Properties
1. The distance of the concatenated code Cout∘Cin is at least dD, that is, it is a code with D' ≥ dD.
Proof: Consider two different messages m1 ≠ m2 ∈ BK. Let Δ denote the distance between two codewords. Then
Thus, there are at least D positions in which the sequence of N symbols of the codewords Cout(m1) and Cout(m2) differ. For these positions, denoted i, we have
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
2. If Cout and Cin are linear block codes, then Cout∘Cin 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 Cout and Cin.
Read more about this topic: Concatenated Error Correction Code
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