Setup reminder: linear block error-correcting code is some linear subspace $C$ in $F_2^N$. (Correcting error means to find a point $c \in C$ which is "nearest" to a given $r$ in $F_2^N$, $r$ is signal with "errors").

Consider two codes which are given as images of some operators: $A: F_2^k \to F_2^L$, $B: F_2^L \to F_2^N$. Define the code which is "superposition" - image of $BA$.

**General question** What can be said about the properties of "superposition" code in terms of properties $A$ and $B$ ? E.g. how to construct decoding algorithm for it ? How to estimate error probability ? How to estimate minimal Hamming distance of this code ?

**Specific question** Consider code $A$ is just adding CRC bit: i.e.
$A: (x_1, ..., x_n) \to (x_1, ..., x_n, \sum x_i)$.

And $B$ is some tail-bited convolutional code (say rate 1/2) which means it is given by $( \bar y ) \to ( T_1 \bar y , T_2 \bar y) $ where $T_1, T_2$ are some circulant matrices with only few non-zero diagonals.

How to estimate minimal Hamming distance of this code ?

How to decode such a code ?

The problem is that Viterbi algorithm used for decoding convolutional codes is based on "local" structure of the circulant matrices, while adding the CRC breaks the locality...

How to estimate error probabilty for such a code ? (Say in AWGN channel)