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The largest minimum weights among quaternary Hermitian linear complementary dual codes are known for dimension 2. In this paper, we give some conditions for the nonexistence of quaternary Hermitian linear complementary dual codes with large minimum weights. As an application, we completely determine the largest minimum weights for dimension 3, by using a classification of some quaternary codes. In addition, for a positive integer s, a maximal entanglement entanglement-assisted quantumarXiv:1904.07517v2 fatcat:fron32eqsbafbkbsexdyrmvxzi