Characterize intermediate permutation-group tightness for self-dual CSS codes

Determine whether permutationally self-dual or strictly self-dual CSS codes exist whose permutation logical groups attain the full symplectic branch $Sp(k,\mathbb{F}_2)$ for $k=6$ or the full orthogonal branch $O(k,\mathbb{F}_2)$ for $6\leq k\leq8$.

Background

For self-dual CSS codes, the paper derives upper bounds on permutation logical groups. It proves that the full symplectic branch cannot be attained for sufficiently large even $k$ and that the full orthogonal branch cannot be attained for $k\geq9$, while constructions attain the branches at smaller values of $k$. The intermediate cases therefore remain unresolved and are needed to complete the finite-$k$ classification.