Minimal generalized computable numberings and families of positive preorders

We study $A$-computable numberings for various natural classes of sets. For an arbitrary oracle $A\geq_T \mathbf{0'}$, an example of an $A$-computable family $S$ is constructed in which each $A$-computable numbering of $S$ has a minimal cover, and at the same time, $S$ does not satisfy the sufficient conditions for the existence of minimal covers specified by S. A. Badaev and S. Yu. Podzorov in [Sib. Math. J., 43, No. 4, 616–622 (2002)]. It is proved that the family of all positive linear preorders has an $A$-computable numbering iff $A' \geq_T \mathbf{0}''$. We obtain a series of results on minimal $A$-computable numberings, in particular, Friedberg numberings and positive undecidable numberings.