To each supersimple 2-(n,4,λ) design D one associates a ‘Conway groupoid’, which may be thought of as a natural generalisation of Conway’s Mathieu groupoid M13 which is constructed from P3.
We show that Sp2m(2) and 22m. Sp2m(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 F2-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 M13 and prove that, for a fixed λ > 0; 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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Part of the work for this paper was completed while Nick Gill was a visiting professor at the Universidad de Costa Rica. He would like to thank the mathematics department there for their warm hospitality. In addition, all three authors would like to thank Pro-fessor Noam Elkies for answering our questions about M13, and an anonymous referee for pointing out several inaccuracies in an earlier version of the paper.
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Gill, N., Gillespie, N.I. & Semeraro, J. Conway Groupoids and Completely Transitive Codes. Combinatorica 38, 399–442 (2018). https://doi.org/10.1007/s00493-016-3433-7
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