Let $N_1,k \ge 1$ be integers and let $N = N_1 k$. Let $G_1,...,G_k$ be an equi-partition of $[N] := \{1,2,\ldots,N\}$. Thus, $|G_j| = N_1$ for all $i$. Let $\mathcal S$ be the transversal of this partition, i.e collection of subsets of $[N]$ which contain exactly one element from each $G_j$. Note that $\mathcal S$ is isomorphic to the cartesian product of the $G_j$'s in an obvious way.
Let $X$ be a Rademacher random variable, i.e $\mathbb P(X=+1) = \mathbb P(X=-1) = 1/2$, and let $X_1,\ldots,X_N$ be iid copies of $X$. Fix $t \ge 0$, and define the random variable $$ Z := \prod_{I \in \mathcal S}(1+t X_I), $$ where $X_I := \prod_{i \in I}^k X_i$.
Question. What is the expectation of $Z$ as a function of $N_1$, $k$ and $t$ ?
Observation
For any subset $\mathcal T$ of $\mathcal S$, let $v(\mathcal T)$ be the $|\mathcal T|k$-tuple made by unpacking all the elements of $\mathcal T$. For any integer $r \in [0,N_1^k]$ let $\mathcal E_r$ be the collection of $r$-element subsets $\mathcal T$ of $\mathcal S$ such that each $n \in [N]$ which appears in $v(\mathcal T)$ does so an even number of times. If $\eta_r := |\mathcal E_r| $, then
$$ \mathbb E\, Z = \sum_{r=0}^{N_1^k} \eta_r t^r, $$
a $N_1^k$-degree polynomial in $t$. Note that $\eta_0 = 1$, $\eta_1 = 0$, $\eta_{N_1^k} = 0$ if $N_1$ is odd, and $\eta_{N_1^k} = 1$ otherwise.
Example
Consider the case where $N_1 = k = 2$. Thus, $N=4$. WLOG, let $G_1 = \{1,2\}$ and $G_2 = \{3,4\}$, then $\mathcal S = \{\{1,3\},\{1,4\},\{2,3\},\{2,4\}\}$. Then, $Z = (1+tX_1X_3)(1+tX_1X_4)(1+tX_2X_3)(1+tX_2X_4)$, which has expectation $1+t^4$. Because $X_i^1=1$ for all $i$ and $\mathbb E[X_i X_{i'}] = 0$ whenever $i \ne i'$.
Another way to see this is to note that $\eta_0 = \eta_4 = 1$ and $\eta_1 = \eta_2 = \eta_3 = 0$.