Hybrid LDPC codes and iterative decoding methods - i3s


Hybrid LDPC codes and iterative decoding methods - i3s

3.3 Machine Learning Methods for Decoder Design 103

Figure 3.5 : Voronoi diagram (or Dirichlet tessellation): the partitioning of a plane with n

points into convex polygons such that each polygon contains exactly one generating point

and every point in a given polygon is closer to its generating point than to any other.

code). Hence, we know theoretically the optimal classifier, which corresponds to implement

a K-dimensional Voronoi partition of the Euclidean space GF(2) N with codewords

as cell centroids, as sketched on figure 3.5. However, implementing this partitioning is

intractable in practice for long codes, and corresponds exactly to implement maximumlikelihood

(ML) decoding. That is why this classification problem is usually solved with

a BP decoder, which actually only implements an approximation of the Voronoi tessellation

frontiers, i.e., of ML decoding. Many previous works [19, 20] have characterized

the phenomenon which arises when BP decoder is used on loopy graphs, and which emphasizes

the difference between ML decoding and BP decoding. ML decoding is always

able to find the codeword closest to the observation (even though it makes errors because

this closest codeword is not the one which has been sent), whereas BP decoding

may converge to fixed points which are not codewords. These points are usually called

pseudo-codewords, and it has been shown [19] that they are of first importance in the loss

of performance of BP decoding compared to maximum-likelihood decoding.

To try to improve the BP decoding, we focus on pseudo-codewords, but indirectly. Indeed,

we make the assumption that pseudo-codewords are the indicators that the frontiers

of the classifier implemented by the BP decoder are not the frontiers of ML decoding.

Hence, we are going to try to find a correction to BP decoding by considering it as a


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