Low complexity decoding algorithms are necessary to meet data rate requirements in excess of 1 Tbps. In this paper, we study one and two bit message passing algorithms for belief propagation decoding of low-density parity-check (LDPC) codes and analyze them by density evolution. The variable nodes exploit soft information from the channel output, and to decrease the data flow, the messages exchanged between check and variable nodes are represented by one or two bits. The newly proposed quaternary message passing (QMP) algorithm is compared asymptotically and in finite length simulations to binary message passing (BMP) and ternary message passing (TMP) for spectrally efficient communication with higher-order modulation and probabilistic amplitude shaping (PAS). To showcase the potential for high throughput forward error correction, spatially coupled LDPC codes and a target spectral efficiency of 3 bits/QAM symbol are considered. Gains of about 0.7 dB and 0.1 dB are observed compared to BMP and TMP, respectively. The gap to unquantized belief propagation decoding is reduced to about 0.75 dB. For smaller code rates, the gain of QMP compared to TMP is more pronounced and amounts to 0.24 dB in the considered example.

This paper investigates the design of non-binary protograph low-density parity-check codes for the Poisson channel with m-ary pulse position modulation. The field order over which the code is constructed is matched to the pulse position modulation order yielding a coded modulation scheme. The optimization of the low-density parity-check code structure is performed via protograph density evolution on a surrogate m-ary erasure channel. The surrogate design is illustrated to be not only accurate, but also robust for a range of practical values of channel background noise and various modulation orders. As a result the proposed codes show excellent performance over the Poisson channel with pulse position modulation outperforming competing schemes. As a side-product of this paper, finite-length benchmarks on the block error probability are provided, together with a union bound to characterize the code performance in the error floor region.

In the framework of error correction in space telecommand (TC) links, the Consultative Committee for Space Data Systems (CCSDS) currently recommends short block-length BCH and binary low-density parity-check (LDPC) codes. Other alternatives have been discarded due to their high decoding complexity, such as non-binary LDPC (NB-LDPC) codes. NB-LDPC codes perform better than their binary counterparts over AWGN and jamming channels, being great candidates for space communications. We show the feasibility of NB-LDPC coding for space TC applications by proposing a highly efficient decoding architecture. The proposed decoder is implemented for a (128,64) NB-LDPC code over GF(16) and the design is particularized for a space-certified Virtex-5QV FPGA. The results prove that NB-LDPC coding is an alternative that outperforms the standardized binary LDPC, with a coding gain of 0.7 dB at a reasonable implementation cost. Given that the maximum rate for TC recommended by the CCSDS is 2 Mbps, the proposed architecture achieves a throughput of 2.03 Mbps using only 9615 LUTs and 5637 FFs (no dedicated memories are used). In addition, this architecture is suitable for any regular (2,4) NB-LDPC (128,64) code over GF(16) independently of the H matrix, allowing flexibility in the choice of the code. This work places NB-LDPC codes as the excellent candidates for future versions of the telecommand uplink standard.

This work considers the design of short non-binary low-density parity-check (LDPC) codes over finite fields of order m, for channels with phase noise. In particular, m-ary differential phase-shift keying (DPSK) modulated code symbols are transmitted over an additive white Gaussian noise (AWGN) channel with Wiener phase noise. At the receiver side, noncoherent detection takes place, with the help of a multi-symbol detection algorithm, followed by a non-binary decoding step. Both the detector and decoder operate on a joint factor graph. As a benchmark, finite length bounds and information rate expressions are computed and compared with the codeword error rate (CER) performance, as well as the iterative threshold of the obtained codes. As a result, performance within 1:2 dB from finite-length bounds is obtained, down to a CER of 10−3.

We consider a set reconciliation setting in which two parties hold similar sets that they would like to reconcile. In particular, we focus on set reconciliation based on invertible Bloom lookup tables (IBLTs), a probabilistic data structure inspired by Bloom filters. IBLT-based set reconciliation schemes have the advantage of exhibiting low computational complexity, however, the schemes available in the literature are known to be far from optimal in terms of communication complexity (overhead). The inefficiency of IBLT-based set reconciliation can be attributed to two facts. First, it requires an estimate of the cardinality of the difference between the sets, which implies an increase in overhead. Second, to cope with uncertainties in the estimation of the cardinality of the set difference, IBLT schemes in the literature oversize the data structures, thus further increasing the overhead. In this work, we present a novel IBLT-based set reconciliation protocol that does not require estimating the cardinality of the set difference. The proposed scheme relies on what we termed multi-edge-type (MET) IBLTs. The simulation results illustrate that the novel scheme outperforms state-of-the-art IBLT-based approaches to set reconciliation in terms of communication cost, i.e., in terms of the number of bits to be exchanged.