# Conway groupoids and completely transitive codes

Nick Gill, Neil I. Gillespie, Jason Semeraro

Research output: Contribution to journalArticlepeer-review

## 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.
Original language English 1-44 44 Combinatorica 13 Feb 2017 https://doi.org/10.1007/s00493-016-3433-7 E-pub ahead of print - 13 Feb 2017

## Keywords

• math.GR
• math.CO
• 20B15, 20B25, 05B05, 94B05

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