Max 1-in-kSAT is the following maximisation problem :
Given $n$ variables $x_1,\dots,x_n$, and $m$ clauses $C_1, \dots, C_m$, find a valuation such that the number of clauses satisfied by exactly one literal is maximal.
In the following paper, there are several inaproximability results about this problem, such as
Theorem 9 : if $ZPP\neq NP$, then for every $\epsilon$, there is a constant $c$ such that there is no polynomial time $(n^{1-\epsilon})$-approximation algorithm for Max 1-in-$(c\log n)$SAT
https://people.eecs.berkeley.edu/~luca/pubs/gt05.pdf
I want to know if this result is still true if we suppose that $m=O(n^{\delta})$ where $\delta$ is any positive constant
Any ideas, remarks, or paper would be useful :)
