Equality Cases in the Tu-Deng Conjecture Bound

The Tu-Deng Conjecture, made in 2011, asserts that for $k\geq2$ and $1\leq t<2^k-1$, the set \[ S_{t,k}=\left\{(a,b)\in\{0,1,\ldots,2^k-2\}^2: a+b\equiv t\pmod{2^k-1},\quad \wt(a)+\wt(b)<k\right\} \] satisfies \[ |S_{t,k}|\leq 2^{k-1}. \] Proofs of the conjecture were given independently by the