The weight enumerator polynomial of a $(n,k)$ binary linear code $\mathcal{C}$ is defined as
$$WE(\mathcal{C}) = \sum_{i=0}^{n}WE_{i}(\mathcal{C}) x^{i}$$
where $$WE_{i}(\mathcal{C}) = \#\{c\in\mathcal{C} : |c| = i\}$$
is the number of codewords in $\mathcal{C}$ of Hamming weight (denoted by $|c|$) $i$.
It is known from:
That computing the weight enumerator polynomial for a binary linear code is a problem in $\mathsf{\#P}$-complete.
I wish to know if anything is known about the case of:
computing the $i^\text{th}$ coefficient of the weight enumerator polynomial where $i = \text{polylog}(n)$ (otherwise a naive enumeration of all binary sequences of weight $i$ and verification of those in the code would be possible).
Computing the weight enumerator, given the promise $WE_{i}(\mathcal{C}) \leq n^\lambda$, for $\lambda$ any constant, independent of $n$.
Does the complexity of the problem remain $\mathsf{\#P}$-hard in both cases ?
