I went through Def 2.1 of this paper, where the approximation version of max-LINSAT is defined as follows.
Let $\mathbb{F}_p$ be a finite field.
Input: For each $i=1,\cdots,m$, let $F_i\subset \mathbb{F}_p$ be an arbitary subset of $\mathbb{F}_p$, which yields a corresponding constraints $$\sum_{j=1}^{n} B_{ij}x_j \in F_i.$$
Goal: The max-LINSAT problem is to find $\textbf{x}\in \mathbb{F}_p^n$ satisfying as many as possible of these $m$ constraints.
Let us define $\alpha = l/m$ as an approximation ratio, where $l$ is the number of satisfied constraints.
Question 1: What is the threshold value of $\alpha$ after which it becomes $\mathsf{NP}$-hard to approximate?
My attempt: I am going through this classical paper by Hastad, where the implication of $\mathsf{PCP}$ on the hardness of approximation for several variants of $\mathsf{SAT}$ is derived. Second row of Table-1 implies $\alpha=1/p+\epsilon$ for $\epsilon>0$ for problem E3-LIN-p.
Here, E3-LIN-p: given a system $m$ of linear equations over $\mathbb{F}_p$ , with exactly $k$ variables in each equation, to find x that maximizes the number of satisfied equations.
I find max-LINSAT very close to E3-Lin-p, as both are about m linear equations over $\mathbb{F}_p$. The only difference is the bound on the number of variables in each equation. One can reduce an instance of max-LINSAT to E3-Lin-p similar to how we reduce $k$-$\mathsf{SAT}$ to 3ドル$-$\mathsf{SAT}$ by introducing new clauses. This is a time- and space-efficient process.
My conclusion: Since the result mentioned in the second row of Table 1 in Hastad's paper is independent of the number of clauses. Thus, the threshold of $\alpha=1/p+\epsilon$ for any $\epsilon>0$ is equally valid for both max-LINSAT and E3-Lin-p.
Question 2: Is the above argument valid?