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{{Use dmy dates|date=July 2014}}
In [[coding theory]], '''concatenated codes''' form a class of [[error-correcting code]]s that are derived by combining an '''inner code''' and an '''outer code'''. They were conceived in 1966 by [[Dave Forney]] as a solution to the problem of finding a code that has both exponentially decreasing error probability with increasing block length and [[polynomial-time]] decoding [[Computational complexity theory|complexity]].<ref name="Forney">
{{cite journal
|author=G. D. Forney
|author-link=Dave Forney
|title=Concatenated codes
|publisher=MIT Press
|location=Cambridge, Massachusetts
|year=1967
}}
</ref>
Concatenated codes became widely used in space communications in the 1970s.

==Background==
The field of [[channel coding]] is concerned with sending a stream of data at the highest possible rate over a given [[communications channel]], and then decoding the original data reliably at the receiver, using encoding and decoding algorithms that are feasible to implement in a given technology.

[[Noisy-channel coding theorem|Shannon's channel coding theorem]] shows that over many common channels there exist channel coding schemes that are able to transmit data reliably at all rates <math>R</math> less than a certain threshold <math>C</math>, called the [[channel capacity]] of the given channel. In fact, the probability of decoding error can be made to decrease exponentially as the block length <math>N</math> of the coding scheme goes to infinity. However, the complexity of a naive optimum decoding scheme that simply computes the likelihood of every possible transmitted codeword increases exponentially with <math>N</math>, so such an optimum decoder rapidly becomes infeasible.

In his [https://web.archive.org/web/20121012080412/http://mitpress.mit.edu/catalog/item/default.asp?tid=5813&ttype=2 doctoral thesis], [[Dave Forney]] showed that concatenated codes could be used to achieve exponentially decreasing error probabilities at all data rates less than capacity, with decoding complexity that increases only polynomially with the code block length.

==Description==
[[File:Concatenated codes diagram.png|thumb|upright=2|Schematic depiction of a concatenated code built upon an inner code and an outer code.]]
[[File:Concatenation of Reed–Solomon code with Hadamard code.svg|thumb|400px|This is a pictorial representation of a code concatenation, and, in particular, the [[Reed–Solomon code]] with n=q=4 and k=2 is used as the outer code and the [[Hadamard code]] with n=q and k=log q is used as the inner code. Overall, the concatenated code is a <math>[q^2,k \log q]</math>-code.]]

Let ''C''''in'' be a [''n'', ''k'', ''d''] code, that is, a [[block code]] of length ''n'', [[dimension (vector space)|dimension]] ''k'', minimum [[Hamming distance]] ''d'', and [[code rate|rate]] ''r'' = ''k''/''n'', over an alphabet ''A'':
:<math>C_{in}: A^k \rightarrow A^n</math>
Let ''C''''out'' be a [''N'', ''K'', ''D''] code over an alphabet ''B'' with |''B''| = |''A''|''k'' symbols:
:<math>C_{out}: B^K \rightarrow B^N</math>
The inner code ''C''''in'' takes one of |''A''|''k'' = |''B''| possible inputs, encodes into an ''n''-tuple over ''A'', transmits, and decodes into one of |''B''| possible outputs. We regard this as a (super) channel which can transmit one symbol from the alphabet ''B''. We use this channel ''N'' times to transmit each of the ''N'' symbols in a codeword of ''C''''out''. The ''concatenation'' of ''C''''out'' (as outer code) with ''C''''in'' (as inner code), denoted ''C''''out''∘''C''''in'', is thus a code of length ''Nn'' over the alphabet ''A'':<ref name="Forney"/>
:<math>C_{out} \circ C_{in}: A^{kK} \rightarrow A^{nN}</math>
It maps each input message ''m'' = (''m''1, ''m''2, ..., ''m''K) to a codeword (''C''''in''(''m''<nowiki>'</nowiki>1), ''C''''in''(''m''<nowiki>'</nowiki>2), ..., ''C''''in''(''m''<nowiki>'</nowiki>N)),
where (''m''<nowiki>'</nowiki>1, ''m''<nowiki>'</nowiki>2, ..., ''m''<nowiki>'</nowiki>N) = ''C''''out''(''m''1, ''m''2, ..., ''m''K).

The ''key insight'' in this approach is that if ''C''''in'' is decoded using a [[maximum likelihood decoding|maximum-likelihood approach]] (thus showing an exponentially decreasing error probability with increasing length), and ''C''''out'' is a code with length ''N'' = 2''nr'' that can be decoded in polynomial time of ''N'', then the concatenated code can be decoded in polynomial time of its combined length ''n''2''nr'' = [[O notation|''O'']](''N''⋅log(''N'')) and shows an exponentially decreasing error probability, even if ''C''''in'' has exponential decoding complexity.<ref name="Forney"/> This is discussed in more detail in section [[#Decoding concatenated codes|Decoding concatenated codes]].

In a generalization of above concatenation, there are ''N'' possible inner codes ''C''''in'',''i'' and the ''i''-th symbol in a codeword of ''C''''out'' is transmitted across the inner channel using the ''i''-th inner code. The [[Justesen code]]s are examples of generalized concatenated codes, where the outer code is a [[Reed–Solomon code]].

==Properties==
'''1.''' The distance of the concatenated code ''C''''out''∘''C''''in'' is at least ''dD'', that is, it is a [''nN'', ''kK'', ''D''<nowiki>'</nowiki>] code with ''D''<nowiki>'</nowiki> ≥ ''dD''.

''Proof:''
Consider two different messages ''m''1 ≠ ''m''2 ∈ ''B''''K''. Let Δ denote the distance between two codewords. Then
:<math>\Delta(C_{out}(m^1), C_{out}(m^2)) \ge D.</math>

Thus, there are at least ''D'' positions in which the sequence of ''N'' symbols of the codewords ''C''''out''(''m''1) and ''C''''out''(''m''2) differ. For these positions, denoted ''i'', we have
:<math>\Delta(C_{in}(C_{out}(m^1)_i), C_{in}(C_{out}(m^2)_i)) \ge d.</math>

Consequently, there are at least ''d''⋅''D'' positions in the sequence of ''n''⋅''N'' symbols taken from the alphabet ''A'' in which the two codewords differ, and hence
:<math>\Delta(C_{in}(C_{out}(m^1)), C_{in}(C_{out}(m^2))) \ge dD.</math>

'''2.''' If ''C''''out'' and ''C''''in'' are [[linear block code]]s, then ''C''''out''∘''C''''in'' is also a linear block code.

This property can be easily shown based on the idea of defining a [[generator matrix]] for the concatenated code in terms of the generator matrices of ''C''''out'' and ''C''''in''.

==Decoding concatenated codes==

A natural concept for a decoding algorithm for concatenated codes is to first decode the inner code and then the outer code. For the algorithm to be practical it must be [[polynomial-time]] in the final block length. Consider that there is a polynomial-time unique decoding algorithm for the outer code. Now we have to find a polynomial-time decoding algorithm for the inner code. It is understood that polynomial running time here means that running time is polynomial in the final block length. The main idea is that if the inner block length is selected to be logarithmic in the size of the outer code then the decoding algorithm for the inner code may run in [[exponential time]] of the inner block length, and we can thus use an exponential-time but optimal [[Decoding methods#Maximum likelihood decoding|maximum likelihood decoder]] (MLD) for the inner code.

In detail, let the input to the decoder be the vector ''y'' = (''y''1, ..., ''y''''N'') ∈ (''A''''n'')''N''. Then the decoding algorithm is a two-step process:
# Use the MLD of the inner code ''C''in to reconstruct a set of inner code words ''y''<nowiki>'</nowiki> = (''y''<nowiki>'</nowiki>1, ..., ''y''<nowiki>'</nowiki>''N''), with ''y''<nowiki>'</nowiki>''i'' = MLD''C''in(''y''i), 1 ≤ ''i'' ≤ ''N''.
# Run the unique decoding algorithm for ''C''out on ''y''<nowiki>'</nowiki>.

Now, the time complexity of the first step is [[O notation|''O'']](''N''⋅exp(''n'')), where ''n'' = ''O''(log(''N'')) is the inner block length. In other words, it is ''N''''O''(1) (i.e., polynomial-time) in terms of the outer block length ''N''. As the outer decoding algorithm in step two is assumed to run in polynomial time the complexity of the overall decoding algorithm is polynomial-time as well.

===Remarks===

The decoding algorithm described above can be used to correct all errors up to less than ''dD''/4 in number. Using [[minimum distance decoding]], the outer decoder can correct all inputs ''y''<nowiki>'</nowiki> with less than ''D''/2 symbols ''y''<nowiki>'</nowiki>''i'' in error. Similarly, the inner code can reliably correct an input ''y''''i'' if less than ''d''/2 inner symbols are erroneous. Thus, for an outer symbol ''y''<nowiki>'</nowiki>''i'' to be incorrect after inner decoding at least ''d''/2 inner symbols must have been in error, and for the outer code to fail this must have happened for at least ''D''/2 outer symbols. Consequently, the total number of inner symbols that must be received incorrectly for the concatenated code to fail must be at least ''d''/2⋅''D''/2 = ''dD''/4.

The algorithm also works if the inner codes are different, e.g., for [[Justesen code]]s. The [[generalized minimum distance decoding|generalized minimum distance algorithm]], developed by Forney, can be used to correct up to ''dD''/2 errors.<ref name="gmd">
{{cite journal
|first=G. David
|last=Forney
|title=Generalized Minimum Distance Decoding
|journal=IEEE Transactions on Information Theory
|volume=12
|issue=2
|pages=125–131
|date=April 1966
|doi=10.1109/TIT.1966.1053873
}}</ref>
It uses [[erasure code|erasure]] information from the inner code to improve performance of the outer code, and was the first example of an algorithm using [[soft-decision decoding]].<ref>{{cite journal
|first1=Christopher C.H.
|last1=Yu
|first2=Daniel J.
|last2=Costello
|title=Generalized Minimum Distance Decoding for ''Q''ary Output Channels
|journal=IEEE Transactions on Information Theory
|volume=26
|issue=2
|pages=238–243
|date=March 1980
|doi=10.1109/TIT.1980.1056148
}}</ref><ref>{{cite journal
|first1=Yingquan
|last1=Wu
|first2=Christoforos
|last2=Hadjicostis
|title=Soft-Decision Decoding of Linear Block Codes Using Preprocessing and Diversification
|journal=IEEE Transactions on Information Theory
|volume=53
|issue=1
|pages=387–393
|date=January 2007
|doi=10.1109/tit.2006.887478
|s2cid=8338433
}}</ref>

==Applications==
Although a simple concatenation scheme was implemented already for the 1971 [[Mariner 8|Mariner]] Mars orbiter mission,<ref name="McEliece"/> concatenated codes were starting to be regularly used for [[Deep Space Network|deep space]] communication with the [[Voyager program]], which launched two [[space probe]]s in 1977.<ref name="deep-space-codes">K. Andrews et al., ''The Development of Turbo and LDPC Codes for Deep-Space Applications'', Proceedings of the IEEE, Vol. 95, No. 11, Nov. 2007.</ref> Since then, concatenated codes became the workhorse for efficient error correction coding, and stayed so at least until the invention of [[turbo codes]] and [[LDPC codes]].<ref name="McEliece"/><ref name="deep-space-codes"/>

Typically, the inner code is not a block code but a [[soft-decision decoder|soft-decision]] [[convolutional code|convolutional]] [[Viterbi decoder|Viterbi-decoded]] code with a short constraint length.<ref name="Odenwalder">
{{cite journal
|author=J. P. Odenwalder
|title=Optimal decoding of convolutional codes
|publisher=[[U.C.L.A.]], Systems Science Dept. (dissertation)
|year=1970
}}
</ref>
For the outer code, a longer hard-decision block code, frequently a [[Reed Solomon code]] with eight-bit symbols, is used.<ref name="Forney"/><ref name="McEliece">
{{cite journal
|author1=Robert J. McEliece
|author-link=Robert McEliece
|author2=Laif Swanson
|title=Reed–Solomon Codes and the Exploration of the Solar System
|publisher=JPL
|date=20 August 1993
}}
</ref>
The larger symbol size makes the outer code more robust to [[error burst]]s that can occur due to channel impairments, and also because erroneous output of the convolutional code itself is bursty.<ref name="Forney"/><ref name="McEliece"/> An [[forward error correction#Interleaving|interleaving layer]] is usually added between the two codes to spread error bursts across a wider range.<ref name="McEliece"/>

The combination of an inner Viterbi convolutional code with an outer Reed–Solomon code (known as an RSV code) was first used in ''[[Voyager 2]]'',<ref name="McEliece"/><ref>R. Ludwig, J. Taylor, [http://descanso.jpl.nasa.gov/DPSummary/Descanso4--Voyager_new.pdf Voyager Telecommunications Manual], [[JPL]] DESCANSO ''(Design and Performance Summary Series)'', March 2002.</ref> and it became a popular construction both within and outside of the space sector. It is still notably used today for [[satellite communication]]s, such as the [[DVB-S]] [[digital television]] broadcast standard.<ref>[http://www.etsi.org/deliver/etsi_en/300400_300499/300421/01.01.02_60/en_300421v010102p.pdf Digital Video Broadcasting (DVB); Framing structure, channel coding and modulation for 11/12 GHz satellite services], [[ETSI]] EN 300 421, V1.1.2, August 1997.</ref>

In a looser sense, any (serial) combination of two or more codes may be referred to as a concatenated code. For example, within the [[DVB-S2]] standard, a highly efficient [[LDPC code]] is combined with an algebraic outer code in order to remove any resilient errors left over from the inner LDPC code due to its inherent [[error floor]].<ref>[http://www.etsi.org/deliver/etsi_en/302300_302399/302307/01.02.01_60/en_302307v010201p.pdf Digital Video Broadcasting (DVB); Second generation framing structure, channel coding and modulation systems for Broadcasting, Interactive Services, News Gathering and other broadband satellite applications (DVB-S2)], [[ETSI]] EN 302 307, V1.2.1, April 2009.</ref>

A simple concatenation scheme is also used on the compact disc (CD), where an interleaving layer between two Reed–Solomon codes of different sizes spreads errors across various blocks.

== Turbo codes: A parallel concatenation approach ==
The description above is given for what is now called a serially concatenated code. [[Turbo code]]s, as described first in 1993, implemented a parallel concatenation of two convolutional codes, with an interleaver between the two codes and an iterative decoder that passes information forth and back between the codes.<ref name="deep-space-codes"/> This design has a better performance than any previously conceived concatenated codes.

However, a key aspect of turbo codes is their iterated decoding approach. Iterated decoding is now also applied to serial concatenations in order to achieve higher coding gains, such as within serially concatenated convolutional codes (SCCCs). An early form of iterated decoding was implemented with two to five iterations in the "Galileo code" of the [[Galileo (spacecraft)|Galileo space probe]].<ref name="McEliece"/>

==See also==
*[[Gilbert–Varshamov bound]]
*[[Justesen code]]
*[[Singleton bound]]
*[[Zyablov bound]]

== References ==
{{reflist|33em}}

== Further reading ==
* {{cite book |author1=Shu Lin |author2=Daniel J. Costello Jr. | title=Error Control Coding: Fundamentals and Applications |url=https://archive.org/details/errorcontrolcodi00lins_044 |url-access=limited | publisher=[[Prentice Hall]] | year=1983 | isbn=978-0-13-283796-5 | pages=[https://archive.org/details/errorcontrolcodi00lins_044/page/n296 278]–280}}
* {{cite book | author=F.J. MacWilliams | authorlink=Jessie MacWilliams |author2=N.J.A. Sloane | title=The Theory of Error-Correcting Codes | url=https://archive.org/details/theoryoferrorcor0000macw | url-access=registration | publisher=North-Holland | year=1977 | isbn=978-0-444-85193-2 | pages=[https://archive.org/details/theoryoferrorcor0000macw/page/307 307–316] }}

== External links ==
* {{scholarpedia|title=Concatenated codes|urlname=Concatenated_codes|curator=[[Dave Forney]]}}
* [https://web.archive.org/web/20110606191907/http://www.cse.buffalo.edu/~atri/courses/coding-theory/fall07.html University at Buffalo Lecture Notes on Coding Theory – Dr. Atri Rudra]

{{DEFAULTSORT:Concatenated Error Correction Codes}}
[[Category:Error detection and correction]]
[[Category:Coding theory]]
[[Category:Finite fields]]
[[Category:Information theory]]

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