WEAK ISOMETRIES OF HAMMING SPACES

In this thesis we study weak isometries of Hamming spaces. These are permutations of a Hamming space that preserve some but not necessarily all distances. We wish to find conditions under which a weak isometry is in fact an isometry. This type of problem was first posed by Beckman and Quarles for Rn. In chapter 2 we give definitions pertinent to our research. The 3rd chapter focuses on some known results in this area with special emphasis on papers by V. Krasin as well as S. De Winter and M. Korb who solved this problem for the Boolean cube, that is, the binary Hamming space. We attempted to generalize some of their methods to the non-boolean case. The 4th chapter has our new results and is split into two major contributions. Our first contribution shows if n=p or p \u3c n2, then every weak isometry of Hnq that preserves distance p is an isometry. Our second contribution gives a possible method to check if a weak isometry is an isometry using linear algebra and graph theory

Weak isometries of Hamming spaces

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

Weak isometries of the Boolean cube

© 2015 Elsevier B.V. All rights reserved. Consider the metric space C consisting of the n-dimensional Boolean cube equipped with the Hamming distance. A weak isometry of C is a permutation of C preserving a given subset of Hamming distances. In Krasin (2006), Krasin showed that in most cases preserving a single Hamming distance forces a weak isometry to be an isometry. In this article we study those weak isometries that are not automatically an isometry, providing a complete classification of weak isometries of C