Low-density Parity-check Code - Code Construction

Code Construction

For large block sizes, LDPC codes are commonly constructed by first studying the behaviour of decoders. As the block size tends to infinity, LDPC decoders can be shown to have a noise threshold below which decoding is reliably achieved, and above which decoding is not achieved. This threshold can be optimised by finding the best proportion of arcs from check nodes and arcs from variable nodes. An approximate graphical approach to visualising this threshold is an EXIT chart.

The construction of a specific LDPC code after this optimisation falls into two main types of techniques:

• Pseudo-random approaches
• Combinatorial approaches

Construction by a pseudo-random approach builds on theoretical results that, for large block size, a random construction gives good decoding performance. In general, pseudo-random codes have complex encoders, however pseudo-random codes with the best decoders can have simple encoders. Various constraints are often applied to help ensure that the desired properties expected at the theoretical limit of infinite block size occur at a finite block size.

Combinatorial approaches can be used to optimise properties of small block-size LDPC codes or to create codes with simple encoders.

Yet another way of constructing LDPC codes is to use finite geometries. This method was proposed by Y. Kou et al. in 2001.