Concatenated error correction code: Difference between revisions

Revision as of 00:32, 5 December 2023

Latest revision as of 22:27, 28 May 2025

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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''∘<math>\circ</math>''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)),

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==Properties==

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

''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''_''out'' and ''C''_''in'' are [[linear block code]]s, then ''C''_''out''∘<math>\circ</math>''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''.