Some Bounds for the Number of Blocks III

Let $\mathcal D=(\Omega, \mathcal B)$ be a pair of $v$ point set $\Omega$ and a set $\mathcal B$ consists of $k$ point subsets of $\Omega$ which are called blocks. Let $d$ be the maximal cardinality of the intersections between the distinct two blocks in $\mathcal B$. The triple $(v,k,d)$ is called the parameter of $\mathcal B$. Let $b$ be the number of the blocks in $\mathcal B$. It is shown that inequality ${v\choose d+2i-1}\geq b\{{k\choose d+2i-1} +{k\choose d+2i-2}{v-k\choose 1}+....$ $.+{k\choose d+i}{v-k\choose i-1} \}$ holds for each $i$ satisfying $1\leq i\leq k-d$, in the paper: Some Bounds for the Number of Blocks, Europ. J. Combinatorics 22 (2001), 91--94, by R. Noda. If $b$ achieves the upper bound, $\mathcal D$ is called a $\beta(i)$ design. In the paper, an upper bound and a lower bound, $\frac{(d+2i)(k-d)}{i}\leq v \leq \frac{(d+2(i-1))(k-d)}{i-1}$, for $v$ of a $\beta(i)$ design $\mathcal D$ are given. In the present paper we consider the cases when $v$ does not achieve the upper bound or lower bound given above, and get new more strict bounds for $v$ respectively. We apply this bound to the problem of the perfect $e$-codes in the Johnson scheme, and improve the bound given by Roos in 1983.Comment: Stylistic corrections are made. References are adde

On relative tt-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.Comment: 17 page

On primitive symmetric association schemes with m_1=3

We classify primitive symmetric association schemes with m_1 = 3. Namely, it is shown that the tetrahedron, i.e., the association scheme of the complete graph K_4, is the unique such association scheme. Our proof of this result is based on the spherical embeddings of association schemes and elementary three dimensional Euclidean geometry

On primitive symmetric association schemes with m_1=3

We classify primitive symmetric association schemes with m_1 = 3. Namely, it is shown that the tetrahedron, i.e., the association scheme of the complete graph K_4, is the unique such association scheme. Our proof of this result is based on the spherical embeddings of association schemes and elementary three dimensional Euclidean geometry

On primitive symmetric association schemes with m_1=3

We classify primitive symmetric association schemes with m_1 = 3. Namely, it is shown that the tetrahedron, i.e., the association scheme of the complete graph K_4, is the unique such association scheme. Our proof of this result is based on the spherical embeddings of association schemes and elementary three dimensional Euclidean geometry

On primitive symmetric association schemes with m_1=3

We classify primitive symmetric association schemes with m_1 = 3. Namely, it is shown that the tetrahedron, i.e., the association scheme of the complete graph K_4, is the unique such association scheme. Our proof of this result is based on the spherical embeddings of association schemes and elementary three dimensional Euclidean geometry

On the Ring of Simultaneous Invariants for the Gleason–MacWilliams Group

AbstractWe construct a canonical generating set for the polynomial invariants of the simultaneous diagonal action (of arbitrary number of l factors) of the two-dimensional finite unitary reflection group G of order 192, which is called the group No. 9 in the list of Shephard and Todd, and is also called the Gleason–MacWilliams group. We find this canonical set in the vector space (⊗i=1lV)G, where V denotes the (dual of the) two-dimensional vector space on which the group G acts, by applying the techniques of Weyl (i.e., the polarization process of invariant theory) to the invariants C [ x, y ]G0of the two-dimensional group G0of order 48 which is the intersection of G and SL(2, C). It is shown that each element in this canonical set corresponds to an irreducible representation which appears in the decomposition of the action of the symmetric group Sl. That is, by letting the symmetric group Slacts on each element of the canonical generating set, we get an irreducible subspace on which the symmetric group Slacts irreducibly, and all these irreducible subspaces give the decomposition of the whole space (⊗i=1lV)G. This also makes it possible to find the generating set of the simultaneous diagonal action (of arbitrary l factors) of the group G. This canonical generating set is different from the homogeneous system of parameters of the simultaneous diagonal action of the group G. We can construct Jacobi forms (in the sense of Eichler and Zagier) in various ways from the invariants of the simultaneous diagonal action of the group G, and our canonical generating set is very fit and convenient for the purpose of the construction of Jacobi forms

A new Euclidean tight 6-design

We give a new example of Euclidean tight 6-design in $\mathbb R^{22}$.Comment: 9 page