2014/10/17 by Nick Gill, Gill, Nick, Neil I. Gillespie +3
Computer Science · Engineering · Mathematics · #05B05 #20B15 #20B25 #94B05 #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems
paper · doi:10.48550/arxiv.1410.4785
openalex publication_date 2014/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 associated to 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 \mathbbF2-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 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 or symmetric group.