  We review connections between coding-theoretic objects and sparse learning
problems. In particular, we show how seemingly different combinatorial objects
such as error-correcting codes, combinatorial designs, spherical codes,
compressed sensing matrices and group testing designs can be obtained from one
another. The reductions enable one to translate upper and lower bounds on the
parameters attainable by one object to another. We survey some of the
well-known reductions in a unified presentation, and bring some existing gaps
to attention. New reductions are also introduced; in particular, we bring up
the notion of minimum "L-wise distance" of codes and show that this notion
closely captures the combinatorial structure of RIP-2 matrices. Moreover, we
show how this weaker variation of the minimum distance is related to
combinatorial list-decoding properties of codes.
