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

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. 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 $\bar H(k,αk)\to \bar H(n,βn)$, 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 $n,k\to\infty$ with a fixed ratio ${n\over k}= ρ$. 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 $n\over k$ -- integer, the ${n\over k}$-fold repetition map achieving $α=β$ is asymptotically optimal. Finally, constraints on configurations of points and hyperplanes in projective spaces over $\mathbb{F}_2$ are derived.