Weak isometries of Hamming spaces

Abstract

Consider any permutation of the elements of a (finite) metric space that preserves a specific distancep. When is such a permutation automatically an isometry of the metric space? In this note we studythis problem for the Hamming spaces H(n,q) both from a linear algebraic and combinatorial pointof view. We obtain some sufficient conditions for the question to have an affirmative answer, as wellas pose some interesting open problems.