Construction of Nested 3-designs

A number of papers have been published on the construction of large sets of t-designs: see Laue, Magliveras & Wassermann, Cameron, Maimani, Omidi & Tayfeh-Rezaie and the references therein.

The method used is that used in the latter of the above, where 3-designs are constructed from PSL(2,q), where q is an odd prime of the form 4s+3: the construction used here is identical, the application differs only in the f numbers as they appear in Table 2. The original f number (giving the number of blocks fixed by a subgroup of PSL(2,q)) is multiplied by the number of nested blocks fixed by that subgroup. For the moment, I have given particular values for these factors, but some general formulae are emerging and may be given at some later date. The examples given below are based on the example given in the same paper.

There appears to be an error or misprint: in Lemma 21, the Mobius function μ(C3,S4) should be +1, and, correspondingly, in Theorem 26, the sign for fk(S4) should be positive.
32 treatments in blocks of 8, with sub-blocks of size 4.
Subgroup Original f Nested f Final f g Orbits Blocks/design Total blocks
C1 10518300 35 368140500 359337120 24149 14880 359337120
C2 1820 11 20020 18336 1146 7440 8526240
C3 45 2 90 40 4 4960 19840
C4 28 3 84 32 4 3720 14880
C8 4 1 4 4 1 1860 1860
D4 28 7 196 162 54 3720 200880
D6 5 2 10 8 8 2480 19840
D8 4 3 12 10 10 1860 18600
A4 1 1 1 0 0 1240 0
S4 1 1 1 1 2 620 1240

The final column adds up to the final value of f for C1.

32 treatments in blocks of 12, with sub-blocks of size 4.
Subgroup Original f Nested f Final f g Orbits Blocks/design Total blocks
C1 225792840 5775 1303953651000 1303451454720 87597544 14880 1303451454720
C2 8008 135 1081080 1070736 66921 7440 497892240
C3 210 27 5670 5200 520 4960 2579200
C4 56 7 392 392 49 3720 182280
C5 15 0 0 0 0 1860 0
D4 56 19 1064 1062 354 3720 1316880
D6 10 9 90 90 90 2480 223200
D10 3 0 0 0 0 1488 0
A4 2 1 2 2 2 1240 2480
A5 1 0 0 0 0 248 0
32 treatments in blocks of 12, with sub-blocks of size 3.
Subgroup Original f Nested f Final f g Orbits Blocks/design Total blocks
C1 225792840 15400 3477209736000 3476612423040 233643308 14880 3476612423040
C2 8008 160 1281280 1273056 79566 7440 591971040
C3 210 37 7770 7640 764 4960 3789440
C4 56 16 896 896 112 3720 416640
C5 15 0 0 0 0 2976 0
D4 56 16 896 888 296 3720 1101120
D6 10 1 10 10 10 2480 24800
D10 3 0 0 0 0 1488 0
A4 2 4 8 8 8 1240 9920
A5 1 0 0 0 0 620 0
Some examples of these designs.

A representation of PSL(2,31) can be generated by the two permutations:

  • [[1, 32], [2, 31], [3, 16], [4, 11], [5, 24], [6, 7], [8, 23], [9, 28], [10, 25], [12, 15], [13, 19], [14, 20], [17, 30], [18, 21], [22, 29], [26, 27]]
  • [[1, 32, 2], [3, 17, 31], [4, 12, 16], [5, 25, 11], [6, 8, 24], [9, 29, 23], [10, 26, 28], [13, 20, 15], [14, 21, 19], [18, 22, 30]]

Using this representation, typical designs can be generated as the orbits of single blocks. λ and μ refer to the occurrence of pairs and triplets of treatments respectively, while the subscripts 1 and 2 refer to main blocks and sub-blocks respectively.

  • v = 32, b = 620, r = 155, k1 = 8, k2 = 4, λ1 = 35, λ2 = 15, μ1 = 7, μ2 = 1: initial block = (15, 17, 20, 31 | 1, 6, 19, 23)
  • v = 32, b = 1240 , r = 465, k1= 12, k2=3 ,λ1 = 165, λ2 = 30, μ1 = 55, μ2 = 1: initial block = (11, 17, 22 | 13, 25, 32 | 10, 30, 31 | 4, 9, 20)
  • v = 32, b= 1240, r = 465, k1 = 12, k2 = 4, λ1 = 165, λ2= 45, μ1 = 55, μ2 = 3: initial block = (9, 11, 31, 32 | 4, 17, 25, 30 | 10, 13, 20, 22)

Some solutions obtained by previous methods are given here.


Last updated: 6th January 2007
By D.H.Rees