On Construction of Orthogonal $d$-ary Operations

Smile Markovski, Aleksandra Mileva

A $d$-hypercube of order $n$ is an $\underbrace{n\times\dots\times n}_d$ 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.