Noncommutative Gr\"obner Bases and Ext groups; Application to the Steenrod Algebra

We consider a theory of noncommutative Gr\"obner bases on decreasingly filtered algebras whose associated graded algebras are commutative. We transfer many algorithms that use commutative Gr\"obner bases to this context. As an important application, we implement very efficient algorithms to compute the Ext groups over the Steenrod algebra $\mathscr{A}$ at the prime $2$. Especially, the cohomology of the Steenrod algebra $Ext_{\mathscr{A}}^{*, *}(\mathbb{F}_2, \mathbb{F}_2)$, which plays an important role in algebraic topology, is calculated up to total degree of 261, including the ring structure in this range.