On metric properties of maps between Hamming spaces and related graph homomorphisms

Abstract

A mapping of k-bit strings into n-bit strings is called an (α,β)-map if k-bit strings which are more than αk apart are mapped to n-bit strings that are more than βn apart in Hamming distance. This is a relaxation of the classical problem of constructing error-correcting codes, which corresponds to α=0. Existence of an (α,β)-map is equivalent to existence of a graph homomorphism H¯(k,αk)→H¯(n,βn), where H(n,d) is a Hamming graph with vertex set {0,1}n and edges connecting vertices differing in d or fewer entries. This paper proves impossibility results on achievable parameters (α,β) in the regime of n,k→∞ with a fixed ratio nk=ρ. This is done by developing a general criterion for existence of graph-homomorphism based on the semi-definite relaxation of the independence number of a graph (known as the Schrijver's θ-function). The criterion is then evaluated using some known and some new results from coding theory concerning the θ-function of Hamming graphs. As an example, it is shown that if β>1/2 and nk – integer, the nk-fold repetition map achieving α=β is asymptotically optimal. Finally, constraints on configurations of points and hyperplanes in projective spaces over F2 are derived.