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Let ''C''<sub>''out''</sub> be a [''N'', ''K'', ''D''] code over an alphabet ''B'' with |''B''| = |''A''|<sup>''k''</sup> symbols:
:<math>C_{out}: B^K \rightarrow B^N</math>
The inner code ''C''<sub>''in''</sub> takes one of |''A''|<sup>''k''</sup> = |''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''<sub>''out''</sub>. The ''concatenation'' of ''C''<sub>''out''</sub> (as outer code) with ''C''<sub>''in''</sub> (as inner code), denoted ''C''<sub>''out''</sub>
:<math>C_{out} \circ C_{in}: A^{kK} \rightarrow A^{nN}</math>
It maps each input message ''m'' = (''m''<sub>1</sub>, ''m''<sub>2</sub>, ..., ''m''<sub>K</sub>) to a codeword (''C''<sub>''in''</sub>(''m''<nowiki>'</nowiki><sub>1</sub>), ''C''<sub>''in''</sub>(''m''<nowiki>'</nowiki><sub>2</sub>), ..., ''C''<sub>''in''</sub>(''m''<nowiki>'</nowiki><sub>N</sub>)),
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==Properties==
'''1.''' The distance of the concatenated code ''C''<sub>''out''</sub>
''Proof:''
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:<math>\Delta(C_{in}(C_{out}(m^1)), C_{in}(C_{out}(m^2))) \ge dD.</math>
'''2.''' If ''C''<sub>''out''</sub> and ''C''<sub>''in''</sub> are [[linear block code]]s, then ''C''<sub>''out''</sub>
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''<sub>''out''</sub> and ''C''<sub>''in''</sub>.
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==See also==
*[[Gilbert–Varshamov bound]]
*[[Justesen code]]
*[[Singleton bound]]
*[[
== References ==
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