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A mapping of $k$-bit strings into $n$-bit strings is called an$(\alpha,\beta)$-map if $k$-bit strings which are more than $\alpha k$ apartare mapped to $n$-bit strings that are more than $\beta n$ apart. This is arelaxation of the classical problem of constructing error-correcting codes,which corresponds to $\alpha=0$. Existence of an $(\alpha,\beta)$-map isequivalent to existence of a graph homomorphism $\bar H(k,\alpha k)\to \barH(n,\beta n)$, 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$(\alpha,\beta)$ in the regime of $n,k\to\infty$ with a fixed ratio ${n\overk}= \rho$. This is done by developing a general criterion for existence ofgraph-homomorphism based on the semi-definite relaxation of the independencenumber of a graph (known as the Schrijver's $\theta$-function). The criterionis then evaluated using some known and some new results from coding theoryconcerning the $\theta$-function of Hamming graphs. As an example, it is shownthat if $\beta>1/2$ and $n\over k$ -- integer, the ${n\over k}$-fold repetitionmap achieving $\alpha=\beta$ is asymptotically optimal.  Finally, constraints on configurations of points and hyperplanes inprojective spaces over $\mathbb{F}_2$ are derived.

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