Designs and binary codes from maximal subgroups and conjugacy classes of ${\rm M}_{11}$

Abstract

By using a method of construction of block-primitive and point-transitive $1$-designs, in this paper we determine all block-primitive and point-transitive $1$-$(v, k, \lambda)$-designs from the maximal subgroups and the conjugacy classes of elements of the small Mathieu group ${\rm M}_{11}.$ We examine the properties of the $1$-$(v, k, \lambda)$-designs and construct the codes defined by the binary row span of their incidence matrices. Furthermore, we present a  number of interesting $\Delta$-divisible binary codes invariant under ${\rm M}_{11}.$