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We introduce and study a family of rate-compatible Low-Density Parity-Check(LDPC) codes characterized by very simple encoders. The design of these codesstarts from simplex codes, which are defined by parity-check matrices having astraightforward 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-levelgranularity of their code rates. We show that, in order to achieve good LDPCcodes, the underlying polynomials, besides being primitive, must meet some morestringent conditions with respect to those of classical punctured simplexcodes. We leverage non-modular Golomb rulers to take the new requirements intoaccount. We characterize the minimum distance properties of PRC-LDPC codes, andstudy and discuss their encoding and decoding complexity. Finally, we assesstheir error rate performance under iterative decoding.

Rate-compatible LDPC Codes based on Primitive Polynomials and Golomb Rulers