Symmetric tensor categories in characteristic 2

We construct and study a nested sequence of finite symmetric tensor categories ${\rm Vec}=$\mathcal{C}$_0\subset $\mathcal{C}$_1\subset\cdots\subset $\mathcal{C}$n\subset\cdots$overafieldofcharacteristic$2$suchthat$$\mathcal{C}${2n}$ are incompressible, i.e., do not admit tensor functors into tensor categories of smaller Frobenius–Perron dimension. This generalizes the category ${\mathcal{C}}_1$ described by Venkatesh and the category ${\mathcal{C}}_2$ defined by Ostrik. The Grothendieck rings of the categories ${\mathcal{C}}_{2n}$ and ${\mathcal{C}}_{2n+1}$ are both isomorphic to the ring of real cyclotomic integers defined by a primitive $2^{n+2}$-th root of unity, ${\mathcal{O}}_n=\mathbb{Z}[2\cos (\pi \mathrm{/}{2}^{n+1})]$.