  We propose dual pairs of $\mathcal{N}=(0,4)$ half-BPS boundary conditions for
3d $\mathcal{N}=4$ Abelian gauge theories related to mirror symmetry and
S-duality by showing the matching of boundary 't Hooft anomalies and
supersymmetric indices. We find simple $\mathcal{N}=(0,4)$ mirror symmetry
between 2d $\mathcal{N}=(0,4)$ Abelian gauge theories and free Fermi multiplets
that generalizes $\mathcal{N}=(0,2)$ Abelian duality. We also propose a
prescription for computing half-index of enriched Neumann boundary condition
including 2d boundary bosonic matters by gauging the 2d boundary flavor
symmetry of Dirichlet boundary condition. By coupling $\mathcal{N}=(0,4)$
half-BPS boundary configurations of 3d $\mathcal{N}=4$ gauge theories to
quarter-BPS corner configurations of 4d $\mathcal{N}=4$ Super Yang-Mills
theories, we further obtain a new type of 4d-3d-2d duality that may involve 3d
non-Abelian gauge symmetry.
