This article extended the properties of the idempotent generator of cyclic codes to polycyclic codes over the finite field ${\mathbb{F}}_q$. In addition, the check matrix of polycyclic codes was provided over ${\mathbb{F}}_q$. Specifically, it has been proven that the constacyclic code is an $\mathrm{M}\mathrm{D}\mathrm{S}$ code over ${\mathbb{F}}_q$ if and only if its annihilator dual code is also an $\mathrm{M}\mathrm{D}\mathrm{S}$ code. Finally, we have provided some examples of good codes.