Weak isometries of Hamming spaces

Yıl 2016, , 209 - 216, 09.08.2016

Ryan Bruner Stefaan De Winter

https://doi.org/10.13069/jacodesmath.67265

Öz

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.

Kaynakça

• P. Abramenko, H. Van Maldeghem, Maps between buildings that preserve a given Weyl distance, Indag. Math. 15(3) (2004) 305–319.
• F. S. Beckman, D. A. Jr. Quarles, On isometries of Euclidean spaces, Proc. Amer. Math. Soc. 4 (1953) 810–815.
• A. Brouwer, A. Cohen, A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, Heidelberg, 1989.
• A. E. Brouwer, M. A. Fiol, Distance-regular graphs where the distance d-graph has fewer distinct eigenvalues, Linear Algebra Appl. 480 (2015) 115–126.
• S. De Winter, M. Korb, Weak isometries of the Boolean cube, Discrete Math. 339(2) (2016) 877–885.
• E. Govaert, H. Van Maldeghem, Distance-preserving maps in generalized polygons. I. Maps on flags, Beitrage Algebra. Geom. 43(1) (2002) 89–110.
• E. Govaert, H. Van Maldeghem, Distance-preserving maps in generalized polygons. II. Maps on points and/or lines, Beitrage Algebra Geom. 43(2) (2002) 303–324.
• V. Yu. Krasin, On the weak isometries of the Boolean cube, Diskretn. Anal. Issled. Oper. Ser. 1 13(4) (2006) 26–32; translation in J. Appl. Ind. Math. 1(4) (2007) 463–467.