Fast, high-performance coders and decoders could be designed.

Accumulate-repeat- accumulate-accumulate (ARAA) codes have been proposed, inspired by the recently proposed accumulate- repeat-accumulate (ARA) codes. These are error-correcting codes suitable for use in a variety of wireless data-communication systems that include noisy channels. ARAA codes can be regarded as serial turbolike codes or as a subclass of low-density parity- check (LDPC) codes, and, like ARA codes they have projected graph or protograph representations; these characteristics make it possible to design high-speed iterative decoders that utilize belief-propagation algorithms. The objective in proposing ARAA codes as a subclass of ARA codes was to enhance the error-floor performance of ARA codes while maintaining simple encoding structures and low maximum variable node degree.

These Block Diagrams and Protographs illustrate the similarities and differences between a simple rate-1/2 ARA code and a simple rate-1/2 ARAA code.
A rate-1/2 classical repeat-and-accumulate (RA) code has a high threshold (3.01 dB). An ARAA code can be viewed as a preceded RA code with puncturing in concatenation with another accumulation, wherein the preceding is also simply an accumulation; these characteristics make it possible to design very fast encoders. The top part of the figure illustrates the simplest example of the encoding process for a rate-1/2 ARA code, its protograph (filled nodes correspond to transmitted code symbols), and the corresponding decoding threshold of 0.516 dB. Other rate-1/2 ARA examples with maximum variable node degree 5 have thresholds as low as 0.26 dB, which can be compared to the Shannon capacity limit of 0.19 dB.

The bottom part of the figure illustrates a simple example of the encoding process for a rate-1/2 ARAA code, its protograph, and the corresponding threshold of 0.654 dB. The protograph of this code is similar to the ARA-code protograph shown in the top part of the figure, except for the additional accumulator stage and fewer parallel edges. The maximum variable node degree (4) of this ARAA protograph is less than that of the ARA protograph, but the total number of nodes is greater than in the ARA protograph.

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