Coding Theorems for Repetition and Superposition Codes over Binary-Input Output-Symmetric Channels

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.