Conway Groupoids and Completely Transitive Codes

Gill, Nick; Gillespie, Neil I. and Semeraro, Jason (2018). Conway Groupoids and Completely Transitive Codes. Combinatorica, 38(2) pp. 399–442.

DOI: https://doi.org/10.1007/s00493-016-3433-7

Abstract

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.