Investigations of quasigroups generated by proper families of Boolean functions of order 2

We analyze all Latin squares of order 4 generated by proper families of Boolean functions. It turns out that all these Latin squares define polynomially incomplete quasigroups. We propose a generalization of the construction based on proper families. As a result, the number of generated Latin squares grows four times and an essential number of the corresponding quasigroups becomes polynomially complete.