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[en] Under two-party deterministic dense coding, Alice communicates (perfectly distinguishable) messages to Bob via a qudit from a pair of entangled qudits in pure state |Ψ>. If |Ψ> represents a maximally entangled state (i.e., each of its Schmidt coefficients is √(1/d)), then Alice can convey to Bob one of d2 distinct messages. If |Ψ> is not maximally entangled, then Ji et al. [Phys. Rev. A 73, 034307 (2006)] have shown that under the original deterministic dense-coding protocol, in which messages are encoded by unitary operations performed on Alice's qudit, it is impossible to encode d2-1 messages. Encoding d2-2 messages is possible; see, for example, the numerical studies by Mozes et al. [Phys. Rev. A 71, 012311 (2005)]. Answering a question raised by Wu et al. [Phys. Rev. A 73, 042311 (2006)], we show that when |Ψ> is not maximally entangled, the communications limit of d2-2 messages persists even when the requirement that Alice encode by unitary operations on her qudit is weakened to allow encoding by more general quantum operators. We then describe a dense-coding protocol that can overcome this limitation with high probability, assuming the largest Schmidt coefficient of |Ψ> is sufficiently close to √(1/d). In this protocol, d2-2 of the messages are encoded via unitary operations on Alice's qudit, and the final (d2-1)-th message is encoded via a non-trace-preserving quantum operation.