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 ${\rm M}_{11}.$
Keywords
primitive designs, linear code, Mathieu group ${\rm M}_{11}$
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