Remoteness of permutation codes

In this paper, we introduce a new parameter of a code, referred to as the remoteness, which can be viewed as a dual to the covering radius. Indeed, the remoteness is the minimum radius needed for a single ball to cover all codewords. After giving some general results about the remoteness, we then focus on the remoteness of permutation codes. We first derive upper and lower bounds on the minimum cardinality of a code with a given remoteness. We then study the remoteness of permutation groups. We show that the remoteness of transitive groups can only take two values, and we determine the remoteness of transitive groups of odd order. We finally show that the problem of determining the remoteness of a given transitive group is equivalent to determining the stability number of a related graph.

► We define the remoteness of a code and study the remoteness of permutation codes.
► We obtain bounds on the minimum cardinality of a code with given remoteness.
► We determine the remoteness of transitive groups of odd order.
► We derive the remoteness of any pair of permutations and of any cyclic group.
► The remoteness problem is related to graph theory and other areas of combinatorics.