I am interested in a general (not necessarily tight) distance lower bound for lifted product codes, specifically the so called 'quasi-Abelian' lifted product (LP) codes. I have seen the nice answer by Panteleev in Distance of the Lifted Product codes and the references therein, which gives such a general bound in terms of distances of related classical codes. This question can be considered a follow up, as I am wondering how this bound is proven using homological arguments.
Here is the setup. Let $G$ be a finite group of order $l$, and let $\mathbb{F}_2[G]$ denote the group algebra over $\mathbb{F}_2$. For simplicity, I will consider the smallest LP codes denoted by $LP(a,b)$ where $a,b\in \mathbb{F}_2[G]$, also known as 2-block group algebra codes (2-BGA) - see https://arxiv.org/pdf/2306.16400. Let $A$ and $B$ denote the $l\times l$ binary matrices given by the left and right regular representations of $a$ and $b$ respectively. Then, $LP(a,b)$ is the code with block parity check matrices $H_X = [A \ B]$, $H_Z = [B^T \ A^T]$. $LP(a,b)$ can also be considered as the code arising from the homological product of the classical group algebra codes with chain complex boundary maps given by multiplication by $a$ and $b$ respectively, where the tensor products in the product complex are over the module $R \equiv \mathbb{F}_2[G]$.
When $G$ is abelian so that $LP(a,b)$ is 'quasi-Abelian', the answer in Distance of the Lifted Product codes states that the distance of the quantum code is lower bounded as
$$d_{LP} \geq \frac{1}{l} \cdot min(d(A), d(A^T), d(B), d(B^T))$$ where the distances on the right are those of the classical codes defined by the associated binary matrices as parity check matrices. There is also a tighter variant of this theorem that also holds for non-Abelian groups given in https://arxiv.org/pdf/2306.16400 (page 8, statement 12), where the factor $\frac{1}{l}$ is replaced by a potentially smaller factor related to the intersection subgroup of $a$ and $b$.
The above distance lower bound can apparently be proven from a Kunneth formula for tensor products over group algebras. The original answer cites the S.M. Loor thesis for the Kunneth formula (page 59, equation 5.39).
My two questions are:
How does this distance lower bound follow from the Kunneth theorem? I can see how the dimension of the quantum code can be derived from the Kunneth expansion for the first Homology group, but bounding the distance is less obvious. In particular, I do not know where the $1/l$ factor comes from.
It appears that the Kunneth theorem as stated in the linked text applies to modules over any finite dimensional group algebra that is 'flat', i.e Torsion free. In particular, Maschke's theorem implies that F_2[G] is flat whenever |G| is odd. Why do we need the restriction to quasi-Abelian LP codes for the above bound to hold, and if this restriction is unnecessary, is there an obvious way to generalize it?
