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Gill, Nick; Gillespie, Neil I. and Semeraro, Jason
(2018).
DOI: https://doi.org/10.1007/s00493-016-3433-7
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.