On construction of orthogonal d-ary operations

A d-hypercube of order n is an n x . . . x n(d) (d times) array with n d elements from a set Q of cardinality n. We recall several connections between d-hypercubes of order n and d-ary operations of order n. We give constructions of orthogonal d-ary operations that generalize a result of Belyavskaya and Mullen. Our main result is a general construction of d-orthogonal d-ary operations from d-ary quasigroups.