  Rate-compatible error-correcting codes (ECCs), which consist of a set of
extended codes, are of practical interest in both wireless communications and
data storage. In this work, we first study the lower bounds for rate-compatible
ECCs, thus proving the existence of good rate-compatible codes. Then, we
propose a general framework for constructing rate-compatible ECCs based on
cosets and syndromes of a set of nested linear codes. We evaluate our
construction from two points of view. From a combinatorial perspective, we show
that we can construct rate-compatible codes with increasing minimum distances.
From a probabilistic point of view, we prove that we are able to construct
capacity-achieving rate-compatible codes.
