Present knowledge of double Youden rectangles (DYRs) with k < v-1 , k =< 15 ------------------------------------------------------------------------------- k x v Do DYRs exist? References (t.a. = to appear) ------------------------------------------------------------------------------- 3 x 7 No Preece [1966] 4 x 7 Yes 1/1 Clarke [1967]; Preece [1991; t.a.] 4 x 13 Yes 1/1 Preece [1982] 5 x 11 Yes 1/1 Preece [1994] 5 x 21 Yes 1/1 Preece, Vowden & Phillips [t.a.] 6 x 11 Yes 1/1 Preece [1991; 1994; t.a.] 6 x 16 No* 6 x 31 Yes 1/1 Preece, Vowden & Phillips [t.a.] 7 x 15 Yes 5/5 Preece [1971; 1992]; Vowden [1994]; Preece, Vowden & Phillips [t.a.]; Phillips & Preece [unpub.] 7 x 22 No~ 7 x 43 No~ 8 x 15 Yes 3/5 Preece [1993]; Preece, Vowden & Phillips [t.a.] 8 x 29 No*~ 8 x 57 Yes 1/1 Preece, Vowden & Phillips [t.a.] 9 x 13 No* 9 x 19 Yes 3/6 Vowden [unpub.]; Preece, Vowden & Phillips [t.a.] 9 x 25 No* 9 x 37 Yes 4/4 Phillips, Preece & Rees [unpub.] 9 x 73 ? 0/1 10 x 16 No* 10 x 19 Yes 2/6 Preece, Vowden & Phillips [t.a.] 10 x 31 ? 0/151 10 x 46 No*~ 10 x 91 ? 0/4 11 x 23 Yes 3/1102 Preece [1971]; Vowden [1994] 11 x 56 ? 0/>=5 11 x 111 No~ 12 x 23 Yes 3/1102 Preece, Vowden & Phillips [t.a.] 12 x 34 No*~ 12 x 45 No* 12 x 67 No*~ 12 x 133 ? 0/>=1 13 x 27 Yes 1/208310 Vowden [unpub.] 13 x 40 ? 0/>=389 13 x 53 No~ 13 x 79 ? 0/>=2 13 x 157 ? 0/>=0 14 x 27 ? 0/208310 14 x 92 No*~ 14 x 183 ? 0/>=1 15 x 31 ? 0/>=1266891 15 x 36 No* 15 x 43 No*~ 15 x 71 No* 15 x 106 No~ 15 x 211 No~ ------------------------------------------------------------------------------- ~ SBIBD does not exist * parameter set is not admissible for a DYR n/Nd If Nd pairwise non-isomorphic SBIBDs exist for a particular parameter set, then n is the number of them for which DYRs have been found _______________________________________________________________________________ biplane.DYRs The possible existence of k x v DYRs with 2 k = 2p+1 and v = 2p +p+1 _____________________________________________________________________________ p k x v Nd SBIBD |A| DYR identifier _____________________________________________________________________________ 1 3 x 4 1 unreduced 24 No 2 5 x 11 1 cyclic 660 Yes 3 7 x 22 0 - - No 4 9 x 37 4 cyclic 333 Yes (use quartic residues of GF(37)) non-self-dual 1512 Yes Hussain (1945, Section 3) solns (A) and (E)' of Atiqullah (1958) soln A of Husain (1961, p. 15) dual of above 1512 Yes Hussain (1945, Section 4) solns (A)' and (E) of Atiqullah (1958) soln B of Husain (1961, p. 15) Assmus et al. (1977) 54 Yes BL(9) of Salwach & Mezzaroba (1978) 5 11 x 56 >= 5 Hall, Lane & Wales (1970) 80640 ? Jonsson (1973) B20 of Salwach & Mezzaroba (1979) B1 of Denniston (1980) Assmus et al. (1977) 288 ? B22 of Salwach & Mezzaroba (1979) B2 of Denniston (1980) B26 of Salwach & Mezzaroba (1979) 144 ? B3 of Denniston (1980) B24 of Salwach & Mezzaroba (1979) 64 ? B4 of Denniston (1980) Janko & van Trung (1986) 24 ? 6 13 x 79 >= 2 non-self-dual 110 ? Aschbacher (1971) Colbourn & Dinitz (1996, p. 81) dual of above 110 ? 7 15 x 106 0 - - No _____________________________________________________________________________