Orthogonal shift systems in the field of $p$-adic numbers

In 2010 S. Albeverio, S. Evdokimov and M. Skopina proved that if the shift system $(\varphi(x\dot-h))$ of a step function $\varphi$ is orthonormal and $\varphi$ generates $p$-adic MRA then its Fourier transform lies in the unit ball. We prove then in some cases the condition "$\varphi$ generates MRA" is possible to be omitted. In general, we indicate the number of linearly independent step-functions, which shifts form an orthonormal system.