Title

WEAK ISOMETRIES OF HAMMING SPACES

Author

Ryan Walter Bruner, Michigan Technological University

Date of Award

2014

Document Type

Master's Thesis

Degree Name

Master of Science in Mathematical Sciences (MS)

College, School or Department Name

Department of Mathematical Sciences

Advisor

Stefaan G De Winter

Abstract

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 < ⁿ₂, then every weak isometry of Hⁿ_q 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.

Recommended Citation

Bruner, Ryan Walter, "WEAK ISOMETRIES OF HAMMING SPACES", Master's Thesis, Michigan Technological University, 2014.

https://doi.org/10.37099/mtu.dc.etds/687