The Building Code allows you to build a small home (one bedroom, one bath, a kitchen). See the Tiny Houses section “Appendix Q” that gives specifics on what codes have been adjusted for this type of build.
Tiny homes on wheels or trailers are prohibited in Louisa County as a single family dwelling per Sec. 86-111: "Use of travel trailers, campers, etc., for dwellings prohibited. Travel trailers, campers, motor homes, tents, camp cabins and the like shall not be used for permanent or full-time dwellings in any district, unless otherwise permitted, whether or not connected to utilities, wells or septic systems. For the purposes of this section, permanent means available for occupancy for 30 consecutive days regardless of actual use."