Complete decoder and uniform recovery

Let $J$ identify occupation bases and set

Retained Kraus operators are $(I_M\otimes ⟨a{|}_K)W^{†}$. For an off-code basis $z_{\ell }$, add $2^{-1/2}|i{⟩}_M⟨z_{\ell }|$, $i=0,1$. Their effects sum to $I_R$, so the decoder is CPTP on every bath input.