Repeat-accumulate (RA) code

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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

Minimum-distance upper bounds [3,4].

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.

Cousin

Quasi-cyclic LDPC (QC-LDPC) code— There exist quasi-cyclic versions of RA codes [4].

Member of code lists

Primary Hierarchy

Classical Domain

Binary Kingdom

Binary codeECC

Binary group-orbit codeECC

Linear binary codeLinear $q$-ary Linear code over $\mathbb{Z}_q$ ECC

Low-density parity-check (LDPC) code$q$-ary LDPC Tanner Linear $q$-ary LRC Distributed-storage ECC

Multi-edge LDPC code

Irregular LDPC code

Irregular repeat-accumulate (IRA) code$q$-ary LDPC Tanner Linear $q$-ary Linear code over $\mathbb{Z}_q$ LRC Distributed-storage Concatenated Generalized concatenated code (GCC) ECC

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$q$-ary LDPC Tanner Linear $q$-ary Linear code over $\mathbb{Z}_q$ LRC Distributed-storage ECC

ARA codes with no pre-encoding accumulator and no post-accumulator puncturing reduce to RA codes.

Repeat-accumulate-accumulate (RAA) code$q$-ary LDPC Tanner Linear $q$-ary Linear code over $\mathbb{Z}_q$ LRC Distributed-storage ECC

RAA codes with no second permutation and accumulator reduce to RA codes.

Repeat-accumulate (RA) code

References

[1]
D. Divsalar, H. Jin, and R. J. McEliece, “Coding theorems for” turbo-like“ codes”, Proceedings of the annual Allerton Conference on Communication control and Computing 36. University Of Illinois, 1998
[2]
S. J. Johnson, “Introducing low-density parity-check codes”, University of Newcastle, Australia 1 (2006): 2006
[3]
J. Chen, R. M. Tanner, J. Zhang, and M. P. C. Fossorier, “Construction of Irregular LDPC Codes by Quasi-Cyclic Extension”, IEEE Transactions on Information Theory 53, 1479 (2007) DOI
[4]
R. M. Tanner, “On quasi-cyclic repeat-accumulate codes”, Proceedings of the 37th Annual Allerton Conference on Communication, Control, and Computing, Monticello, IL, Sept. 1999. 37, 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]
D. J. C. MacKay, Information Theory, Inference and Learning Algorithms (Cambridge University Press, 2003)

Page edit log

Victor V. Albert (2026-06-08) — most recent
Victor V. Albert (2023-05-04)

Cite as:

“Repeat-accumulate (RA) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/ra, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/tanner/irregular/ra/ra.yml.