  Sequence generators obtained by linear recursions over the two-element field
$\mathbb{F}_2$, i.e., $\mathbb{F}_2$-linear generators, are widely used as
pseudorandom number generators. For example, the Mersenne Twister MT19937 is
one of the most successful applications. An advantage of such generators is
that we can assess them quickly by using theoretical criteria, such as the
dimension of equidistribution with $v$-bit accuracy. To compute these
dimensions, several polynomial-time lattice reduction algorithms have been
proposed in the case of $\mathbb{F}_2$-linear generators.
  In this paper, in order to assess non-random bit patterns in dimensions that
are higher than the dimension of equidistribution with $v$-bit accuracy,we
focus on the relationship between points in the Couture--L'Ecuyer dual lattices
and $\mathbb{F}_2$-linear relations on the most significant $v$ bits of output
sequences, and consider a new figure of merit $N_v$ based on the minimum weight
of $\mathbb{F}_2$-linear relations whose degrees are minimal for $v$. Next, we
numerically show that MT19937 has low-weight $\mathbb{F}_2$-linear relations in
dimensions higher than 623, and show that some output vectors with specific
lags are rejected or have small $p$-values in the birthday spacings tests. We
also report that some variants of Mersenne Twister, such as WELL generators,
are significantly improved from the perspective of $N_v$.
