Abstract

Proof: Table 4 gives the required results to find the SBIBD. To read the Table we first use base sequences BS(2m + 1), X, Y, Z, W of lengths m + 1, m + 1, m, m found in Cohen eta! [3], Koukouvinos eta! [16], Koukouvinos, Kounias and Sotiraglou [20], Yang [41] and C. Yang and J. Yang (see [18], [19]) (private communication to the authors April1989). The base sequences are used to form suitable sequences SS(2m + 1), A = HX + Y), B = t(XY), C = t(z + W), D = t(ZW) of lengths m + 1, m + 1, m, m. These are then used to form Tsequences of length 25(2m + 1) corresponding to the decomposition where :r1, :r2, :r3, :r4 are given in the column headed “t =2m+ 1" in Table 4. The decomposition 4w = w? + w~ +w~ + w~ is achieved by using w1, w2, w3, W4 in the column headed "4w =Williamson”. The Williamson matrices, found by Baumert and Hall, are from [29, p389] except for 4.23 = 32 + 32 + 72 + 52 whose first rows are: The method used uses the notation of Jenkins eta! [11].