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Register a new userRelative t-designs in binary Hamming association scheme H(n,2)
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A relative t-design in the binary Hamming association schemes H(n,2) is equivalent to a weighted regular t-wise balanced design, i.e., certain combinatorial t-design which allow different sizes of blocks and a weight function on blocks. In this paper, we study relative t-designs in H(n,2), putting emphasis on Fisher type inequalities and the existence of tight relative t-designs. We mostly consider relative t-designs on two shells. We prove that if the weight function is constant on each shell of a relative t-design on two shells then the subset in each shell must be a combinatorial (t-1)-design. This is a generalization of the result of Kageyama who proved this under the stronger assumption that the weight function is constant on the whole block set. Using this, we define tight relative t-designs for odd t, and a strong restriction on the possible parameters of tight relative t-designs in H(n,2). We obtained a new family of such tight relative t-designs, which were unnoticed before. We will give a list of feasible parameters of such relative 3-designs with n up to 100, and then we discuss the existence and/or the non-existence of such tight relative 3-designs. We also discuss feasible parameters of tight relative 4-designs on two shells in H(n,2) with n up 50. In this study we come up with the connection on the topics of classical design theory, such as symmetric 2-designs (in particular 2-(4u-1,2u-1,u-1) Hadamard designs) and Driessens result on the non-existence of certain 3-designs. We believe the Problem 1 and Problem 2 presented in Section 5.2 open a new way to study relative t-designs in H(n,2). We conclude our paper listing several open problems.
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Relative $t$-designs in the $n$-dimensional hypercube $mathcal{Q}_n$ are equivalent to weighted regular $t$-wise balanced designs, which generalize combinatorial $t$-$(n,k,lambda)$ designs by allowing multiple block sizes as well as weights. Partly motivated by the recent study on tight Euclidean $t$-designs on two concentric spheres, in this paper we discuss tight relative $t$-designs in $mathcal{Q}_n$ supported on two shells. We show under a mild condition that such a relative $t$-design induces the structure of a coherent configuration with two fibers. Moreover, from this structure we deduce that a polynomial from the family of the Hahn hypergeometric orthogonal polynomials must have only integral simple zeros. The Terwilliger algebra is the main tool to establish these results. By explicitly evaluating the behavior of the zeros of the Hahn polynomials when they degenerate to the Hermite polynomials under an appropriate limit process, we prove a theorem which gives a partial evidence that the non-trivial tight relative $t$-designs in $mathcal{Q}_n$ supported on two shells are rare for large $t$.
A set $X$ in the Euclidean space $mathbb{R}^d$ is called an $m$-distance set if the set of Euclidean distances between two distinct points in $X$ has size $m$. An $m$-distance set $X$ in $mathbb{R}^d$ is said to be maximal if there does not exist a vector $x$ in $mathbb{R}^d$ such that the union of $X$ and ${x}$ still has only $m$ distances. Bannai--Sato--Shigezumi (2012) investigated the maximal $m$-distance sets which contain the Euclidean representation of the Johnson graph $J(n,m)$. In this paper, we consider the same problem for the Hamming graph $H(n,m)$. The Euclidean representation of $H(n,m)$ is an $m$-distance set in $mathbb{R}^{m(n-1)}$. We prove that the maximum $n$ is $m^2 + m - 1$ such that the representation of $H(n,m)$ is not maximal as an $m$-distance set. Moreover we classify the largest $m$-distance sets which contain the representation of $H(n,m)$ for $mleq 4$ and any $n$. We also classify the maximal $2$-distance sets in $mathbb{R}^{2n-1}$ which contain the representation of $H(n,2)$ for any $n$.
It is inevitable that the $L_{g}(s)$ association scheme with $ggeq 3, sgeq g+2$ is a pseudo-$L_{g}(s)$ association scheme. On the contrary, although $s^2$ treatments of the pseudo-$L_{g}(s)$ association scheme can form one $L_{g}(s)$ association scheme, it is not always an $L_{g}(s)$ association scheme. Mainly because the set of cardinality $s$, which contains two first-associates treatments of the pseudo-$L_{g}(s)$ association scheme, is non-unique. Whether the order $s$ of a Latin square $mathbf{L}$ is a prime power or not, the paper proposes two new conditions in order to extend a $POL(s,w)$ containing $mathbf{L}$. It has been known that a $POL(s,w)$ can be extended to a $POL(s,s-1)$ so long as Brucks cite{brh} condition $sgeq frac{(s-1-w)^4-2(s-1-w)^3+2(s-1-w)^2+(s-1-w)}{2}$ is satisfied, Brucks condition will be completely improved through utilizing six properties of the $L_{w+2}(s)$ association scheme in this paper. Several examples are given to elucidate the application of our results.
In 1963, Shrikhande and Raghavarao published a recursive construction for designs that starts with a resolvable design (the master design) and then uses a second design (the indexing design) to take certain unions of blocks in each parallel class of the master design. Several variations of this construction have been studied by different authors. We revisit this construction, concentrating on the case where the master design is a resolvable BIBD and the indexing design is a 3-design. We show that this construction yields a 3-design under certain circumstances. The resulting 3-designs have block size k = v/2 and they are resolvable. We also construct some previously unknown simple designs by this method.
Let $mathcal{S}_n$ denote the set of permutations of ${1,2,dots,n}$. The function $f(n,s)$ is defined to be the minimum size of a subset $Ssubseteq mathcal{S}_n$ with the property that for any $rhoin mathcal{S}_n$ there exists some $sigmain S$ such that the Hamming distance between $rho$ and $sigma$ is at most $n-s$. The value of $f(n,2)$ is the subject of a conjecture by Kezdy and Snevily, which implies several famous conjectures about latin squares. We prove that the odd $n$ case of the Kezdy-Snevily Conjecture implies the whole conjecture. We also show that $f(n,2)>3n/4$ for all $n$, that $s!< f(n,s)< 3s!(n-s)log n$ for $1leq sleq n-2$ and that [f(n,s)>leftlfloor frac{2+sqrt{2s-2}}{2}rightrfloor frac{n}{2}] if $sgeq 3$.
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