Finer characterization of bounded languages described by GF(2)-grammars

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