### 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.

#### Full Text:

PDFDOI: http://dx.doi.org/10.13069/jacodesmath.67265

#### References

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.