# Concatenated code[1]

Also known as Serially concatenated code.

## Description

A code whose encoding mapping is a composition of two mappings: first the message set is mapped onto the code space of the outer code, then each coordinate of the outer code is mapped on the code space of the inner code. In the basic construction, the outer code's alphabet is the finite field \(GF(p^m)\) and the \(m\)-dimensional inner code is over over the field \(GF(p)\). The construction is not limited to linear codes.

## Rate

There exist bounds on distance and rate of concatenated codes with a fixed outer and random inner code [2,3].

## Decoding

Generalized minimum-distance decoder [4].

## Parent

## Children

- Irregular repeat-accumulate (IRA) code — IRA codes can be interpreted as serial concatenated codes [5].
- Tensor-product code — Tensor-product codes can be viewed as serial concatenated codes [6].
- Binary balanced spherical code — A binary balanced spherical code can be thought of as a concatenation of a constant-weight binary outer code with a shifted and scaled BPSK-like inner code.
- Polyphase code — A polyphase code can be thought of as a concatenation of a \(q\)-ary outer code with a PSK inner code.

## Cousins

- Concatenated quantum code — Quantum codes can be concatenated with classical codes to yield good quantum codes [7].
- Hsu-Anastasopoulos LDPC (HA-LDPC) code — HA-LDPC codes are a concatenation of an LDPC and an LDGM code.
- Generalized RS (GRS) code — Concatenations of GRS codes with random linear codes almost surely attain the GV bound [8].

## References

- [1]
- G. D. Forney, Jr (1966). Concatenated Codes. MIT Press, Cambridge, MA.
- [2]
- A. Barg, J. Justesen, and C. Thommesen, “Concatenated codes with fixed inner code and random outer code”, IEEE Transactions on Information Theory 47, 361 (2001) DOI
- [3]
- D. Doron, J. Mosheiff, and M. Wootters, “When Do Low-Rate Concatenated Codes Approach The Gilbert-Varshamov Bound?”, (2024) arXiv:2405.08584
- [4]
- G. Forney, “Generalized minimum distance decoding”, IEEE Transactions on Information Theory 12, 125 (1966) DOI
- [5]
- S. Benedetto et al., “Serial concatenation of interleaved codes: performance analysis, design, and iterative decoding”, IEEE Transactions on Information Theory 44, 909 (1998) DOI
- [6]
- A. Barg and G. Zemor, “Concatenated Codes: Serial and Parallel”, IEEE Transactions on Information Theory 51, 1625 (2005) DOI
- [7]
- M. Grassl, P. W. Shor, and B. Zeng, “Generalized concatenation for quantum codes”, 2009 IEEE International Symposium on Information Theory (2009) arXiv:0905.0428 DOI
- [8]
- C. Thommesen, “The existence of binary linear concatenated codes with Reed - Solomon outer codes which asymptotically meet the Gilbert- Varshamov bound”, IEEE Transactions on Information Theory 29, 850 (1983) DOI

## Page edit log

- Alexander Barg (2024-08-03) — most recent
- Victor V. Albert (2022-03-22)

## Cite as:

“Concatenated code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/concatenated