80 Chapitre 2 : HybridLDPC Codes 10 0 E b /N 0 10 −1 10 −2 Frame Error Rate 10 −3 10 −4 10 −5 MET LDPC code HybridLDPC quasi−cyclic GF(2) code THC (1,6/7) 8 M=3 R=1/6 10 −6 −1 −0.5 0 0.5 1 1.5 Figure 2.8 : Comparison of hybrid LDPC code with punctured Turbo Hadamard (PTH) taken from  and other powerful codes, for code rate 1/6. The PTH code has K bit = 999 information bits, and the other codes have K bit = 1024 information bits. N iter = 50 for the PTH code, and N iter = 200 for the other codes. on Hadamard codes are obtained with no complexity increase. Indeed, the complexity of these codes is dominated by the complexity of the fast Hadamard transform, which is O(r ·2 r ) , where r is the order of the Hadamard code. The complexity of hybrid LDPCcodes is dominated by the fast Fourier transform at check nodes O(q log(q)), where q is the maximum group order. The complexity of Hadamard type codesand hybrid LDPCcodes is therefore equivalent. However, contrary to TH codes, one should note that hybrid LDPCcodes are suitable for decoding with reduced complexity and no loss, as described in . 2.6 Conclusions In this work, asymptotic analysis of a new class of non-binary LDPCcodes, named hybrid LDPCcodes, has been carried out. Specific properties of considered hybrid LDPC code ensembles, like the Linear-Map invariance, have been studied to be able to derive both stability condition and EXIT charts. The stability condition of such hybrid LDPC ensembles shows interesting advantages over non-binary codes. Study of the condition allows to conclude that there exist many cases where any fixed point of density evolution for hybrid LDPCcodes can be stable at lower SNR than for non-binary codes. The EXIT charts
2.7 Proofs of theorems in Chapter 2 81 analysis is performed on the BIAWGN channel, whereas studies of generalized LDPCcodes usually consider the BEC [30, 29]. In order to optimize the distributions of hybrid LDPC ensembles, we have investigated how to project the message densities on only one scalar parameter using a Gaussian approximation. The accuracy of such an approximation has been studied, and used to lead to two kinds of EXIT charts of hybrid LDPCcodes: multi-dimensional and mono-dimensional EXIT charts. Distribution optimization allows to get finite length codes with very low connection degrees and better waterfall region than protograph or multi-edge type LDPCcodes. Moreover, hybrid LDPCcodes are well fitted for the cycle cancellation presented in , thanks to the specific structure of the linear maps. The resulting codes appear to have, additionally to a better waterfall region, a very low error-floor for code rate one-half and codeword length lower than three thousands bits, thereby competing with multi-edge type LDPC. Thus, hybrid LDPCcodes allow to achieve an interesting trade-off between good error-floor performance and good waterfall region with non-binary codes techniques. We have also shown that hybrid LDPCcodes can be very good candidates for efficient low rate coding schemes. For code rate one sixth, they compare very well to existing Turbo Hadamard or Zigzag Hadamard codes. In particular, hybrid LDPCcodes exhibit very good minimum distances and error floor properties. As future work, it would be of first interest to allow degree one variable nodes in the representation of hybrid LDPCcodes, by, e.g., adopting a multi-edge type representation . As shown in , this would allow to have better decoding thresholds, in particular for low rate codes. This would give rise to the study and optimization, with the same tools, of non-binary protograph based or multi-edge type LDPCcodes. However, the extension may be theoretically not completely straightforward as the non-zero values have to be carefully handled to define the code ensemble. On the other hand, it would be interesting to study hybrid LDPCcodes on other channels. Let us mention that we made some experiments on an AWGN channel with 16- QAM modulation. We restricted the connection profile to be regular, in order to not bias the results by the absence of special allocation on differently protected symbols. Only two group orders where allowed to avoid correlation between channel LLRs: G(16) and G(256). The optimization of fractions of variable nodes in these two different orders have been done. The results where slightly degraded compared to a (2, 4) GF(256) LDPCcodes. A study of these codes on the BEC would be also interesting, according to what has been done for D-GLDPCcodes on the BEC . 2.7 Proofs of theorems in Chapter 2 Lemma 5 Let P e (t) (x) denote the conditional error probability after the t th BP decoding iteration of a GF(q) LDPC code, assuming that codeword x was sent. If the channel is symmetric, then P e (t) (x) is independent of x.