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Register a new userMaximal $m$-distance sets containing the representation of the Hamming graph $H(n,m)$
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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 $n$.
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A finite subset $X$ of the Euclidean space is called an $m$-distance set if the number of distances between two distinct points in $X$ is equal to $m$. An $m$-distance set $X$ is said to be maximal if any vector cannot be added to $X$ while maintaining the $m$-distance condition. We investigate a necessary and sufficient condition for vectors to be added to a regular simplex such that the set has only $2$ distances. We construct several $d$-dimensional maximal $2$-distance sets that contain a $d$-dimensional regular simplex. In particular, there exist infinitely many maximal non-spherical $2$-distance sets that contain both the regular simplex and the representation of a strongly resolvable design. The maximal $2$-distance set has size $2s^2(s+1)$, and the dimension is $d=(s-1)(s+1)^2-1$, where $s$ is a prime power.
In this paper we extend the classical notion of digraphical and graphical regular representation of a group and we classify, by means of an explicit description, the finite groups satisfying this generalization. A graph or digraph is called regular if each vertex has the same valency, or, the same out-valency and the same in-valency, respectively. An m-(di)graphical regular representation (respectively, m-GRR and m-DRR, for short) of a group G is a regular (di)graph whose automorphism group is isomorphic to G and acts semiregularly on the vertex set with m orbits. When m=1, this definition agrees with the classical notion of GRR and DRR. Finite groups admitting a 1-DRR were classified by Babai in 1980, and the analogue classification of finite groups admitting a 1-GRR was completed by Godsil in 1981. Pivoting on these two results in this paper we classify finite groups admitting an m-GRR or an m-DRR, for arbitrary positive integers m. For instance, we prove that every non-identity finite group admits an m-GRR, for every m>4.
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.
A binary poset code of codimension M (of cardinality 2^{N-M}, where N is the code length) can correct maximum M errors. All possible poset metrics that allow codes of codimension M to be M-, (M-1)- or (M-2)-perfect are described. Some general conditions on a poset which guarantee the nonexistence of perfect poset codes are derived; as examples, we prove the nonexistence of R-perfect poset codes for some R in the case of the crown poset and in the case of the union of disjoin chains. Index terms: perfect codes, poset codes
$H_q(n,d)$ is defined as the graph with vertex set ${mathbb Z}_q^n$ and where two vertices are adjacent if their Hamming distance is at least $d$. The chromatic number of these graphs is presented for various sets of parameters $(q,n,d)$. For the $4$-colorings of the graphs $H_2(n,n-1)$ a notion of robustness is introduced. It is based on the tolerance of swapping colors along an edge without destroying properness of the coloring. An explicit description of the maximally robust $4$-colorings of $H_2(n,n-1)$ is presented.
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