Hypergraph product codes are constructed from two classical codes.
I have come across several works (1 2 3) that use the classical expander code constructed from the (3,4)-regular Tanner graph.
Classical expander codes are good codes, as they have constant rate and linear distance asymptotically, but why choose the (3,4)-regular construction specifically? Why not use, for example, a (2,3)- or (2,4)-regular graph—which offer better rates—or some other good classical code?
Found the explanation in Grospellier's thesis.
Expander codes have an expansion parameter $\gamma$ defined in Definition 8.1 here.
We know these codes achieve linear distance asymptotically if $\gamma > 1/2$.
However, for expander codes constructed from $(d_v,d_c)$-regular bipartite graph, Lemma 3.11 here shows: $$ 1-\gamma > \max\left(\frac{1}{d_v},\frac{1}{d_c}\right)\;. $$ Which requires $d_v \ge 3$ to satisfy $\gamma > 1/2$.
The rate of the code is given by $1-d_v/d_c$, so out of all expander codes with $\gamma > 1/2$, the $(3,4)$-regular construction has the lowest stabilizer weight (which is 7).