Bounds on the Error Probability of Raptor Codes under Maximum Likelihood Decoding

In this paper upper and lower bounds on the probability of decoding failure under maximum likelihood decoding are derived for different (nonbinary) Raptor code constructions. In particular four different constructions are considered; (i) the standard Raptor code construction, (ii) a multi-edge type construction, (iii) a construction where the Raptor code is nonbinary but the generator matrix of the LT code has only binary entries, (iv) a combination of (ii) and (iii). The latter construction resembles the one employed by RaptorQ codes, which at the time of writing this article represents the state of the art in fountain codes. The bounds are shown to be tight, and provide an important aid for the design of Raptor codes.Comment: Submitted for revie

Raptor Codes in the Low SNR Regime

In this paper, we revisit the design of Raptor codes for binary input additive white Gaussian noise (BIAWGN) channels, where we are interested in very low signal to noise ratios (SNRs). A linear programming degree distribution optimization problem is defined for Raptor codes in the low SNR regime through several approximations. We also provide an exact expression for the polynomial representation of the degree distribution with infinite maximum degree in the low SNR regime, which enables us to calculate the exact value of the fractions of output nodes of small degrees. A more practical degree distribution design is also proposed for Raptor codes in the low SNR regime, where we include the rate efficiency and the decoding complexity in the optimization problem, and an upper bound on the maximum rate efficiency is derived for given design parameters. Simulation results show that the Raptor code with the designed degree distributions can approach rate efficiencies larger than 0.95 in the low SNR regime.Comment: Submitted to the IEEE Transactions on Communications. arXiv admin note: text overlap with arXiv:1510.0772

Raptor Codes for BIAWGN Channel: SNR Mismatch and the Optimality of the Inner and Outer Rates

Fountain codes are a class of rateless codes with two interesting properties, first, they can generate potentially limitless numbers of encoded symbols given a finite set of source symbols, and second, the source symbols can be recovered from any subset of encoded symbols with cardinality greater than the number of source symbols. Raptor codes are the first implementation of fountain codes with linear complexity and vanishing error floors on noisy channels. Raptor codes are designed by the serial concatenation of an inner Luby trans-form (LT) code, the first practical realization of fountain codes, and an outer low-density parity-check (LDPC) code. Raptor codes were designed to operate on the binary erasure channel (BEC), however, since their invention they received considerable attention in or-der to improve their performance on noisy channels, and especially additive white Gaussiannoise (AWGN) channels. This dissertation considers two issues that face Raptor codes on the binary input additive white Gaussian noise (BIAWGN) channel: inaccurate estimation of signal to noise ratio (SNR) and the optimality of inner and outer rates. First, for codes that use a belief propagation algorithm (BPA) in decoding, such as Raptor codes on the BIAWGN channel, accurate estimation of the channel SNR is crucial to achieving optimal performance by the decoder. A difference between the estimated SNR and the actual channel SNR is known as signal to noise ratio mismatch (SNRM). Using asymptomatic analysis and simulation, we show the degrading effects of SNRM on Raptor codes and observe that if the mismatch is large enough, it can cause the decoding to fail. Using the discretized density evolution (DDE) algorithm with the modifications required to simulate the asymptotic performance in the case of SNRM, we determine the decoding threshold of Raptor codes for different values of SNRM ratio. Determining the threshold under SNRM enables us to quantify its effects which in turn can be used to reach important conclusions about the effects of SNRM on Raptor codes. Also, it can be used to compare Raptor codes with different designs in terms of their tolerance to SNRM. Based on the threshold response to SNRM, we observe that SNR underestimation is slightly less detrimental to Raptor codes than SNR overestimation for lower levels of mismatch ratio, however, as the mismatch increases, underestimation becomes more detrimental. Further, it can help estimate the tolerance of a Raptor code, with certain code parameters when transmitted at some SNR value, to SNRM. Or equivalently, help estimate the SNR needed for a given code to achieve a certain level of tolerance to SNRM. Using our observations about the performance of Raptor codes under SNRM, we propose an optimization method to design output degree distributions of the LT part that can be used to construct Raptor codes with more tolerance to high levels of SNRM. Second, we study the effects of choosing different values of inner and outer code rate pairs on the decoding threshold and performance of Raptor codes on the BIAWGN channel. For concatenated codes such as Raptor codes, given any instance of the overall code rate R, different inner (Ri) and outer (Ro) code rate combinations can be used to share the available redundancy as long asR=RiRo. Determining the optimal inner and outer rate pair can improve the threshold and performance of Raptor codes. Using asymptotic analysis, we show the effect of the rate pair choice on the threshold of Raptor codes on the BIAWGN channel and how the optimal rate pair is decided. We also show that Raptor codes with different output degree distributions can have different optimal rate pairs, therefore, by identifying the optimal rate pair we can further improve the performance and avoid suboptimal use of the code. We make the observation that as the outer rate of Raptor codes increases the potential of achieving better threshold increases, and provide the reason why the optimal outer rate of Raptor codes cannot occur at lower values. Finally, we present an optimization method that considers the optimality of the inner and outer rates in designing the output degree distribution of the inner LT part of Raptor codes. The designed distributions show improvement in both the decoding threshold and performance compared to other code designs that do not consider the optimality of the inner and outer rates

Rate adaptive binary erasure quantization with dual fountain codes

In this contribution, duals of fountain codes are introduced and their use for lossy source compression is investigated. It is shown both theoretically and experimentally that the source coding dual of the binary erasure channel coding problem, binary erasure quantization, is solved at a nearly optimal rate with application of duals of LT and raptor codes by a belief propagation-like algorithm which amounts to a graph pruning procedure. Furthermore, this quantizing scheme is rate adaptive, i.e., its rate can be modified on-the-fly in order to adapt to the source distribution, very much like LT and raptor codes are able to adapt their rate to the erasure probability of a channel