# Repeat-accumulate (RA) code[1]

## Description

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

## Protection

## Rate

RA codes are not asymptotically good [5].

## Encoding

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.

## Parents

- Irregular repeat-accumulate (IRA) code — IRA codes for which the outer code is a 1-in-3 repetition code reduce to RA codes.
- Accumulate-repeat-accumulate (ARA) code — ARA codes with no pre-encoding acumulator and no post-acumulator puncturing reduce to RA codes.
- Repeat-accumulate-accumulate (RAA) code — RAA codes with no second permutation and acumulator reduce to RA codes.

## Cousin

- Quasi-cyclic LDPC (QC-LDPC) code — There exist quasi-cyclic versions of RA codes [7].

## References

- [1]
- Divsalar, Dariush, Hui Jin, and Robert J. McEliece. "Coding theorems for" turbo-like" codes." Proceedings of the annual Allerton Conference on Communication control and Computing. Vol. 36. University Of Illinois, 1998.
- [2]
- Johnson, Sarah J. "Introducing low-density parity-check codes." University of Newcastle, Australia 1 (2006): 2006.
- [3]
- J. Chen et al., “Construction of Irregular LDPC Codes by Quasi-Cyclic Extension”, IEEE Transactions on Information Theory 53, 1479 (2007) DOI
- [4]
- Tanner, R. Michael. "On quasi-cyclic repeat-accumulate codes." Proc. 37th Allerton Conf., Monticello, IL, Sept. 1999. 1999.
- [5]
- L. Bazzi, M. Mahdian, and D. A. Spielman, “The Minimum Distance of Turbo-Like Codes”, IEEE Transactions on Information Theory 55, 6 (2009) DOI
- [6]
- David J. C. MacKay. Information Theory, Inference and Learning Algorithms. Cambridge university press, 2003.
- [7]
- 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.

## Page edit log

- Victor V. Albert (2023-05-04) — most recent

## Cite as:

“Repeat-accumulate (RA) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/ra