Distance preserving maps and combinatorial joint source-channel coding for large alphabets

In this paper we present several results regarding distance preserving maps between nonbinary Hamming spaces and combinatorial (adversarial) joint source-channel coding. In an (α, β)-map from one Hamming space to another, any two sequences that are at least α relative distance apart, are mapped to sequences that are relative distance at least β apart. The motivation to study such maps come from (D, δ)-joint source-channel coding (JSCC) schemes, where any encoded sequence must be recovered within a relative distortion D, even in the presence of δ proportion of adversarial errors. We provide bounds on the parameters of both (α, β)-maps and (D,δ)-JSCC for nonbinary alphabets. We also provide constructive schemes for both, that are optimal for many cases.