Non-binary multi-edge type ensembles of low-density parity-check codes are analyzed in terms of non-binary codeword weight distribution and its growth rate. In particular, an exact expression of the growth rate for small weights is developed. As a side result, the stopping set distributions of these ensembles are developed. Examples of weight distributions are provided, showing that the derived closed-form expressions can be easily evaluated. The obtained results can thus be exploited to analyze and design non-binary low-density parity-check codes that fall within the multi-edge type framework such as, but not limited to, protograph-based codes.

The non-binary codeword weight distribution and its growth rate are developed for non-binary multi-edge type ensembles of low-density parity-check codes. Moreover, an analysis of the growth rate for small weights is provided. The derived expressions can serve as powerful and flexible tools to analyze and design the non-binary low-density parity-check codes that fall within the multi-edge type framework.

The code design of non-binary low-density paritycheck codes for the erasure channel, under maximum a posteriori decoding, is addressed. In particular, a partially structured ensemble of codes, characterized by a careful control of the amount and of the connectivity of the variable nodes of small degrees, is proposed. The identified ensemble of codes is analyzed in terms of asymptotic thresholds and weight distribution and it is shown that codes from the ensemble provide a remarkable trade-off between waterfall performance, error floor, and decoding complexity. As an example, the performance curve of a short (256,128) code on the memoryless 16-ary erasure channel tightly approaches the Singleton bound at least down to a codeword error rate of 10-9, at low decoding complexity.

The finite-length design of non-binary low-density parity-check (LDPC) codes for the q-ary erasure channel under maximum a posteriori (MAP) decoding is addressed. A low-complexity MAP decoding algorithm is reviewed for which a code design strategy is proposed. In particular, it is illustrated how a judicious code design permits to find a trade-off between performance in terms of codeword error rate (CER) and decoding complexity. As an example, the performance curve of a short (400, 200) code on the memoryless 4-ary erasure channel tightly approaches the Singleton bound at least down to a CER of 10-8.

We introduce a class of structured (protograph-based) doubly generalized low-density parity-check (DGLDPC) codes. The proposed class leads to quasi-cyclic codes for which efficient encoder and decoder implementations are possible. In particular, we illustrate that on the additive white Gaussian noise (AWGN) channel, the introduced structured DGLDPC codes do not lose in performance with respect to unstructured DGLDPC codes. Furthermore, a sufficient condition on the code graph is derived, which allows unveiling the quasi-cyclic nature of the designed codes.

The performance and the decoding complexity of a novel coding scheme based on the concatenation of maximum distance separable (MDS) codes and linear random fountain codes are investigated. Differently from Raptor codes (which are based on a serial concatenation of a high-rate outer block code and an inner Luby-transform code), the proposed coding scheme can be seen as a parallel concatenation of a MDS code and a linear random fountain code, both operating on the same finite field. Upper and lower bounds on the decoding failure probability under maximum-likelihood (ML) decoding are developed. It is shown how, for example, the concatenation of a (15,10) Reed-Solomon (RS) code and a linear random fountain code over a finite field of order 16, {F}_{16}, brings to a decoding failure probability 4 orders of magnitude lower than the one of a linear random fountain code for the same receiver overhead in a channel with a erasure probability of ε=5\cdot10^{-2}. It is illustrated how the performance of the novel scheme approaches that of an idealized fountain code for higher-order fields and moderate erasure probabilities. An efficient decoding algorithm is developed for the case of a (generalized) RS code.

This paper investigates non-binary low-density parity-check (LDPC) erasure correcting codes suitable to guarantee reliable transmission in wireless communications systems. In particular, irregular repeat-accumulate (IRA) codes are considered, characterized by linear-time encoding complexity. The performance of non-binary IRA codes is compared with their binary counterparts on the packet erasure channel (PEC), with considerable advantages for the non-binary construction. Particularly, it is illustrated that the performance of short-block-length erasure correcting IRA codes over Galois fields (GFs) of order q >; 2 approaches, under maximum-likelihood (ML) decoding, the performance of ideal maximum distance separable (MDS) codes. This is especially appealing in the context of satellite communications, where efficient codes are required to cope with small link margins.