Hello there, I am currently reading the Protocol Description in the Halo 2 Book, but I have some trouble finding the link to other descriptions on Halo 2 I have read. I recognize concepts from the Halo paper and from the remainder of the Halo 2 Book, but there are some deviations that I have not found described anywhere else:

g' seems to correspond to \text{gate}_0(x) + \cdots + y^i \cdot \text{gate}_i(x) + \cdots from the vanishing argument. But what are these challenges c_j about that it depends on? a_j' seems to depend on c_1, \ldots, c_{j1}, but these seem to be neither reflected in the description on circuit commitments nor in the code (crate
halo2_proofs
,plonk/prover.rs
). 
The polynomial to evaluate \boldsymbol{b} has changed from \prod_{j=1}^{k}(u_i+u_{i}^{1}X^{2^j1}) to \prod_{j=0}^{k1}(1+u_{k1j}X^{2^j}). The idea seems to be the same, but why did this change?
I would be happy if someone could give me some hints on this!