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 error-correcting codes problem (α = 0). The question is equivalent to existence of graph homomorphisms between certain graphs on the hypercube. Tools based on Schrijver’s θ-function are developed for testing when such homomorphisms are possible. For n > k the non-existence results on (α, β) are proved by invoking the asymptotic results on θ-function of McEliece, Rodemich, Rumsey and Welch (1977), Samorodnitsky (2001) as well as an exact solution of Delsarte’s linear program for d > n/2. Among other things, these bounds show that for β > 1/2 and n/k – integer, the repetition map achieving α = β is best possible. For n < k a quantitative version of the no-homomorphism lemma is used together with Kleitman’s theorem, which precisely characterizes the diameter-volume tradeoff in Hamming space. Finally, the question of constructing good linear (α, β) maps is shown to be equivalent to finding certain extremal configurations of points in (finite) projective spaces. Consequently, implications of our results for projective geometry over F2 is given. Email address: [email protected] (Yury Polyanskiy) Preprint submitted to Elsevier March 11, 2015