Conway groupoids and completely transitive codes

TL;DR

This paper explores the construction of Conway groupoids from certain combinatorial designs, revealing their connection to well-known groups and introducing new families of completely transitive codes with specific parameters.

Contribution

It introduces new Conway groupoids associated with specific designs, links them to classical groups, and constructs new completely transitive codes with particular properties.

Findings

Conway groupoids related to Sp_{2m}(2) and 2^{2m}.Sp_{2m}(2) are identified.

A new family of completely transitive codes with minimum distance 4 and covering radius 3 is constructed.

An alternative construction for a known family of completely transitive codes is provided.

Abstract

To each supersimple $2-(n,4,\lambda)$ 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 matrix of the other design gives an alternative construction to 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 or symmetric group.