Enumerating semifields

In 1960, Kleinfeld published representatives for all of the isomorphism classes of 16 element semifields, [E. Kleinfeld, Techniques for Enumerating Veblen-Wedderburn Systems, J. ACM 7 (1960) 330–337]. It is not entirely clear how Kleinfeld generated some of his results, but it is likely that it was similar to the approach that Walker used in 1962 to generate representative for the isotopism classes of 32 element semifields, [R. J. Walker, Determination of Division Algebras With 32 Elements, Proc. Sympos. Appl. Math. XV (1963) 8385]. This paper introduces an alternative notation for publication which is both simple and practical, and which leads to an alternative method which was used to verify Kleinfeldʼs results.