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 $(\alpha,\beta)$-map if $k$-bit strings which are more than $\alpha k$ apart are mapped to $n$-bit strings that are more than $\beta n$ apart. This is a relaxation of the classical problem of constructing error-correcting codes, which corresponds to $\alpha=0$. Existence of an $(\alpha,\beta)$-map is equivalent to existence of a graph homomorphism $\bar H(k,\alpha k)\to \bar H(n,\beta n)$, 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 ${n\over k}= \rho$. 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 $\theta$-function). The criterion is then evaluated using some known and some new results from coding theory concerning the $\theta$-function of Hamming graphs. As an example, it is shown that if $\beta>1/2$ and $n\over k$ -- integer, the ${n\over k}$-fold repetition map achieving $\alpha=\beta$ is asymptotically optimal. Finally, constraints on configurations of points and hyperplanes in projective spaces over $\mathbb{F}_2$ are derived.