70 Chapitre 2 : HybridLDPC Codes By simplifying equation (2.12), x (t+1) e can be expressed by a recursion in terms of x (t) e as: x (j,l) (t) ( ) ) ) cv,k = Jc (J c (1 −1 − J c (j − 1)Jc −1 (1 − x (t) e , q l ), q l , q l , q k ; ⎛ ⎛ ⎛ ⎞⎞⎞ x (t+1) e = ∑ Π(i, k) ⎝1 − log(q k) ⎝1 − J v ⎝σ 2 , (i − 1)Jc −1 ( ∑ Π(j|i, k)x (j,l) (t) cv,k , qk )1 qk −1, q k ⎠⎠⎠ . log(q l ) i,k j (2.17) Thus, the condition for successful decoding of hybrid LDPCcodes in that specific case is ∀t ≥ 0, x (t+1) e > x (t) e (2.18) In that case, the optimization procedure aims at finding distribution Π(i, k|j, l) for given Π(j, l). We see on equation (2.17) that x (t+1) e depends linearly on Π(i, k), turning the optimization problem into a linear programming problem. We may jointly optimize th whole distribution Π(i, k), but we rather prefer to present in the next sections two different methods. In each case, one of the two sets of parameters, Π(i) or Π(k), is set a priori. Set group-order profile, open connexion profile The first way to optimize Π(i, k) is to set the different group orders, and then find the best connection profile of variable nodes for each group. Starting from Π(i, j, k, l), the decomposition we use is the following: Π(i, j, k, l) = Π(i, k, l)Π(j) = Π(i, k)Π(l|i, k)Π(j) = Π(i|k)Π(k)Π(l|k)Π(j) Actually, we do not set the proportion of edges in G(q k ) exactly, but the proportion of variable nodes in G(q k ). We put the redundancy (check nodes and corresponding variable nodes) in the highest order group G(q red ) = G(q max ), corresponding to a proportion α red of variable nodes, and the information variable nodes in two lower order groups G(q info1 ) and G(q info2 ), corresponding to proportions α info1 and α info2 . Hence, the proportion which is optimized is Π(i|k). This means that, for each group order k of variable node, we look for the best connection profile for these variable nodes in G(q k ). Thus, we optimize as many connection profiles as the number of different group orders of variable nodes. This is performed in a single optimization procedure by concatenating Π(i|k), ∀(i, k) in a single vector. In this way, this vector of profiles will hence contain: First profile: ∀i = 2 . . .d vmax , Π(i, red) (2.19) Second profile: ∀i = 2 . . .d vmax , Π(i, info1) Third profile: ∀i = 2 . . .d vmax , Π(i, info2) Equation (2.17) reduces to: x (t+1) e x (t+1) e = = F(x (t) e , Π(i, k), σ 2 ) (2.20) ⎛ ⎛ ⎛ ⎞⎞⎞ ∑ ∑ Π(i, k) ⎝1 − log(q k) ⎝1 − J v ⎝σ 2 , (i − 1)Jc −1 ( ∑ Π(j)x (j,red) (t) cv,k , qk )1 qk −1, q k ⎠⎠⎠ log(q k=red,info1,info2 i red ) j

2.3 Distributions optimization 71 Due to the fact that we a priori set the group orders of variable nodes necessarily equal or lower than check nodes group orders they are connected, the rate of the hybrid bipartite graph, whose nodes are in different order groups, is higher than the code rate (i.e., the actual rate of the transmission). Setting the proportion of variable nodes in G(q k ) for all k also sets the rate of the graph, which becomes the target graph rate in the optimization procedure. From the target code rate R target , we can compute the target graph rate, denoted by R graph by: R graph = ∑ R target α k log 2 (q k ) k=info1,info2 ∑ R target α k log 2 (q k ) + 1 − R target α red log 2 (q red ) k=info1,info2 (2.21) The result of the optimization is finally the set of the three profiles Π(i, k), ∀(i, k) ∈ [1, d vmax ] × [red,info1,info2], for which the following constraints are fulfilled at lowest SNR: ∑ Proportion constraint: ∀i = 2... d vmax , Π(i, red) + Π(i, info1) + Π(i, info2) = 1 Code rate constraint: ∀k = red,info1,info2, i ∑ i Π(i,k) i = α k ∑ 1 − R graph j Π(j) j Sorting constraint: Π(i,j,k,l) = 0, ∀(i,j,k,l) such that q k > q l (2.22) Successful decoding condition: x (t+1) e = F(x (t) e ,Π(i,k),σ2 ) > x (t) e Set connexion profile, open group-order profile Another way to optimize hybrid LDPC ensembles is to set the connection profile and optimize the group orders of variable nodes. As in the previous section, we set the check node parameters (group order G(q red ) and connection profile), independently of the variable nodes parameters. This time, the decomposition of Π(i, j, k, l) is: Π(i, j, k, l) = Π(i, k|l)Π(j)Π(l) Similarly to equation (2.19), we aim at optimizing several group order profiles, as many as the number of different variable node connection degrees. In a finite length performance purpose, we start from an ultra-sparse Tanner graph with a regular connection profile (e.g. (d v = 2, d c = 3)). Hence the previous decomposition falls into: Π(i, j, k, l) = δ(i, d v )δ(j, d c )Π(k)δ(l, red) Since the group order profile of the redundancy is set, the result of the optimization will be the group order profiles of information variable nodes. We denote by I the indexes of