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.

翻译：我们引入四进制修正四μ-循环码作为四循环码的一种修正。给出了四进制修正四μ-循环Hermitian自对偶码的基本性质。我们还构造了具有较大最小权重的四进制修正四μ-循环Hermitian自对偶码。首次构造了两个四进制Hermitian自对偶[56,28,16]码。这些码改进了此前所有四进制（线性）[56,28]码中最大最小权重的已知下界。此外，这些码隐含了量子[[56,0,16]]码的存在性。