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.

翻译：本文引入并研究了一类具有极简编码器的速率兼容低密度奇偶校验（LDPC）码族。这类码的设计始于单形码，其奇偶校验矩阵由本原多项式系数导出，结构简洁。因此，我们将新码命名为"本原速率兼容LDPC（PRC-LDPC）码"。通过对这些码进行打孔操作，可获得比特级的码率粒度。研究表明，为获得性能优良的LDPC码，所采用的多项式除本原性外，还需满足比经典打孔单形码更严格的条件。我们利用非模Golomb标尺来满足这些新需求。本文刻画了PRC-LDPC码的最小距离特性，研究并讨论了其编码与解码复杂度。最后，针对迭代解码下的误码率性能进行了评估。