Remoteness of permutation codes

TL;DR

This paper introduces the concept of remoteness in codes, explores its properties especially in permutation codes, and establishes bounds and exact values for various classes of groups, linking it to graph stability problems.

Contribution

It defines the remoteness parameter as a dual to covering radius, analyzes its bounds in permutation codes, and characterizes remoteness for transitive groups, including a graph-theoretic equivalence.

Findings

Remoteness can only take two values for transitive groups.

Remoteness of transitive groups of odd order is explicitly determined.

Determining remoteness is equivalent to finding the stability number of a related graph.

Abstract

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.