Repeat-accumulate (RA) code[1] 


An LDPC code whose parity-check matrix has weight-two columns arranged in a step-like pattern for its last columns [2].


Minimum-distance upper bounds [3,4].


RA codes are not asymptotically good [5].


An encoder for an RA code acting on a string \((c_1c_2\cdots c_K)\) of logical bits begins by repeating each bit three times to obtain the length-\(3K\) bitstring \((c_1 c_1 c_1 c_2 c_2 c_2 \cdots c_K c_K c_K)\), permuting using a random permutation to obtain a bitstring \(u\), and applying the mod-two accumulated sum (or accumulator) to obtain [6; Ch. 49] \begin{align} (u_{1},u_{1}+u_{2},\cdots,u_{1}+\cdots+u_{3K})~. \tag*{(1)}\end{align} The first repeating step is effectively using a 1-in-3 repetition code, which can be thought of as the outer code in this concatenated construction.




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Johnson, Sarah J. "Introducing low-density parity-check codes." University of Newcastle, Australia 1 (2006): 2006.
J. Chen et al., “Construction of Irregular LDPC Codes by Quasi-Cyclic Extension”, IEEE Transactions on Information Theory 53, 1479 (2007) DOI
Tanner, R. Michael. "On quasi-cyclic repeat-accumulate codes." Proc. 37th Allerton Conf., Monticello, IL, Sept. 1999. 1999.
L. Bazzi, M. Mahdian, and D. A. Spielman, “The Minimum Distance of Turbo-Like Codes”, IEEE Transactions on Information Theory 55, 6 (2009) DOI
David J. C. MacKay. Information Theory, Inference and Learning Algorithms. Cambridge university press, 2003.
M. R. Tanner, "On quasi-cyclic repeat-accumulate codes." PROCEEDINGS OF THE ANNUAL ALLERTON CONFERENCE ON COMMUNICATION CONTROL AND COMPUTING. Vol. 37. The University; 1998, 1999.
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“Repeat-accumulate (RA) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.
@incollection{eczoo_ra, title={Repeat-accumulate (RA) code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Repeat-accumulate (RA) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.