We present the theory of selfdistributive quasigroups and the construction of non-affine selfdistributive quasigroup of size 216 that was presented by Onoi in 1970 and which was the least known example of such structure of size 2k . Based on this construction, we introduce the notion of Onoi structures and Onoi mappings between them which generalizes Onoi's construction and which allows us to construct non-affine selfdistributive quasigroups of size 22k for k ≥ 3. We present and implement algorithm for finding central extensions of self- distributive quasigroups which enables us to classify non-affine selfdistributive quasigroups of size 2k and prove that those quasigroup exists exactly for k ≥ 6, k ̸= 7. We use this algorithm also in order to better understand the structure of non-affine selfdistributive quasigroups of size 26 . 1