangle is considered. All possible fusions of this scheme are listed, and a full description for those of classes 2 and 3 is given. Furthermore, it is…

Let $\mathscr{Q}(m,q)$ and $\mathscr{S}(m,q)$ be the sets of quadratic forms and symmetric bilinear forms on an $m$-dimensional vector space over $\mathbb{F}_q$, respectively. The orbits of $\mathscr{Q}(m,q)$ and $\mathscr{S}(m,q)$ under a…

H. Cohn et. al. proposed an association scheme of 64 points in R^{14} which is conjectured to be a universally optimal code. We show that this scheme has a generalization in terms of Kerdock codes, as well as in terms of maximal real…

The previous constructions of quadrature amplitude modulation (QAM) Golay complementary sequences (GCSs) were generalized as $4^q $-QAM GCSs of length $2^{m}$ by Li \textsl{et al.} (the generalized cases I-III for $q\ge 2$) in 2010 and Liu…

We investigate intriguing sets of an association scheme introduced by Penttila and Williford (2011) that was the basis for their construction of primitive cometric association schemes that are not $P$-polynomial nor the dual of a…

We prove that if X is a spherical t-design and s-distance set with $t\geq 2s-3$, then X has the structure of Q-polynomial association scheme of class s. Also, we describe the parameters of the association scheme.

Delsarte-Goethals-Seidel showed that if $X$ is a spherical $t$-design with degree $s$ satisfying $t\geq 2s-2$, $X$ carries the structure of an association scheme. Also Bannai-Bannai showed that the same conclusion holds if $X$ is an…

A 3-dimensional quasiperiodic lattice, with overlapping unit cells and periodic in one direction, is constructed using grid and projection methods pioneered by de Bruijn. Each unit cell consists of 26 points, of which 22 are the vertices of…

We construct a pair of non-commutative rank 8 association schemes from a rank 3 non-symmetric association scheme. For the pair, two association schemes have the same character table but different Frobenius-Schur indicators. This situation…

We provide a method to construct $t$-designs from weighing matrices and association schemes. One instance of our method can produce a $3$-design from any (symmetric or skew-symmetric) conference matrix, thereby providing a partial answer to…

We classify the Q-polynomial association schemes with $m_{1} = 4$ which are partially metric with respect to the nearest neighbourhood relation. An association scheme is partially metric with respect to a relation $R_1$ if the scheme graph…

n [D. de Caen, E.R. van Dam. Fissioned triangular schemes via the cross-ratio, {Europ. J. Combin.}, 22 (2001) 297-301], de Caen and van Dam constructed a fission scheme $\FT(q+1)$ of the triangular scheme on $\PG(1,q)$. This fission scheme…

This paper presents four new classes of binary quantum codes with minimum dis- tance 3 and 4, namely Class-I, Class-II, Class-III and Class-IV. The classes Class-I and Class-II are constructed based on self-dual orientable embeddings of the…

The classification problem of $P$- and $Q$-polynomial association schemes has been one of the central problems in algebraic combinatorics. Generalizing the concept of $P$- and $Q$-polynomial association schemes to multivariate cases, namely…

By applying Chinese remainder theorem for coprime polynomials and the "creative microscoping" method recently introduced by the author and Zudilin, we establish parametric generalizations of three $q$-supercongruences modulo the fourth…