26 Chapitre 1 : Introduction to binary and non-binary LDPCcodes parity-check equation is fulfilled if N−1 ∑ j=0 h ij x j = 0 (1.1) where additions and multiplications are performed over GF(q). The degree of connection of a variable node (the same for a check node) is the number of edges linked to this node. A node is said “i connected” or “of degree i” if it is connected to i edges. Figure (1.1) sums up these notions. H= h 00 h 01 h 02 0 0 0 h 20 0 0 h 23 0 h 25 0 h 11 0 h 13 h 14 0 0 0 h 32 0 h 34 h 35 Variable nodes V0 V1 V2 V3 V 4 V 5 Check nodes Figure 1.1 : Parity-check matrix of a non-binary LDPC code and its bipartite graph. A code is irregular if it is not regular. The usual parametrization of irregular LDPCcodes is done by means of polynomials , sometimes referred to as edgewise parametrization: • Polynomial associated to variable nodes: λ(x) = d∑ vmax i=2 λ i x i−1 where λ i is the proportion of edges of the graph connected to degree i variable nodes, and d vmax is the maximum degree of a variable node. • Polynomial associated to check nodes: ρ(x) = d∑ cmax j=2 ρ j x j−1 where ρ j is the proportion of edges of the graph connected to degree j check nodes, and d cmax is the maximum degree of a check node.
1.3 General notation 27 When the parity-check matrix of the code, whose graph parameters are λ(x) and ρ(x), is full rank, then those two quantities are related to the code rate by: R = 1 − ∑ dcmax j=2 ρ j /j ∑ dvmax i=2 λ i /i (1.2) There is also a dual parametrization of the previous one, referred to as nodewise parametrization : • Polynomial associated to data nodes: ˜λ(x) = d∑ vmax i=2 ˜λ i x i−1 where ˜λ i is the proportion of degree i variable nodes. • Polynomial associated to check nodes: ˜ρ(x) = d∑ cmax j=2 ˜ρ j x j−1 where ˜ρ j is the proportion of degree j check nodes. The transitions from one parametrization to another are given by: ˜λ i = ∑ λ i/i k λ k/k λ i = i˜λ i ∑ , ˜ρ j = ∑ ρ j/j k ρ k/k k k˜λ , ρ j = j˜ρ j ∑ k k k˜ρ k (1.3) Thus, a ensemble of irregular LDPCcodes is parametrized by (N, λ(x), ρ(x)). The regular case is a particular case of this parametrization where λ(x) and ρ(x) are monomials. Figure 1.2 is a graphical representation for this kind of code. 1.3 General notation Throughout the thesis, vectors are denoted by boldface notations, e.g. x. Random variables are denoted by upper-case letters, e.g. X and their instantiations in lower-case, e.g. x. The characterization and the optimization of non-binary LDPCcodes are based on DE equations, assuming that the codes are decoded using iterative BP . An important difference between non-binary and binary BP decoders is that the former uses multidimensional vectors as messages, rather than scalar values. There are two possible representations for the messages: plain-density probability vectors or Log-Density-Ratio