Effects of shortening and puncturing on codes

Question

Given a binary block code $C=[n,k]$ of codeword length $n$, and dimension $k$. Suppose I've determined these properties for it : $d_{min}$ (minimum distance), $N_{dmin}$ (number of codewords at $d_{min}$), $G_{min}$ (minimum girth), and $N_{four}$ (number of 4-cylces in its Tanner graph). These are all commonly defined properties so I won't redefine them here. Now suppose that I have a new code derived from $C$ by shortening $N_s$ bits and puncturing $N_p$ bits resulting in a new $C'=[n-N_s-N_p,k-N_s]$ code what can be said about this new code? How would $d_{min}$,$G_{min}$,...change? Bounds are fine if no exact treatment is known. Any reference or survey articles would be really appreciated.

Answer 1

Here is an answer that says (at least for $d_{\mathrm{min}}$) we cannot say much meaningful unless we know more about the specific code. For a moment let's forget about shortening and concentrate of minimum distance and puncturing. Puncturing will either leave minimum distance fixed or decrease by exactly $1$. Both are possible; so, we can say $d_{\mathrm{min}} - N_p \leq d'_{\mathrm{min}} \leq d_{\mathrm{min}}$ in the case of only puncturing.

Since shortening is expurgating (removing codewords) which may increase the minimum distance followed by puncturing which may decrease the minimum distance it is hard to say what will happen. Consider $C = \{0^n,10^{n-1},01^{n-1},1^n\}$ which has minimum distance $d'_{\mathrm{min}} = 1$. We can shorten $C$ to obtain $C' = \{0^{n-1}, 1^{n-1}\}$ which has $d'_{\mathrm{min}} = n-1$. We will still have a lower bound $d_{\mathrm{min}} - N_s - N_p \leq d'_{\mathrm{min}}$, but $d'_{\mathrm{min}}$ can become large.

I will think more if something can be said about your other quantities.