  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.
