Projective polynomials, a projection construction and a family of semifields

The projection construction is a method to blend two or more semifields of the same order into a possibly new one. Early constructions which may be interpreted in this manner include the Hughes–Kleinfeld semifields and the Dickson semifields. In those cases the ingredients are isotopic versions of field multiplication. In the present paper we describe the theory of projective polynomials over finite fields in terms of an underlying non-associative algebra closely related to Knuth semifields. Using this theory and the projection construction with field multiplication and Albert twisted fields as ingredients we define a family of presemifields \(B(p,m,s,l,C_1,C_2)\) in odd characteristic \(p.\) In the remainder of the paper the properties of those (pre)semifields are studied. The family contains the Budaghyan–Helleseth families of commutative semifields but also many semifields which are not isotopic to commutative.