Generalized Combining of Elements From Finite Fields

Fisher's Contributions to Agricultural Statistics

“…The extension to symmetrical prime, and prime power, factorial designs was made independently by four people- Bose and Kishen (1940), Fisher (1942Fisher ( , 1945, Bose (1947), and Kempthorne (1947). Extensions, using ring theory, to asymmetrical prime power factorials have been given by White and Hultquist (1965) and Raktoe (1969Raktoe ( , 1970. Worthley and Banerjee (1974) have further extended the ring theory methods to any asymmetrical factorial.…”

A Calculus for Factorial Arrangements

“…The results of White and Hultquist were generalized by Raktoe (1969Raktoe ( , 1970 to factorial experiments for which each factor occurs at one of several prime-powered numbers of levels. Further results were reported by Worthley and Banerjee (1974) and Sihota and Banerjee (1981).…”

“…All but the latest of these were reviewed and discussed by Raktoe, Rayner and Chalton (1978). Each of the above generalizations of the classical method, with the exception of a generalization of Raktoe (1970), is equivalent to applying the classical method of Bose (1947) separately to each symmetric subexperiment. investigated the relationship among the various methods and showed that, barring that due to Raktoe (1970) for experiments involving factors with numbers of levels being different powers of the same prime, each produce classes of designs which form subsets of a more general class namely the general classical class of designs.…”

“…A general classical design is determined by the subgroups At = {(0,0),(1,1),(2,2)} and A2 = {(0),(1)}. Since (1,1) generates At and (1) generates A 2 , by Definition 7.2.1, & had shown, the class of general classical designs includes the classes of designs obtained by the methods of White and Hultquist (1965), Raktoe (1969), Worthley and Banerjee (1974), Sihota and Banerjee (1981) and the method of Raktoe (1970) restricted to experiments where the numbers of levels of different factors are either equal or powers of different primes. We denote an algebra of order 8 by A (8 ) and an element of A (8) by a (8).…”

“…AB remarks, the class of general classical designs does not contain the class of designs generated by the method of Raktoe (1970) for experiments where the numbers of levels of different factors are powers of the same prime since the elements of OF(p n) do not yield the necessary decomposition into a sum of elements of subfields.…”