Biased Linearity Testing in the 1% Regime

We study linearity testing over the $p$-biased hypercube $(\{0,1\}^n, \mu_p^{\otimes n})$ in the 1% regime. For a distribution $\nu$ supported over $\{x\in \{0,1\}^k:\sum_{i=1}^k x_i=0 \text{ (mod 2)} \}$, with marginal distribution $\mu_p$ in each coordinate, the corresponding $k$-query linearity test $\text{Lin}(\nu)$ proceeds as follows: Given query access to a function $f:\{0,1\}^n\to \{-1,1\}$, sample $(x_1,\dots,x_k)\sim \nu^{\otimes n}$, query $f$ on $x_1,\dots,x_k$, and accept if and only if $\prod_{i\in [k]}f(x_i)=1$. Building on the work of Bhangale, Khot, and Minzer (STOC '23), we show, for $0 < p \leq \frac{1}{2}$, that if $k \geq 1 + \frac{1}{p}$, then there exists a distribution $\nu$ such that the test $\text{Lin}(\nu)$ works in the 1% regime; that is, any function $f:\{0,1\}^n\to \{-1,1\}$ passing the test $\text{Lin}(\nu)$ with probability $\geq \frac{1}{2}+\epsilon$, for some constant $\epsilon > 0$, satisfies $\Pr_{x\sim \mu_p^{\otimes n}}[f(x)=g(x)] \geq \frac{1}{2}+\delta$, for some linear function $g$, and a constant $\delta = \delta(\epsilon)>0$. Conversely, we show that if $k < 1+\frac{1}{p}$, then no such test $\text{Lin}(\nu)$ works in the 1% regime. Our key observation is that the linearity test $\text{Lin}(\nu)$ works if and only if the distribution $\nu$ satisfies a certain pairwise independence property.