42 Chapitre 1 : Introduction to binary and non-binary LDPCcodes 1 EXIT curves in three Galois fields 0.9 0.8 0.7 0.6 x out 0.5 0.4 0.3 0.2 0.1 x out =x in GF(2) GF(8) GF(256) 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x in Figure 1.5 : EXIT curves of (2,4) GF(2), GF(8) and GF(256) regular codes. The SNR is 0.7dB. 1.6.2 Structured ensembles As pointed out in the introduction, a very efficient way to design code ensembles with iterativedecoding performance close to capacity and low error-floor, is to choose the permutations in a structured way. Indeed, the aforementioned representation of LDPCcodes defines only the connection degrees of variable and check nodes, but any variable node can be connected to any check node. A structured code ensemble has a representation which defines to which type of check nodes each type of variable node can be connected. LDPCcodes with a detailed representation have been introduced in . Some structured code ensembles have been under the scope of many studies these last years: irregular repeataccumulate (IRA) codes , protograph-based LDPCcodes  and multi-edge type LDPC . The design of good D-GLDPCcodes have been addressed for the BEC in . These techniques lead to codes with good code properties in terms, for instance, of girth of the bipartite graph and possibility to perform the encoding procedure efficiently. For a comprehensive survey of the design of those kinds of LDPCcodes, we refer the reader to . 1.7 Proof of theorems in Chapter 1 Lemma 3.Let P e (l) (x) denote the conditional error probability after the l-th BP decoding iteration of a GF(q) LDPC code, assuming that codeword x was sent. If the channel is symmetric, then P e (l) (x) is independent of x. Proof: The proof has the same structure as the proof of Lemma 1 in . Thus, we do not detail it, but instead refer the reader to  and rather only give the key elements.
1.7 Proof of theorems in Chapter 1 43 The notations are the same as in . • Check node symmetry: For any sequence (b 1 , . . .,b dc−1) in GF(q), we have Ψ (l) c (m +b 1 1 , . . .,m +b dc−1 d c−1 ) = Ψ (l) c (m 1 , . . .,m dc−1) +b 1+···+b dc−1 • Variable node symmetry: We also have, for any b ∈ GF(q): Ψ (l) v (m+b 0 ,m+b 1 , . . .,m+b d v−1 ) = Ψ(l) v (m 1, . . .,m dc−1) +b With same notation as in , we define y = z +x , where x is a vector of size q, denoting an arbitrary codeword over GF(q). y and z are sets of vectors, and each element y t corresponds to y t = z +xt t . Still with same notations as in , we easily prove that: m (0) ij (y) = m(0) ij (z)+x i ; . We also prove that, since x is a codeword, then ∑ k:∃e=(v k ,c j ) x k = 0. Hence, as in , we conclude that m (l+1) ji thanks to the check node symmetry, and (y) = m (l+1) ji (z) +x i m (l+1) ij thanks to the variable node symmetry. (y) = m (l+1) ij (z) +x i Lemma 3.If the channel is symmetric, then, under the all-zero codeword assumption, the initial message density P 0 in LDR form is symmetric: P 0 (W = w) = e w i P 0 (W = w +i ) Proof: Let us define y by y = LDR −1 (w). If we call x noisy the noisy observation of the sent symbol value, by following the notation of , we have w = L(x noisy ). ( ) Hence, the i th component of y is y i = P(x noisy ∈ L −1 (w)|x = i), and w i = log y0 y i = ) log also. ( P(xnoisy ∈L −1 (w)|x=0) P(x noisy ∈L −1 (w)|x=i) Given the symmetry of the channel, let us prove that P 0 (W = w) satisfies equation □