  We introduce and study a family of rate-compatible Low-Density Parity-Check
(LDPC) codes characterized by very simple encoders. The design of these codes
starts from simplex codes, which are defined by parity-check matrices having a
straightforward form stemming from the coefficients of a primitive polynomial.
For this reason, we call the new codes Primitive Rate-Compatible LDPC
(PRC-LDPC) codes. By applying puncturing to these codes, we obtain a bit-level
granularity of their code rates. We show that, in order to achieve good LDPC
codes, the underlying polynomials, besides being primitive, must meet some more
stringent conditions with respect to those of classical punctured simplex
codes. We leverage non-modular Golomb rulers to take the new requirements into
account. We characterize the minimum distance properties of PRC-LDPC codes, and
study and discuss their encoding and decoding complexity. Finally, we assess
their error rate performance under iterative decoding.
