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In [[coding theory]], '''concatenated codes''' form a class of [[Error detection and correction|error-correcting codes]] which are derived by combining an '''inner code''' and an '''outer code'''.
The [[Compact Disc]] and [[Voyager Program]] spacecraft use concatenated error correction technologies. Concatenated error codes may have been introduced with spacecraft missions that were involved with mapping the Moon and Mars. In typical telecommunications systems before these data intensive missions -- only one hard error correction layer was used.▼
==Description==
Let ''C'' be a code with length ''N'', size ''K'' and minimum distance ''D'' over an alphabet ''A'' with ''k'' symbols. Let ''I'' be another code with length ''n'', size ''k'' and minimum distance ''d'' over an alphabet ''B''.
The inner code ''I'' takes one of ''k'' possible inputs, encodes onto an ''n''-tuple from ''B'', transmits, and decodes into one of ''k'' possible outputs. We regard this as a channel which can transmit one symbol from the alphabet ''A'', also of size ''k''. We use this channel ''N'' times to transmit each of the ''N'' symbols in a codeword of ''C''. The ''concatenation'' of ''C'' (as outer code) with ''I'' as (inner code) is thus a code of length ''Nn'' over the alphabet ''B''.
In a generalisation, there are ''N'' possible inner codes ''I''<sub>''i''</sub> and the ''i''-th symbol in a codeword of ''C'' is transmitted across the inner channel using the ''i''-th inner code. The [[Justesen code]]s are examples of generalised concatenated codes, where the outer codes is a [[Reed-Solomon code]].
==Applications==
▲The [[Compact Disc]] and [[Voyager Program]] spacecraft use concatenated error correction technologies. Concatenated error codes may have been introduced with spacecraft missions that were involved with mapping the Moon and Mars{{cn}}. In typical telecommunications systems before these data intensive missions -- only one hard error correction layer was used.
Concatenating [[error correction]] methods is useful in cases where the telecommunications link may be problematical. Typically a soft inner code (Viterbi) is used concatenated to a hard outer code (Reed-Solomon).
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Concatenating codes can provide performance near that of [[Turbo code]]s but with an increase in decoder complexity.
== References ==
* {{cite book | author=F.J. MacWilliams | authorlink=Jessie MacWilliams | coauthors=N.J.A. Sloane | title=The Theory of Error-Correcting Codes | publisher=North-Holland | date=1977 | isbn=0-444-85193-3 | pages=307-316 }}
{{Error correction}}
[[Category:Error detection and correction]]
[[Category:Coding theory]]
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