Conway groupoids and completely transitive codes

Nick Gill, Neil I. Gillespie, Jason Semeraro

Allbwn ymchwil: Cyfraniad at gyfnodolynErthygladolygiad gan gymheiriaid

14 Wedi eu Llwytho i Lawr (Pure)

Crynodeb

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.
Iaith wreiddiolSaesneg
Tudalennau (o-i)1-44
Nifer y tudalennau44
CyfnodolynCombinatorica
Dyddiad ar-lein cynnar13 Chwef 2017
Dynodwyr Gwrthrych Digidol (DOIs)
StatwsE-gyhoeddi cyn argraffu - 13 Chwef 2017

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