Conway Groupoids and Completely Transitive Codes

To each supersimple $2-(n,4,\lambda)$ design \De one associates a `Conway groupoid,' which may be thought of as a natural generalisation of Conway's Mathieu groupoid $M_{13}$ which is constructed from $\mathbb{P}_3$. We show that \Sp_{2m}(2) and 2^{2m}.\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 matrix of the other design gives an alternative construction of a previously known family of completely transitive codes. We also give a new characterization of $M_{13}$ and prove that, for a fixed $\lambda > 0,$ there are finitely many Conway groupoids for which the set of morphisms does not contain all elements of the full alternating group

Conway groupoids, regular two-graphs and supersimple designs

A $2-(n,4,\lambda)$ design $(\Omega, \mathcal{B})$ is said to be supersimple if distinct lines intersect in at most two points. From such a design, one can construct a certain subset of Sym$(\Omega)$ called a "Conway groupoid". The construction generalizes Conway's construction of the groupoid $M_{13}$. It turns out that several infinite families of groupoids arise in this way, some associated with 3-transposition groups, which have two additional properties. Firstly the set of collinear point-triples forms a regular two-graph, and secondly the symmetric difference of two intersecting lines is again a line. In this paper, we show each of these properties corresponds to a group-theoretic property on the groupoid and we classify the Conway groupoids and the supersimple designs for which both of these two additional properties hold.Comment: 17 page

Some examples related to Conway Groupoids

We discuss the recently introduced notion of a Conway Groupoid. In particular we consider various generalisations of the concept including infinite analogues

Some examples related to Conway Groupoids and their generalisations

We discuss the recently introduced notion of a Conway Groupoid. In particular we consider various generalisations of the concept including infinite analogues

Generating groups using hypergraphs

To a set B of 4-subsets of a set Ω of size n, we introduce an invariant called the ‘hole stabilizer’ which generalizes a construction of Conway, Elkies and Martin of the Mathieu group M12 based on Lloyd's ‘15-puzzle’. It is shown that hole stabilizers may be regarded as objects inside an objective partial group (in the sense of Chermak). We classify pairs (Ω,B) with a trivial hole stabilizer, and determine all hole stabilizers associated to 2-(n,4,λ) designs with λ⩽2⁠