Theory of factorial designs of the parallel flats type. I: the coefficient matrix

Let GF(s) be the finite field with s elements.(Thus, when s=3, the elements of GF(s) are 0, 1 and 2.)Let A(r×n), of rank r, and ci(i=1,…,f), (r×1), be matrices over GF(s). (Thus, for n=4, r=2, f=2, we could have A=[11100121], c1=[10], c2=[02].) Let Ti (i=1,…,f) be the flat in EG(n, s) consisting of the set of all the sn−r solutions of the equations At=ci, wheret′=(t1,…,tn) is a vector of variables.(Thus, EG(4, 3) consists of the 34=81 points of the form (t1,t2,t3,t4), where t's take the values 0,1,2 (in GF(3)). The number of solutions of the equations At=ci is sn−r, where r=Rank(A), and the set of such solutions is said to form an (n−r)-flat, i.e. a flat of (n−r) dimensions. In our example, both T1 and T2 are 2-flats consisting of 34−2=9 points each. The flats T1,T2,…,Tf are said to be parallel since, clearly, no two of them can have a common point. In the example, the points of T1 are (1000), (0011), (2022), (0102), (2110), (1121), (2201), (1212) and (0220). Also, T2 consists of (0002), (2010), (1021), (2101), (1112), (0120), (1200), (0211) and (2222).) Let T be the fractional design for a sn symmetric factorial experiment obtained by taking T1,T2,…,Tf together. (Thus, in the example, 34=81 treatments of the 34 factorial experiment correspond one-one with the points of EG(4,3), and T will be the design (i.e. a subset of the 81 treatments) consisting of the 18 points of T1 and T2 enumerated above.)