Binary protograph codes are state-of-the-art codes over the binary field Galois Field GF(2). These codes were invented by JPL in 2003 and proposed to CCSDS (Consultative Committee for Space Data Systems) recently as a channel-coding standard for space communications. It is expected that the proposed codes for non-binary signaling such as M-PSK (M-ary phase-shift keying), M-QAM (multi-level quadrature amplitude modulation), and PPM (pulse position modulation) will have better performance. The predicted advantage is based on gap to capacity, larger minimum distance based on GV bound, and larger girth in their graph representation for improved iterative decoding performance. Similar to Reed-Solomon codes (RS is a non-binary code but cannot be soft-decoded using BP (Belief Propagation), these codes will outperform the binary protograph codes on burst error and burst era-sure channels.

Obtaining non-binary codes combined with non-binary modulation that can operate at very low threshold close to the theoretical channel capacity thresholds is very desirable for many applications for high-data-rate transmission over a given allocated bandwidth and quantum limited optical channels. The goal is to design such a non-binary coded modulation system for a given throughput with the lowest possible signal-to-noise ratio for not only an additive White Gaussian noise channel, but also for other types of channel models used in wireless communications, satellite communications, and optical communications.

One of the most popular codes to be used for such applications is the low-density parity-check (LDPC) code, and one of the most promising recent methods of designing LDPC codes is via so-called protographs. Protograph methods allow the design of LDPC codes that are modular, easy to encode, and have excellent performance.

A protograph-based design advances the state of the art in the ensemble design of structured non-binary LDPC codes. A protograph is a Tanner graph of a relatively small size used to generate larger graphs of various sizes. The construction of protograph-based codes is based on applying “copy-and-permute” operations on the constituent protographs. In this work, the generalized version of PEXIT analysis called non-binary PEXIT (NB-PEXIT) is introduced. Here, a protograph-based design of LDPC code ensembles with near-capacity thresholds in Galois fields of different sizes is proposed. Using this new proposed tool, protograph-based non-binary codes with the lowest thresholds for various non-binary modulations schemes were discovered.

This work was done by Dariush Divsalar of Caltech, and Lara Dolecek and Ben-Yue Chang of UCLA for NASA’s Jet Propulsion Laboratory. For more information, contact This email address is being protected from spambots. You need JavaScript enabled to view it.. NPO-48361

NASA Tech Briefs Magazine

This article first appeared in the February, 2015 issue of NASA Tech Briefs Magazine.

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