  In this paper, we prove new relations between the bias of multilinear forms,
the correlation between multilinear forms and lower degree polynomials, and the
rank of tensors over $GF(2)= \{0,1\}$. We show the following results for
multilinear forms and tensors.
  1. Correlation bounds : We show that a random $d$-linear form has
exponentially low correlation with low-degree polynomials. More precisely, for
$d \ll 2^{o(k)}$, we show that a random $d$-linear form $f(X_1,X_2, \dots, X_d)
: \left(GF(2)^{k}\right)^d \rightarrow GF(2)$ has correlation $2^{-k(1-o(1))}$
with any polynomial of degree at most $d/10$. This result is proved by giving
near-optimal bounds on the bias of random $d$-linear form, which is in turn
proved by giving near-optimal bounds on the probability that a random rank-$t$
$d$-linear form is identically zero.
  2. Tensor-rank vs Bias : We show that if a $d$-dimensional tensor has small
rank, then the bias of the associated $d$-linear form is large. More precisely,
given any $d$-dimensional tensor $$T :\underbrace{[k]\times \ldots
[k]}_{\text{$d$ times}}\to GF(2)$$ of rank at most $t$, the bias of the
associated $d$-linear form $$f_T(X_1,\ldots,X_d) := \sum_{(i_1,\dots,i_d) \in
[k]^d} T(i_1,i_2,\ldots, i_d) X_{1,i_1}\cdot X_{1,i_2}\cdots X_{d,i_d}$$ is at
least $\left(1-\frac1{2^{d-1}}\right)^t$.
  The above bias vs tensor-rank connection suggests a natural approach to
proving nontrivial tensor-rank lower bounds for $d=3$. In particular, we use
this approach to prove that the finite field multiplication tensor has tensor
rank at least $3.52 k$ matching the best known lower bound for any explicit
tensor in three dimensions over $GF(2)$.
