We introduce quaternary modified four \(\mu \) -circulant codes as a modification of four circulant codes. We give basic properties of quaternary modified four \(\mu \) -circulant Hermitian self-dual codes. We also construct quaternary modified four \(\mu \) -circulant Hermitian self-dual codes having large minimum weights. Two quaternary Hermitian self-dual [56, 28, 16] codes are constructed for the first time. These codes improve the previously known lower bound on the largest minimum weight among all quaternary (linear) [56, 28] codes. In addition, these codes imply the existence of a quantum [[56, 0, 16]] code.
