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 G F (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 2 o ( k ) , we show that a random d -linear form f ( X 1 X 2 X d ) : G F (2 ) k d G F (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 : [ k ] [ k ] d times G F (2) of rank at most t , the bias of the associated d -linear form f T ( X 1 X d ) : = ( i 1 i d ) [ k ] d T ( i 1 i 2 i d ) X 1 i 1 X 1 i 2 X d i d is at least 1 − 1 2 d − 1 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 G F (2) .