# 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.

## Abstract

To each supersimple design one associates a `Conway groupoid,' which may be thought of as a natural generalisation of Conway's Mathieu groupoid which is constructed from .

We show that and 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 -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 and prove that, for a fixed there are finitely many Conway groupoids for which the set of morphisms does not contain all elements of the full alternating group.

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