Weak isometries of Hamming spaces

Abstract

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