On the order of recursive differentiability of finite binary quasigroups

Abstract

The recursive derivatives of an algebraic operation are defined in [1], where they appear as control mappings of complete recursive codes. It is proved in [1], in particular, that the recursive derivatives of order up to r of a finite binary quasigroup (Q, ·) are quasigroup operations if and only if (Q, ·) defines a recursive MDS-code of length r + 3. The author of the present note gives an algebraic proof of an equivalent statement: a finite binary quasigroup (Q, ·) is recursively r-differentiable (r ≥ 0) if and only if the system consisting of its recursive derivatives of order up to r and of the binary selectors, is orthogonal. This involves the fact that the maximum order of recursive differentiability of a finite binary quasigroup of order q does not exceed q − 2.