This manuscript dates from 1997 and is unpublished. It is submitted to the arXiv in order to be accessible to researchers. (Several have asked for it, as it is referred to in various places.) It contains 18 pages
We view a design $\mathcal{D}$ as a set of $k$-subsets of a fixed set $X$ of $v$ points. A $k$-subset of $X$ is at distance $i$ from $\mathcal{D}$ if it intersects some $k$-set in $\mathcal{D}$ in $k-i$ points, and no subset in more than $k-i$ points. Thus $\mathcal{D}$ determines a partition by distance of the $k$-subsets of $X$. We say $\mathcal{D}$ is completely transitive if the cells of this partition are the orbits of the automorphism group of $\mathcal{D}$ in its induced action on the $k$-subsets of $X$. This paper initiates a study of completely transitive designs $\mathcal{D}$. A classification is given of all examples for which the automorphism group is not primitive on $X$. In the primitive case the focus is on examples with the property that any two distinct $k$-subsets in $\mathcal{D}$ have at most $k-3$ points in common. Here a reduction is given to the case where the automorphism group is $2$-transitive on $X$. New constructions are given by classifying all examples for some families of $2$-transitive groups, leaving several unresolved cases.
In this paper new infinite families of linear binary completely transitive codes are presented. They have covering radius $ρ= 3$ and 4, and are a half part of the binary Hamming and the binary extended Hamming code of length $n=2^m-1$ and $2^m$, respectively, where $m$ is even. From these new completely transitive codes, in the usual way, i.e., as coset graphs, new presentations of infinite families of distance transitive coset graphs of diameter three and four, respectively, are constructed.
Given a parity-check matrix $H_m$ of a $q$-ary Hamming code, we consider a partition of the columns into two subsets. Then, we consider the two codes that have these submatrices as parity-check matrices. We say that anyone of these two codes is the supplementary code of the other one. We obtain that if one of these codes is a Hamming code, then the supplementary code is completely regular and completely transitive. If one of the codes is completely regular with covering radius $2$, then the supplementary code is also completely regular with covering radius at most $2$. Moreover, in this case, …
A code $C$ in the Hamming metric, that is, is a subset of the vertex set $V\varGamma$ of the Hamming graph $\varGamma=H(m,q)$, gives rise to a natural distance partition $\{C,C_1,\ldots,C_ρ\}$, where $ρ$ is the covering radius of $C$. Such a code $C$ is called completely transitive if the automorphism group $\rm{Aut}(C)$ acts transitively on each of the sets $C$, $C_1$, \ldots, $C_ρ$. A code $C$ is called $2$-neighbour-transitive if $ρ\geq 2$ and $\rm{Aut}(C)$ acts transitively on each of $C$, $C_1$ and $C_2$. Let $C$ be a completely transitive code in a binary ($q=2$) Hamming graph having ful…
In his doctoral thesis, Snover proved that any binary $(m,256,δ)$ code is equivalent to the Nordstrom-Robinson code or the punctured Nordstrom-Robinson code for $(m,δ)=(16,6)$ or $(15,5)$ respectively. We prove that these codes are also characterised as \emph{completely regular} binary codes with $(m,δ)=(16,6)$ or $(15,5)$, and moreover, that they are \emph{completely transitive}. Also, it is known that completely transitive codes are necessarily completely regular, but whether the converse holds has up to now been an open question. We answer this by proving that certain completely regular cod…
The completely regular codes in Hamming graphs have a high degree of combinatorial symmetry and have attracted a lot of interest since their introduction in 1973 by Delsarte. This paper studies the subfamily of completely transitive codes, those in which an automorphism group is transitive on each part of the distance partition. This family is a natural generalisation of the binary completely transitive codes introduced by Sole in 1990. We take the first step towards a classification of these codes, determining those for which the automorphism group is faithful on entries.
The symmetric $2$-$(v,k,λ)$ designs, with $k>λ\left(λ-3 \right)/2$, admitting a flag-transitive, point-imprimitive automorphism group are completely classified: they are the known $2$-designs with parameters $(16,6,2),(45,12,3),(15,8,4)$ or $(96,20,4)$.
To each supersimple $2-(n,4,λ)$ design $\mathcal{D}$ one associates a `Conway groupoid,' which may be thought of as a natural generalisation of Conway's Mathieu groupoid associated to $M_{13}$ which is constructed from $\mathbb{P}_3$. We show that $\operatorname{Sp}_{2m}(2)$ and $2^{2m}.\operatorname{Sp}_{2m}(2)$ naturally occur as Conway groupoids associated to certain designs. It is shown that the incidence matrix associated to one of these designs generates a new family of completely transitive $\mathbb{F}_2$-linear codes with minimum distance 4 and covering radius 3, whereas the incidence …
We describe two similar but independently-coded computations used to construct a complete catalogue of the transitive groups of degree less than $48$, thereby verifying, unifying and extending the catalogues previously available. From this list, we construct all the vertex-transitive graphs of order less than $48$. We then present a variety of summary data regarding the transitive groups and vertex-transitive graphs, focussing on properties that seem to occur most frequently in the study of groups acting on graphs. We illustrate how such catalogues can be used, first by finding a complete list…
For a given transitive binary relation e on a set E, the transitive closures of open (i.e., co-transitive in e) sets, called the regular closed subsets, form an ortholattice Reg(e), the extended permutohedron on e. This construction, which contains the poset Clop(e) of all clopen sets, is a common generalization of known notions such as the generalized permutohedron on a partially ordered set on the one hand, and the bipartition lattice on a set on the other hand. We obtain a precise description of the completely join-irreducible (resp., completely meet-irreducible) elements of Reg(e) and the …
We study transitivity properties of graphs with more than one end. We completely classify the distance-transitive such graphs and, for all $k \geq 3$, the $k$-CS-transitive such graphs.
The study of locally s-distance transitive graphs initiated by the authors in previous work, identified that graphs with a star quotient are of particular interest. This paper shows that the study of locally s-distance transitive graphs with a star quotient is equivalent to the study of a particular family of designs with strong symmetry properties that we call nicely affine and pairwise transitive. We show that a group acting regularly on the points of such a design must be abelian and give a general construction for this case.
In this paper we construct structures from Mathieu group $M_{11}$. We classify transitive $t$-designs with 11, 12 and 22 points admitting a transitive action of Mathieu group $M_{11}$. Thereby we proved the existence of designs with parameters 3-(22,7,18) and found first simple designs with parameters 4-(11,5,6) and 5-(12,6,6). Additionally, we proved the existence of $2$-designs with certain parameters having 55 and 66 points. Furthermore, we classified strongly regular graphs on at most 450 vertices admitting a transitive action of the Mathieu group $M_{11}$.
We consider codes of length $m$ over an alphabet of size $q$ as subsets of the vertex set of the Hamming graph $Γ=H(m,q)$. A code for which there exists an automorphism group $X\leq Aut(Γ)$ that acts transitively on the code and on its set of neighbours is said to be neighbour transitive, and were introduced by the authors as a group theoretic analogue to the assumption that single errors are equally likely over a noisy channel. Examples of neighbour transitive codes include the Hamming codes, various Golay codes, certain Hadamard codes, the Nordstrom Robinson codes, certain permutation codes …
The pairs $(\mathcal{D},G)$, where $\mathcal{D}$ is a non-trivial $2$-$(k^{2},k,λ)$ design, with $λ\mid k$, and $G$ is a flag-transitive automorphism group of $\mathcal{D}$ of affine type such that $G \nleq A ΓL_{1}(k^{2})$, are classified.
In this paper, for any prime $p$, we propose the notion of a $p$-transitive association scheme. This notion aims to generalize the fact that the regular module of a group algebra of a finite group has a unique trivial submodule to the case of the regular modules of modular adjacency algebras. We completely determine the $p$-transitive quasi-thin association schemes and the $p$-transitive association schemes with thin thin residue by their structure theory properties.
Let $q$ be a prime power and $V\cong{\mathbb F}_q^n$. A $t$-$(n,k,λ)_q$ design, or simply a subspace design, is a pair ${\mathcal D}=(V,{\mathcal B})$, where ${\mathcal B}$ is a subset of the set of all $k$-dimensional subspaces of $V$, with the property that each $t$-dimensional subspace of $V$ is contained in precisely $λ$ elements of ${\mathcal B}$. Subspace designs are the $q$-analogues of balanced incomplete block designs. Such a design is called block-transitive if its automorphism group ${\rm Aut}({\mathcal D})$ acts transitively on ${\mathcal B}$. It is shown here that if $t\geq 2$ and…
A Steiner quadruple system of order v is a 3-(v,4,1) design, and will be denoted SQS(v). Using the classification of finite 2-transitive permutation groups all SQS(v) with a flag-transitive automorphism group are completely classified, thus solving the "still open and longstanding problem of classifying all flag-transitive 3-(v,k,1) designs" for the smallest value of k. Moreover, a generalization of a result of H. Lueneburg (1965, Math. Z. 89, 82-90) is achieved.
Block-transitive Steiner $t$-designs form a central part of the study of highly symmetric combinatorial configurations at the interface of several disciplines, including group theory, geometry, combinatorics, coding and information theory, and cryptography. The main result of the paper settles an important open question: There exist no non-trivial examples with $t=7$ (or larger). The proof is based on the classification of the finite 3-homogeneous permutation groups, itself relying on the finite simple group classification.
We classify binary completely regular codes of length $m$ with minimum distance $δ$ for $(m,δ)=(12,6)$ and $(11,5)$. We prove that such codes are unique up to equivalence, and in particular, are equivalent to certain Hadamard codes. We prove that the automorphism groups of these Hadamard codes, modulo the kernel of a particular action, are isomorphic to certain Mathieu groups, from which we prove that completely regular codes with these parameters are necessarily completely transitive.
It is shown that, apart from the smallest Ree group, a flag-transitive automorphism group $G$ of a $2$-$(k^{2}, k, λ)$ design D, with $λ\mid k$, is either an affine group or an almost simple classical group. Moreover, when $G$ is the smallest Ree group, $\mathcal{D}$ is isomorphic either to the $2$-$(62, 6, 2)$ design or to one of the three $2$- $(62, 6, 6)$ designs constructed in this paper. All the four $2$-designs have the $36$ secants of a nondegenerate conic $\mathcal{C}$ of $PG_{2}(8)$ as a point set and 6-sets of secants in a remarkable configuration as a block set.
Let $\mathcal{D}=\left(\mathcal{P},\mathcal{B} \right)$ be a symmetric $2$-$(v,k,λ)$ design admitting a flag-transitive, point-imprimitive automorphism group $G$ that leaves invariant a non-trivial partition $Σ$ of $\mathcal{P}$. Praeger and Zhou \cite{PZ} have shown that, there is a constant $k_{0}$ such that, for each $B \in \mathcal{B}$ and $Δ\in Σ$, the size of $\left\vert B \cap Δ\right \vert$ is either $0$ or $k_{0}$. In the present paper we show that, if $k>λ\left(λ-3 \right)/2$ and $k_{0} \geq 3$, $\mathcal{D}$ is isomorphic to one of the known flag-transitive, point-imprimitive symmet…
We consider an experimental design problem for permutations: given a fixed set $X$, and an integer $t$, construct a list $L$ of permutations of $X$ such that every ordered $t$-tuple of distinct elements of $X$ occurs as a consecutive subsequence of exactly one permutation in $L$. In this paper we focus on solutions based on sharply transitive group actions, in effect generalizing Gordon's notion of group sequencing. We give an explicit construction when $|X|$ is prime for the case $t=3$, and analyze a branching algorithm for the general case which produces, for example, a rare design with $t=6…
Tetravalent graphs admitting a half-arc-transitive subgroup of automorphisms, that is a subgroup acting transitively on its vertices and its edges but not on its arcs, are investigated. One of the most fruitful approaches for the study of structural properties of such graphs is the well known paradigm of alternating cycles and their intersections which was introduced by Marušič 20 years ago. In this paper a new parameter for such graphs, giving a further insight into their structure, is introduced. Various properties of this parameter are given and the parameter is completely determined for th…
Analysing 2-$(v,k,2)$ designs admitting a flag-transitive almost simple automorphism group with socle $PSL(2,q)$ by means of conics and hyperovals of $PG(2,q)$
In this paper we complete a classification of finite linear spaces $\cS$ with line size at most 12 admitting a line-transitive point-imprimitive subgroup of automorphisms. The examples are the Desarguesian projective planes of orders $4,7, 9$ and 11, two designs on 91 points with line size 6, and 467 designs on 729 points with line size 8.
In this paper we develop several general methods for analysing flag-transitive point-imprimitive $2$-designs, which give restrictions on both the automorphisms and parameters of such designs. These constitute a tool-kit for analysing these designs and their groups. We apply these methods to complete the classification of flag-transitive, point-imprimitive $2$-$(v,k,λ)$ designs with $λ$ at most $4$.
This paper is devoted to the classification of flag-transitive 2-(v,k,2) designs. We show that apart from two known symmetric 2-(16,6,2) designs, every flag-transitive subgroup G of the automorphism group of a nontrivial 2-(v,k,2) design is primitive of affine or almost simple type. Moreover, we classify the 2-(v,k,2) designs admitting a flag transitive almost simple group G with socle PSL(n,q) for some n \geq 3. Alongside this analysis, we give a construction for a flag-transitive 2-(v,k-1,k-2) design from a given flag-transitive 2-(v,k,1) design which induces a 2-transitive action on a line.…
We give a construction of a family of designs with a specified point-partition, and determine the subgroup of automorphisms leaving invariant the point-partition. We give necessary and sufficient conditions for a design in the family to possess a flag-transitive group of automorphisms preserving the specified point-partition. We give examples of flag-transitive designs in the family, including a new symmetric $2$-$(1408,336,80)$ design with automorphism group $2^{12}:((3\cdot\mathrm{M}_{22}):2)$, and a construction of one of the families of the symplectic designs (the designs $S^-(n)$) exhibit…
This paper studies flag-transitive point-primitive non-symmetric $2$-($v,k,2$) designs. We prove that if $\mathcal{D}$ is a non-trivial non-symmetric $2$-$(v,k,2)$ design admitting a flag-transitive point-primitive automorphism group $G$ with $Soc(G)=A_{n}$ for $n\geq5$, then $\mathcal{D}$ is a $2$-$(6,3,2)$ or $2$-$(10,4,2)$ design.
In this article, we study $2$-designs with prime replication number admitting a flag-transitive automorphism group. The automorphism groups of these designs are point-primitive of almost simple or affine type. We determine $2$-designs with prime replication number admitting an almost simple automorphism group.
Among the properties of homogeneity of incidence structures flag-transitivity obviously is a particularly important and natural one. Consequently, in the last decades also flag-transitive Steiner tdesigns (i.e. flag-transitive t-(v,k,1) designs) have been investigated, whereas only by the use of the classification of the finite simple groups has it been possible in recent years to essentially characterize all flag-transitive Steiner 2-designs. However, despite the finite simple group classification, for Steiner t-designs with parameters t > 2 such characterizations have remained challenging op…
This paper considers flag-transitive $4$-$(q+1,k,λ)$ designs with $λ\geq5$ and $q+1>k>4$. Let the automorphism group of a design $\cal D$ be a simple group $G=PSL(2,q)$. Depend on the fact that the setwise stabilizer $G_B$ must be one of twelve kinds of subgroups, up to isomorphism we get the following two results. (i) If $10\geq λ\geq 5$, then except $(G,G_x,G_B,k,λ)=(PSL(2,761),{E_{761}}\rtimes {C_{380}},S_4,24,7)$ or $(PSL(2,512),{E_{512}}\rtimes {C_{511}},{D_{18}},18,8)$ undecided, $\cal D$ is a $4$-$(24,8,5)$, $4$-$(9,8,5)$, $4$-$(8,6,6)$, $4$-$(10,9,6)$, $4$-$(9,6,10)$, $4$-$(9,7,10)$, $…
We solve the long-standing open problem of classifying all 3-(v,k,1) designs with a flag-transitive group of automorphisms (cf. A. Delandtsheer, Geom. Dedicata 41 (1992), p. 147; and in: "Handbook of Incidence Geometry", ed. by F. Buekenhout, Elsevier Science, Amsterdam, 1995, p. 273; but presumably dating back to 1965). Our result relies on the classification of the finite 2-transitive permutation groups.
In this article, we study $2$-designs with $\gcd(r,λ)=1$ admitting a flag-transitive almost simple automorphism group with socle a finite simple exceptional group of Lie type. We obtain four infinite families of such designs and provide some examples in each of these families.
A permutation group is called semiprimitive if each of its normal subgroups is either transitive or semiregular. Given nontrivial finite transitive permutation groups $L_1$ and $L_2$ with $L_1$ not semiprimitive, we construct an infinite family of rank two amalgams of permutation type $[L_1,L_2]$ and Borel subgroups of strictly increasing order. As an application, we show that there is no bound on the order of edge-stabilisers in locally $[L_1,L_2]$ graphs. We also consider the corresponding question for amalgams of rank $k\geq 3$. We completely resolve this by showing that the order of the Bo…
This paper deals with block-transitive $t$-$(v,k,λ)$ designs in affine spaces for large $t$, with a focus on the important index $λ=1$ case. We prove that there are no non-trivial 5-$(v,k,1)$ designs admitting a block-transitive group of automorphisms that is of affine type. Moreover, we show that the corresponding non-existence result holds for 4-$(v,k,1)$ designs, except possibly when the group is one-dimensional affine. Our approach involves a consideration of the finite 2-homogeneous affine permutation groups.
We classify the pairwise transitive 2-designs, that is, 2-designs such that a group of automorphisms is transitive on the following five sets of ordered pairs: point-pairs, incident point-block pairs, non-incident point-block pairs, intersecting block-pairs and non-intersecting block-pairs. These 2-designs fall into two classes: the symmetric ones and the quasisymmetric ones. The symmetric examples include the symmetric designs from projective geometry, the 11-point biplane, the Higman-Sims design, and designs of points and quadratic forms on symplectic spaces. The quasisymmetric examples aris…
In this paper, we present a classification of $2$-designs with $\gcd(r,λ)=1$ admitting flag-transitive automorphism groups. If $G$ is a flag-transitive automorphism group of a non-trivial $2$-design $\mathcal{D}$ with $\gcd(r,λ)=1$, then either $(\mathcal{D},G)$ is one of the known examples described in this paper, or $\mathcal{D}$ has $q = p^{d}$ points with $p$ prime and $G$ is a subgroup of $AΓL_{1}(q)$.
In this article, we investigate $2$-$(v,k,λ)$ designs with $\gcd(r,λ)=1$ admitting flag-transitive automorphism groups $G$. We prove that if $G$ is an almost simple group, then such a design belongs to one of the seven infinite families of $2$-designs or it is one of the eleven well-known examples. We describe all these examples of designs. We, in particular, prove that if $\mathcal{D}$ is a symmetric $(v,k,λ)$ design with $\gcd(k,λ)=1$ admitting a flag-transitive automorphism group $G$, then either $G\leq AΓL_{1}(q)$ for some odd prime power $q$, or $\mathcal{D}$ is a projective space or the …
