  GF(2)-grammars are a somewhat recently introduced grammar family that have
some unusual algebraic properties and are closely connected to unambiguous
grammars. In "Bounded languages described by GF(2)-grammars", Makarov proved a
necessary condition for subsets of $a_1^* a_2^* \cdots a_k^*$ to be described
by some GF(2)-grammar. By extending these methods further, we prove an even
stronger upper bound for these languages. Moreover, we establish a lower bound
that closely matches the proven upper bound. Also, we prove the exact
characterization for the special case of linear GF(2)-grammars. Finally, by
using the previous result, we show that the class of languages described by
linear GF(2)-grammars is not closed under GF(2)-concatenation
