In a recent paper , White and Hultquist extended the use of finite fields for the construction of confounding plans to include "asymmetrical" or "mixed" factorials. The technique, in their own words, was to define addition and multiplication of elements from distinct finite fields by mapping these elements into a finite commutative ring containing subrings isomorphic to each of the fields in question. The standard techniques were then applied to the asymmetrical case illustrating the procedure with a numerical example for $3^2 \times 5$ factorial experiment. Later, Raktoe  also provided an equivalent theoretical basis for the results obtained by White and Hultquist , worked out a generalization of the technique and illustrated his procedure with an example of $2^2 \times 3 \times 5$ factorial experiment. It appears, however, that to provide a basis of the required calculus covering confounding plans of "mixed factorials" of the types discussed by them, it may not be necessary to invoke the properties of finite fields and to combine them. Instead, properties of finite multiplicative groups may be sufficient to construct such confounding plans. The aim of the present note is to indicate that this alternative approach, when it exists, is structurally identical with the procedure as outlined in , and that this methodology is simple, taking as it does only the properties of multiplicative groups. The procedure has been illustrated with reference to the same example of $3^2 \times 5$ factorial design as discussed in full in . The correspondence relationships between the levels of the factors and the elements of the group may be so worked out that the complete model for the $3^2 \times 5$ experiment as provided in Table 4.1 of  would come out exactly the same by this alternative approach. The procedure of White and Hultquist would require that the number of levels of a factor be prime. But by the procedure presented here, it would be possible to cover mixed factorials of other types where the number of levels of a factor may not be a prime number. Analysis of variance is not attempted here, as such analysis can be carried out following the procedures as given by White and Hultquist .
K. S. Banerjee. "Some Observations on the White-Hultquist Procedure for the Construction of Confounding Plans for Mixed Factorial Designs." Ann. Math. Statist. 41 (3) 1108 - 1113, June, 1970. https://doi.org/10.1214/aoms/1177696993