Short block codes are in demand by international space agencies for transmission of command and control data to spacecraft in the uplink channel. Codes that outperform currently used binary codes with the same block lengths and code rates are desirable. A design method constructs non-binary LDPC (low-density parity-check) codes both for unconstrained non-binary protograph-based codes and the new class of constrained protograph-based codes. The non-binary short block codes outperform the binary version by at least 1 dB for the same block size and code rate.
The design method for short blocks is based on ensuring a large enough girth and a suitable edge label assignment to achieve a large code minimum distance. For a given protograph, a derived graph by copy-and-permute operations was obtained using the circulant PEG (Progressive Edge Growth) algorithm such that the girth is as large as possible. The selection of the original protograph is guided by an EXIT analysis starting with a protograph having a competitive threshold at the prescribed alphabet size. After lifting the protograph to a desired size to obtain a large girth, the cycle distribution of the graph is computed to select the best one among the resulting graph candidates. The assignment of the non-binary labels from a given Galois field GF(q) is selective rather than random. The labels are selected so that the binary image of the check node with the assigned labels produces the largest possible minimum Hamming distance for that check node. For each q, all possible label sets are generated that produce the largest minimum distance for the binary images of the checks with associated labels. A modified cycle cancelation method was proposed that, in addition to the shortest cycles, eliminates other cycles as well.
The constrained protograph-based LDPC codes are a new class of codes proposed for low-complexity implementation of decoders. High-performance codes were developed for unconstrained non-binary protograph-based codes.
This work was done by Dariush Divsalar of Caltech, and Lara Dolecek and Ben-Yue Chang of UCLA for NASA’s Jet Propulsion Laboratory.