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.

DOI: https://doi.org/10.1007/s00493-016-3433-7

Abstract

To each supersimple $2-(n,4,\lambda)$ design $\De$ one associates a `Conway groupoid,' which may be thought of as a natural generalisation of Conway's Mathieu groupoid $M_{13}$ which is constructed from $\mathbb{P}_3$.

We show that $\Sp_{2m}(2)$ and $2^{2m}.\Sp_{2m}(2)$ 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 of 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 group.

Viewing alternatives

Download history

Metrics

Public Attention

Altmetrics from Altmetric

Number of Citations

Citations from Dimensions

Item Actions

Export

About