Well-spaced labelings of points in rectangular grids (Q2706183)

It is a natural question, to ask for a map from one metric space to another that maps close points of the first metric into far ones of the second space. In the paper this very problem is studied for \(L^1\)-spaces on (\(d\)-dimensional) grids. The domain is always \(1\)-dimensional and the image space is of \(2\) or more dimensions. By using so-called mixed radix vector mappings (that maps an integer \(m\) into a vector that is a linear combination of given \(d\)-dimensional vectors with coefficients coming from the mixed radix expansion of \(m\)), the author shows that if one chooses the \(d\)-dimensional vectors and the basis of the mixed radix expansion in a certain way then the map is an almost optimal one for \(d=2\) and well-spaced for higher dimensions. As possible applications of this method, the storage of error-correcting codes and the construction of holographis memories are mentioned.