Uncoverings-by-bases for base-transitive permutation groups

Abstract

An uncovering-by-bases for a group G acting on a finite set ? is a set $$\mathcal{U}$$ of bases for G such that any r-subset of ? is disjoint from at least one base in $$\mathcal{U}$$ , where r is a parameter dependent on G. They have applications in the decoding of permutation groups when used as error-correcting codes, and are closely related to covering designs. We give constructions for uncoverings-by-bases for many families of base-transitive group (i.e. groups which act transitively on their irredundant bases), including a general construction which works for any base-transitive group with base size 2, and some more specific constructions for other groups. In particular, those for the groups GL (3,q) and AGL (2,q) make use of the theory of finite fields. We also describe how the concept of uncovering-by-bases can be generalised to matroid theory, with only minor modifications, and give an example of this.