For a generic stabilizer code, there's a standard form for $H,X,Z$; see for example previuos post.
For CSS codes, there are at least these simplifications : $B=0$; $C_2=0$; I think $A_1$ and $D$ can also be transformed to $0$ but I'm not sure. There could also be other simplifications. Has anyone worked out the details or knows of a reference for such a standard form?
Certainly $B=C_2=0$ for CSS codes. But for the other two.
A CSS code is made out of two classical codes, with parity check matrices $H_X, H_Z$, and these matrices are the submatrices of the quantum code's stabilizer generator matrix.
Classical codes can also be transformed into equivalent codes by row operations on the parity-check matrices.
Let's imagine if $H_i$ is a $k_i\times n$ matrix for $i=X,Z$, and represents a $n$ bit code. Now, if $H_i$ could be simplified further so that entire columns could be made zero, then it is actually a code on fewer than $n$ bits! That is clearly not possible.
We have the matrix $$ H = \left(\begin{array}{ccc|ccc} I & A_1 & A_2 & 0 & 0 & 0 \\ 0 & 0 & 0 & D & I & E_2 \end{array}\right). $$
How do we intend to turn any of the submatrices $0$ with row operations? Take $D$ for instance. There are no $1$ in any row above it. So we can't do any operations that remove all $1$ from every column of $D$. We will always be left with at least one $1$ in each column no matter what row operations we do.
