Parametric representation of Boolean functions over a quasifield of \(8\)th order (Q2730572)

The authors define a linear form over the quasifield \(\langle R_{k},\oplus,\otimes\rangle\), \(R_{k}=\{0,1,\ldots,k-1\}\) with variables \(x_1,x_2,\ldots,x_{n}\) and with arrangement of brackets \(\sigma=(i_0,i_1,\ldots,i_{n-1})\) as the sum \(w_0\oplus w_1\otimes x_1\oplus\ldots\oplus w_{n} \otimes x_{n}\) in which the bracket arrangement is given by the permutation \(\sigma\). Let be \(\phi:R_{k}\to R_2\). The representation \(f(x_1,x_2, \ldots,x_{n})=\phi(w_0\oplus w_1\otimes x_1\oplus\ldots\oplus w_{n} \otimes x_{n})\) of a Boolean function \(f\) is called parametric representation of \(f\). This paper deals with the dependence of the maximal number of Boolean functions of three variables which permit a parametric representation over a quasifield of \(8\)th order by the method of bracket arrangement in weighted sums.