Boundary effect of deterministic dense coding
  We present a rigorous proof of an interesting boundary effect of deterministic dense coding first observed by Mozes et al. [Phys. Rev. A 71, 012311 (2005)]. Namely, it is shown that $d^2-1$ cannot be the maximal alphabet size of any isometric deterministic dense coding schemes utilizing $d$-level partial entanglement. 
Zhengfeng Ji, Yuan Feng, Runyao Duan, Mingsheng Ying