This paper is concerned with a class of low density generator matrix codes (LDGM), called repetition and superposition (RaS) codes, which have been proved to be capacity-achieving over binary-input output-symmetric (BIOS) channels in terms of bit-error rate (BER). We prove with a recently proposed framework that the RaS codes are also capacity-achieving over BIOS channels in terms of frame-error rate (FER). With this new framework, the theorem for the RaS codes can be generalized to source coding and joint source and channel coding (JSCC). In particular, we prove with this framework that the corresponding low-density parity-check (LDPC) codes, as an enlarged ensemble of quasi-cyclic LDPC (QC-LDPC) codes, can also achieve the capacity. To further improve the iterative decoding performance, we consider the convolutional RaS (Conv-RaS) code ensemble and prove it to be capacity-achieving over BIOS channels in terms of the first error event probability. The construction of Conv-RaS codes is flexible with rate (defined as the ratio of the input length to the encoding output length) ranging from less than one (typically for channel codes) to greater than one (typically for source codes), which can be implemented as a universal JSCC scheme, as confirmed by simulations.

翻译：本文研究一类称为重复与叠加（RaS）码的低密度生成矩阵（LDGM）码，该码已被证明在二进制输入对称（BIOS）信道上能够以误比特率（BER）达到信道容量。我们采用最近提出的框架证明了RaS码在BIOS信道上以误帧率（FER）同样能够达到信道容量。借助这一新框架，RaS码的定理可推广至信源编码及联合信源信道编码（JSCC）。特别地，我们利用该框架证明了相应的低密度奇偶校验（LDPC）码（作为准循环LDPC（QC-LDPC）码的扩展集合）也能达到信道容量。为进一步改善迭代译码性能，我们考虑了卷积RaS（Conv-RaS）码集合，并证明其在BIOS信道上以首次错误事件概率衡量也能达到信道容量。Conv-RaS码的构造具有灵活性，其码率（定义为输入长度与编码输出长度之比）可从小于1（典型信道编码）到大于1（典型信源编码）变化，可作为一种通用JSCC方案实现，仿真结果验证了这一点。