The augmented message-matrix approach to deterministic dense coding   theory
  A method is presented for producing analytical results applicable to the standard two-party deterministic dense coding protocol, wherein communication of K perfectly distinguishable messages is attainable with the aid of K selected local unitary operations on one qudit from a pair of entangled qudits of equal dimension d in a pure state. The method utilizes the properties of a (d^2)x(d^2) unitary matrix whose initial columns represent message states of the system used for communication, augmented by sufficiently many additional orthonormal column vectors so that the resulting matrix is unitary. Using the unitarity properties of this augmented message-matrix, we produce simple proofs of previously established results including (i) an upper bound on the value of the square of the largest Schmidt coefficient, given by d/K, and (ii) the impossibility of finding a pure state that can enable transmission of K=d^2-1 messages but not d^2. Additional results obtained using the method include proofs that when K=d+1 the upper bound on the square of the largest Schmidt coefficient (i) always reduces to at least (1/2)[1+sqrt{(d-2)/(d+2)}], and (ii) reduces to (d-1)/d in the special case that the identity and shift operators are two of the selected local unitaries. 
E. Gerjuoy, H. T. Williams, P. S. Bourdon