  A lower bound on the number of uncorrectable errors of weight half the
minimum distance is derived for binary linear codes satisfying some condition.
The condition is satisfied by some primitive BCH codes, extended primitive BCH
codes, Reed-Muller codes, and random linear codes. The bound asymptotically
coincides with the corresponding upper bound for Reed-Muller codes and random
linear codes. By generalizing the idea of the lower bound, a lower bound on the
number of uncorrectable errors for weights larger than half the minimum
distance is also obtained, but the generalized lower bound is weak for large
weights. The monotone error structure and its related notion larger half and
trial set, which are introduced by Helleseth, Kl{\o}ve, and Levenshtein, are
mainly used to derive the bounds.
