On generalized Howell designs with block size three

Abstract

In this paper, we examine a class of doubly resolvable combinatorial objects. Let $t, k, \lambda , s$ and v be nonnegative integers, and let X be a set of v symbols. A generalized Howell design, denoted t-$\mathrm {GHD}_{k}(s,v;\lambda )$, is an $s\times s$ array, each cell of which is either empty or contains a k-set of symbols from X, called a block, such that: (i) each symbol appears exactly once in each row and in each column (i.e. each row and column is a resolution of X); (ii) no t-subset of elements from X appears in more than $\lambda$ cells. Particular instances of the parameters correspond to Howell designs, doubly resolvable balanced incomplete block designs (including Kirkman squares), doubly resolvable nearly Kirkman triple systems, and simple orthogonal multi-arrays (which themselves generalize mutually orthogonal Latin squares). Generalized Howell designs also have connections with permutation arrays and multiply constant-weight codes. In this paper, we concentrate on the case that $t=2$, $k=3$ and $\lambda =1$, and write $\mathrm {GHD}(s,v)$. In this case, the number of empty cells in each row and column falls between 0 and $(s-1)/3$. Previous work has considered the existence of GHDs on either end of the spectrum, with at most 1 or at least $(s-2)/3$ empty cells in each row or column. In the case of one empty cell, we correct some results of Wang and Du, and show that there exists a $\mathrm {GHD}(n+1,3n)$ if and only if $n \ge 6$, except possibly for $n=6$. In the case of two empty cells, we show that there exists a $\mathrm {GHD}(n+2,3n)$ if and only if $n \ge 6$. Noting that the proportion of cells in a given row or column of a $\mathrm {GHD}(s,v)$ which are empty falls in the interval [0, 1 / 3), we prove that for any $\pi \in [0,5/18]$, there is a $\mathrm {GHD}(s,v)$ whose proportion of empty cells in a row or column is arbitrarily close to $\pi$.

Appendices

Appendix 1: Starters and adders for small $\mathrm {GHD}(n+1,3n)$

First we give those obtained by transitive starters and adders:

Example 7.1

For $n=14$:

$$\begin{aligned} \begin{array}{llllll} 0_0 3_0 5_0 [10] &{} 1_1 3_1 7_1 [ 5] &{} \infty _0 4_0 4_1 [12] &{} \infty _1 8_0 9_1 [ 1] &{} \infty _2 9_0 11_1 [ 2] &{} \infty _3 11_0 14_1 [ 8] \\ \infty _4 13_0 2_1 [ 9] &{} \infty _5 10_0 0_1 [ 4] &{} \infty _6 14_0 5_1 [13] &{} \infty _7 6_0 13_1 [11] &{} \infty _8 1_0 10_1 [ 7] &{} \infty _9 2_0 12_1 [ 3] \\ \infty _{10} 12_0 8_1 [ 6] &{} \infty _{11} 7_0 6_1 [14] \end{array} \end{aligned}$$

Example 7.2

For $n=20$:

$$\begin{aligned} \begin{array}{llllll} 0_0 1_0 3_0 [14] &{} 2_1 6_1 8_1 [ 7] &{} \infty _0 8_0 9_1 [ 2] &{} \infty _1 9_0 11_1 [12] &{} \infty _2 7_0 10_1 [11] &{} \infty _3 11_0 15_1 [ 5] \\ \infty _4 13_0 18_1 [17] &{} \infty _5 18_0 3_1 [ 4] &{} \infty _6 10_0 17_1 [10] &{} \infty _7 14_0 1_1 [18] &{} \infty _8 12_0 0_1 [16] &{} \infty _9 15_0 4_1 [ 8] \\ \infty _{10} 5_0 16_1 [ 1] &{} \infty _{11} 16_0 7_1 [ 3] &{} \infty _{12} 20_0 12_1 [13] &{} \infty _{13} 4_0 19_1 [20] &{} \infty _{14} 19_0 14_1 [15] &{} \infty _{15} 17_0 13_1 [ 9] \\ \infty _{16} 2_0 20_1 [ 6] &{} \infty _{17} 6_0 5_1 [19] \end{array} \end{aligned}$$

Example 7.3

For $n=26$:

$$\begin{aligned} \begin{array}{llllll} 0_0 6_0 10_0 [18] &{} 3_1 11_1 16_1 [ 9] &{} \infty _0 8_0 8_1 [19] &{} \infty _1 14_0 15_1 [17] &{} \infty _2 22_0 24_1 [25] &{} \infty _3 11_0 14_1 [ 2] \\ \infty _4 13_0 17_1 [16] &{} \infty _5 17_0 22_1 [24] &{} \infty _6 7_0 13_1 [15] &{} \infty _7 2_0 9_1 [ 4] &{} \infty _8 24_0 5_1 [12] &{} \infty _9 1_0 10_1 [11] \\ \infty _{10} 15_0 25_1 [20] &{} \infty _{11} 12_0 23_1 [ 7] &{} \infty _{12} 19_0 4_1 [ 6] &{} \infty _{13} 16_0 2_1 [21] &{} \infty _{14} 20_0 7_1 [ 1] &{} \infty _{15} 18_0 6_1 [ 5] \\ \infty _{16} 23_0 12_1 [ 3] &{} \infty _{17} 3_0 20_1 [14] &{} \infty _{18} 9_0 1_1 [23] &{} \infty _{19} 25_0 18_1 [13] &{} \infty _{20} 5_0 26_1 [10] &{} \infty _{21} 26_0 21_1 [ 8] \\ \infty _{22} 4_0 0_1 [26] &{} \infty _{23} 21_0 19_1 [22] \end{array} \end{aligned}$$

Example 7.4

For $n=32$:

$$\begin{aligned} \begin{array}{llllll} 0_0 4_0 7_0 [22] &{} 5_1 6_1 11_1 [11] &{} \infty _0 25_0 25_1 [ 8] &{} \infty _1 8_0 9_1 [17] &{} \infty _2 16_0 18_1 [25] &{} \infty _3 11_0 14_1 [ 1] \\ \infty _4 13_0 17_1 [14] &{} \infty _5 17_0 22_1 [29] &{} \infty _6 6_0 12_1 [15] &{} \infty _7 1_0 8_1 [ 3] &{} \infty _8 2_0 10_1 [32] &{} \infty _9 10_0 19_1 [13] \\ \infty _{10} 3_0 13_1 [28] &{} \infty _{11} 18_0 29_1 [18] &{} \infty _{12} 19_0 31_1 [21] &{} \infty _{13} 21_0 1_1 [23] &{} \infty _{14} 22_0 4_1 [26] &{} \infty _{15} 20_0 3_1 [ 4] \\ \infty _{16} 23_0 7_1 [16] &{} \infty _{17} 12_0 30_1 [31] &{} \infty _{18} 14_0 0_1 [ 5] &{} \infty _{19} 15_0 2_1 [ 2] &{} \infty _{20} 5_0 26_1 [27] &{} \infty _{21} 27_0 16_1 [20] \\ \infty _{22} 9_0 32_1 [ 7] &{} \infty _{23} 24_0 15_1 [ 6] &{} \infty _{24} 29_0 21_1 [24] &{} \infty _{25} 30_0 23_1 [12] &{} \infty _{26} 26_0 20_1 [ 9] &{} \infty _{27} 32_0 27_1 [19] \\ \infty _{28} 28_0 24_1 [10] &{} \infty _{29} 31_0 28_1 [30] \end{array} \end{aligned}$$

Example 7.5

For $n=38$:

$$\begin{aligned} \begin{array}{llllll} 0_0 6_0 7_0 [26] &{} 8_1 14_1 24_1 [13] &{} \infty _0 20_0 20_1 [20] &{} \infty _1 24_0 25_1 [ 1] &{} \infty _2 17_0 19_1 [ 4] &{} \infty _3 26_0 29_1 [ 2] \\ \infty _4 19_0 23_1 [23] &{} \infty _5 23_0 28_1 [30] &{} \infty _6 25_0 31_1 [19] &{} \infty _7 2_0 9_1 [29] &{} \infty _8 3_0 12_1 [27] &{} \infty _9 5_0 15_1 [ 7] \\ \infty _{10} 10_0 21_1 [31] &{} \infty _{11} 18_0 30_1 [25] &{} \infty _{12} 29_0 3_1 [ 9] &{} \infty _{13} 35_0 10_1 [24] &{} \infty _{14} 28_0 4_1 [28] &{} \infty _{15} 30_0 7_1 [17] \\ \infty _{16} 33_0 11_1 [22] &{} \infty _{17} 14_0 32_1 [21] &{} \infty _{18} 37_0 17_1 [12] &{} \infty _{19} 32_0 13_1 [36] &{} \infty _{20} 34_0 16_1 [32] &{} \infty _{21} 12_0 34_1 [33] \\ \infty _{22} 22_0 6_1 [14] &{} \infty _{23} 16_0 1_1 [34] &{} \infty _{24} 11_0 36_1 [11] &{} \infty _{25} 13_0 0_1 [ 5] &{} \infty _{26} 38_0 26_1 [38] &{} \infty _{27} 9_0 37_1 [ 6] \\ \infty _{28} 15_0 5_1 [37] &{} \infty _{29} 8_0 38_1 [16] &{} \infty _{30} 1_0 33_1 [ 8] &{} \infty _{31} 27_0 22_1 [35] &{} \infty _{32} 31_0 27_1 [ 3] &{} \infty _{33} 21_0 18_1 [18] \\ \infty _{34} 4_0 2_1 [15] &{} \infty _{35} 36_0 35_1 [10] \end{array} \end{aligned}$$

Example 7.6

For $n=41$:

$$\begin{aligned} \begin{array}{llllll} 0_0 24_0 32_0 [28] &{} 10_1 11_1 13_1 [14] &{} \infty _0 26_0 26_1 [ 8] &{} \infty _1 8_0 9_1 [ 1] &{} \infty _2 16_0 18_1 [ 5] &{} \infty _3 29_0 32_1 [10] \\ \infty _4 18_0 22_1 [15] &{} \infty _5 22_0 27_1 [36] &{} \infty _6 17_0 23_1 [33] &{} \infty _7 35_0 0_1 [11] &{} \infty _8 4_0 12_1 [31] &{} \infty _9 5_0 14_1 [19] \\ \infty _{10} 6_0 16_1 [32] &{} \infty _{11} 9_0 20_1 [39] &{} \infty _{12} 3_0 15_1 [17] &{} \infty _{13} 11_0 24_1 [20] &{} \infty _{14} 33_0 5_1 [35] &{} \infty _{15} 2_0 17_1 [41] \\ \infty _{16} 21_0 37_1 [34] &{} \infty _{17} 31_0 6_1 [40] &{} \infty _{18} 15_0 33_1 [29] &{} \infty _{19} 30_0 7_1 [24] &{} \infty _{20} 25_0 3_1 [16] &{} \infty _{21} 19_0 40_1 [30] \\ \infty _{22} 14_0 36_1 [13] &{} \infty _{23} 23_0 4_1 [22] &{} \infty _{24} 20_0 2_1 [37] &{} \infty _{25} 36_0 19_1 [38] &{} \infty _{26} 1_0 29_1 [18] &{} \infty _{27} 10_0 39_1 [12] \\ \infty _{28} 12_0 1_1 [ 2] &{} \infty _{29} 38_0 28_1 [27] &{} \infty _{30} 40_0 31_1 [ 7] &{} \infty _{31} 7_0 41_1 [23] &{} \infty _{32} 41_0 34_1 [26] &{} \infty _{33} 27_0 21_1 [ 9] \\ \infty _{34} 13_0 8_1 [ 4] &{} \infty _{35} 34_0 30_1 [ 6] &{} \infty _{36} 28_0 25_1 [25] &{} \infty _{37} 37_0 35_1 [ 0] &{} \infty _{38} 39_0 38_1 [ 3] \end{array} \end{aligned}$$

Example 7.7

For $n=44$:

$$\begin{aligned} \begin{array}{llllll} 0_0 2_0 10_0 [30] &{} 16_1 30_1 36_1 [15] &{} \infty _0 11_0 11_1 [28] &{} \infty _1 25_0 26_1 [31] &{} \infty _2 26_0 28_1 [ 2] &{} \infty _3 21_0 24_1 [5] \\ \infty _4 16_0 20_1 [ 7] &{} \infty _5 27_0 32_1 [17] &{} \infty _6 29_0 35_1 [32] &{} \infty _7 1_0 8_1 [35] &{} \infty _8 4_0 12_1 [23] &{} \infty _9 8_0 17_1 [43] \\ \infty _{10} 19_0 29_1 [33] &{} \infty _{11} 33_0 44_1 [22] &{} \infty _{12} 30_0 42_1 [16] &{} \infty _{13} 36_0 4_1 [12] &{} \infty _{14} 38_0 7_1 [29] &{} \infty _{15} 39_0 9_1 [10] \\ \infty _{16} 34_0 5_1 [ 4] &{} \infty _{17} 37_0 10_1 [27] &{} \infty _{18} 32_0 6_1 [42] &{} \infty _{19} 20_0 40_1 [13] &{} \infty _{20} 42_0 18_1 [24] &{} \infty _{21} 15_0 37_1 [26] \\ \infty _{22} 9_0 33_1 [38] &{} \infty _{23} 14_0 39_1 [39] &{} \infty _{24} 40_0 21_1 [19] &{} \infty _{25} 7_0 34_1 [36] &{} \infty _{26} 3_0 31_1 [21] &{} \infty _{27} 18_0 2_1 [44] \\ \infty _{28} 41_0 27_1 [41] &{} \infty _{29} 6_0 38_1 [ 3] &{} \infty _{30} 13_0 1_1 [37] &{} \infty _{31} 24_0 13_1 [34] &{} \infty _{32} 35_0 25_1 [25] &{} \infty _{33} 12_0 3_1 [ 8] \\ \infty _{34} 31_0 23_1 [11] &{} \infty _{35} 22_0 15_1 [ 9] &{} \infty _{36} 28_0 22_1 [ 6] &{} \infty _{37} 5_0 0_1 [20] &{} \infty _{38} 23_0 19_1 [40] &{} \infty _{39} 17_0 14_1 [18] \\ \infty _{40} 43_0 41_1 [14] &{} \infty _{41} 44_0 43_1 [ 1] \end{array} \end{aligned}$$

Now the intransitive starters and adders:

Example 7.8

For $n=10$:

$$\begin{aligned} \begin{array}{llllll} 0_2 6_2 7_2 [ 0] &{} 7_0 8_0 3_2 [ 8] &{} 5_1 6_1 1_2 [ 2] &{} 9_0 6_0 8_2 [ 4] &{} 3_1 0_1 2_2 [ 6] &{} 3_0 1_0 4_2 [ 1] \\ 4_1 2_1 5_2 [ 9] &{} 4_0 0_0 8_1 [ 7] &{} 5_0 1_1 7_1 [ 3] &{} 2_0 9_1 9_2 [ R] &{} 9_0 2_1 9_2 [ C] \end{array} \end{aligned}$$

Example 7.9

For $n=12$:

$$\begin{aligned} \begin{array}{llllll} 3_2 6_2 10_2 [ 0] &{} 6_0 10_0 9_0 [ 4] &{} 10_1 2_1 1_1 [ 8] &{} 5_0 0_1 9_2 [ 2] &{} 2_0 7_1 11_2 [10] &{} 7_0 3_1 1_2 [ 1] \\ 4_0 8_1 2_2 [11] &{} 11_0 1_0 0_2 [ 5] &{} 4_1 6_1 5_2 [ 7] &{} 8_0 9_1 4_2 [ 3] &{} 0_0 11_1 7_2 [ 9] &{} 3_0 5_1 8_2 [ R] \\ 5_0 3_1 8_2 [ C] \end{array} \end{aligned}$$

Example 7.10

For $n=16$:

$$\begin{aligned} \begin{array}{llllll} 6_2 12_2 15_2 [ 0] &{} 0_0 2_0 7_0 [10] &{} 10_1 12_1 1_1 [ 6] &{} 4_0 10_0 5_2 [ 4] &{} 8_1 14_1 9_2 [12] &{} 8_0 9_1 1_2 [13] \\ 6_0 5_1 14_2 [ 3] &{} 3_0 6_1 8_2 [15] &{} 5_0 2_1 7_2 [ 1] &{} 12_0 3_1 0_2 [11] &{} 14_0 7_1 11_2 [ 5] &{} 13_0 1_0 4_2 [14] \\ 11_1 15_1 2_2 [ 2] &{} 11_0 0_1 10_2 [ 9] &{} 9_0 4_1 3_2 [ 7] &{} 15_0 13_1 13_2 [ R] &{} 13_0 15_1 13_2 [ C] \end{array} \end{aligned}$$

Example 7.11

For $n=18$:

$$\begin{aligned} \begin{array}{llllll} 1_2 2_2 12_2 [ 0] &{} 0_0 8_0 7_0 [12] &{} 12_1 2_1 1_1 [ 6] &{} 4_0 10_0 3_2 [ 4] &{} 8_1 14_1 7_2 [14] &{} 12_0 15_1 13_2 [ 1] \\ 16_0 13_1 14_2 [17] &{} 1_0 0_1 5_2 [ 5] &{} 5_0 6_1 10_2 [13] &{} 14_0 3_1 6_2 [ 3] &{} 6_0 17_1 9_2 [15] &{} 3_0 5_1 0_2 [ 8] \\ 13_0 11_1 8_2 [10] &{} 9_0 11_0 17_2 [16] &{} 7_1 9_1 15_2 [ 2] &{} 17_0 4_1 11_2 [11] &{} 15_0 10_1 4_2 [ 7] &{} 2_0 16_1 16_2 [ R] \\ 16_0 2_1 16_2 [ C] \end{array} \end{aligned}$$

Example 7.12

For $n=22$:

$$\begin{aligned} \begin{array}{llllll} 0_2 7_2 13_2 [ 0] &{} 0_0 4_0 7_0 [16] &{} 16_1 20_1 1_1 [ 6] &{} 8_0 14_0 11_2 [10] &{} 18_1 2_1 21_2 [12] &{} 10_0 11_1 4_2 [ 5] \\ 16_0 15_1 9_2 [17] &{} 19_0 0_1 6_2 [ 9] &{} 9_0 6_1 15_2 [13] &{} 2_0 9_1 14_2 [ 3] &{} 12_0 5_1 17_2 [19] &{} 1_0 19_1 18_2 [ 2] \\ 21_0 3_1 20_2 [20] &{} 11_0 21_1 12_2 [18] &{} 17_0 7_1 8_2 [ 4] &{} 20_0 3_0 5_2 [14] &{} 12_1 17_1 19_2 [ 8] &{} 15_0 13_1 1_2 [15] \\ 6_0 8_1 16_2 [ 7] &{} 5_0 14_1 3_2 [21] &{} 13_0 4_1 2_2 [ 1] &{} 18_0 10_1 10_2 [ R] &{} 10_0 18_1 10_2 [ C] \end{array} \end{aligned}$$

Example 7.13

For $n=28$:

$$\begin{aligned} \begin{array}{llllll} 4_2 12_2 17_2 [ 0] &{} 0_0 8_0 5_0 [16] &{} 16_1 24_1 21_1 [12] &{} 20_0 26_1 23_2 [ 8] &{} 6_0 0_1 3_2 [20] &{} 22_0 5_1 16_2 [19] \\ 24_0 13_1 7_2 [ 9] &{} 9_0 12_1 27_2 [23] &{} 7_0 4_1 22_2 [ 5] &{} 10_0 17_1 18_2 [ 1] &{} 18_0 11_1 19_2 [27] &{} 13_0 17_0 2_2 [18] \\ 3_1 7_1 20_2 [10] &{} 15_0 25_1 1_2 [ 4] &{} 1_0 19_1 5_2 [24] &{} 25_0 2_1 21_2 [21] &{} 23_0 18_1 14_2 [ 7] &{} 3_0 15_1 24_2 [17] \\ 4_0 20_1 13_2 [11] &{} 21_0 2_0 9_2 [ 6] &{} 27_1 8_1 15_2 [22] &{} 11_0 12_0 10_2 [26] &{} 9_1 10_1 8_2 [ 2] &{} 27_0 1_1 11_2 [15] \\ 16_0 14_1 26_2 [13] &{} 26_0 22_1 0_2 [25] &{} 19_0 23_1 25_2 [ 3] &{} 14_0 6_1 6_2 [ R] &{} 6_0 14_1 6_2 [ C] \end{array} \end{aligned}$$

Example 7.14

For $n=30$:

$$\begin{aligned} \begin{array}{llllll} 3_2 7_2 10_2 [ 0] &{} 0_0 8_0 5_0 [18] &{} 18_1 26_1 23_1 [12] &{} 18_0 24_1 22_2 [28] &{} 22_0 16_1 20_2 [ 2] &{} 28_0 9_1 15_2 [27] \\ 6_0 25_1 12_2 [ 3] &{} 27_0 0_1 2_2 [23] &{} 23_0 20_1 25_2 [ 7] &{} 10_0 17_1 11_2 [17] &{} 4_0 27_1 28_2 [13] &{} 9_0 13_0 21_2 [ 6] \\ 15_1 19_1 27_2 [24] &{} 19_0 29_1 26_2 [22] &{} 21_0 11_1 18_2 [ 8] &{} 7_0 12_1 23_2 [21] &{} 3_0 28_1 14_2 [ 9] &{} 1_0 13_1 16_2 [ 1] \\ 14_0 2_1 17_2 [29] &{} 15_0 24_0 4_2 [20] &{} 5_1 14_1 24_2 [10] &{} 25_0 26_0 9_2 [26] &{} 21_1 22_1 5_2 [ 4] &{} 29_0 1_1 19_2 [11] \\ 12_0 10_1 0_2 [19] &{} 2_0 6_1 1_2 [ 5] &{} 11_0 7_1 6_2 [25] &{} 20_0 3_1 29_2 [14] &{} 17_0 4_1 13_2 [16] &{} 16_0 8_1 8_2 [ R] \\ 8_0 16_1 8_2 [ C] \end{array} \end{aligned}$$

Example 7.15

For $n=36$:

$$\begin{aligned} \begin{array}{llllll} 6_2 18_2 35_2 [ 0] &{} 0_0 4_0 7_0 [26] &{} 26_1 30_1 33_1 [10] &{} 34_0 4_1 3_2 [14] &{} 18_0 12_1 17_2 [22] &{} 32_0 7_1 33_2 [35] \\ 6_0 31_1 32_2 [ 1] &{} 35_0 2_1 15_2 [25] &{} 27_0 24_1 4_2 [11] &{} 28_0 35_1 19_2 [31] &{} 30_0 23_1 14_2 [ 5] &{} 21_0 25_1 28_2 [ 8] \\ 33_0 29_1 0_2 [28] &{} 25_0 1_1 31_2 [30] &{} 31_0 19_1 25_2 [ 6] &{} 15_0 20_1 2_2 [27] &{} 11_0 6_1 29_2 [ 9] &{} 3_0 13_1 1_2 [ 7] \\ 20_0 10_1 8_2 [29] &{} 19_0 28_1 21_2 [20] &{} 12_0 3_1 5_2 [16] &{} 17_0 18_1 13_2 [34] &{} 16_0 15_1 11_2 [ 2] &{} 13_0 27_1 24_2 [19] \\ 10_0 32_1 7_2 [17] &{} 9_0 22_0 34_2 [12] &{} 21_1 34_1 10_2 [24] &{} 29_0 2_0 12_2 [15] &{} 8_1 17_1 27_2 [21] &{} 26_0 9_1 30_2 [32] \\ 5_0 22_1 26_2 [ 4] &{} 1_0 23_0 9_2 [13] &{} 14_1 0_1 22_2 [23] &{} 8_0 14_0 23_2 [33] &{} 5_1 11_1 20_2 [ 3] &{} 24_0 16_1 16_2 [ R] \\ 16_0 24_1 16_2 [ C] \end{array} \end{aligned}$$

Example 7.16

For $n=46$:

$$\begin{aligned} \begin{array}{llllll} 4_2 17_2 35_2 [ 0] &{} 0_0 4_0 7_0 [36] &{} 36_1 40_1 43_1 [10] &{} 44_0 4_1 45_2 [22] &{} 26_0 20_1 21_2 [24] &{} 40_0 5_1 44_2 [25] \\ 30_0 19_1 23_2 [21] &{} 45_0 2_1 1_2 [45] &{} 1_0 44_1 0_2 [ 1] &{} 42_0 3_1 11_2 [ 3] &{} 6_0 45_1 14_2 [43] &{} 37_0 41_1 27_2 [ 2] \\ 43_0 39_1 29_2 [44] &{} 21_0 33_1 12_2 [16] &{} 3_0 37_1 28_2 [30] &{} 17_0 22_1 5_2 [11] &{} 33_0 28_1 16_2 [35] &{} 13_0 23_1 20_2 [39] \\ 16_0 6_1 13_2 [ 7] &{} 25_0 34_1 7_2 [32] &{} 20_0 11_1 39_2 [14] &{} 31_0 32_1 37_2 [28] &{} 14_0 13_1 19_2 [18] &{} 41_0 9_1 30_2 [13] \\ 22_0 8_1 43_2 [33] &{} 15_0 32_0 42_2 [ 6] &{} 21_1 38_1 2_2 [40] &{} 19_0 24_0 36_2 [ 5] &{} 24_1 29_1 41_2 [41] &{} 38_0 7_1 18_2 [ 4] \\ 11_0 42_1 22_2 [42] &{} 27_0 9_0 40_2 [37] &{} 18_1 0_1 31_2 [ 9] &{} 8_0 18_0 38_2 [17] &{} 25_1 35_1 9_2 [29] &{} 36_0 12_1 6_2 [27] \\ 39_0 17_1 33_2 [19] &{} 10_0 31_1 34_2 [20] &{} 5_0 30_1 8_2 [26] &{} 28_0 1_1 15_2 [34] &{} 35_0 16_1 3_2 [12] &{} 2_0 15_1 24_2 [ 8] \\ 23_0 10_1 32_2 [38] &{} 12_0 14_1 10_2 [15] &{} 29_0 27_1 25_2 [31] &{} 34_0 26_1 26_2 [ R] &{} 26_0 34_1 26_2 [ C] \end{array} \end{aligned}$$

Appendix 2: Starters and adders for small $\mathrm {GHD}(n+2,3n)$

First we give those obtained by transitive starters and adders:

Example 8.1

For $n=8$:

$$\begin{aligned} \begin{array}{llllllll} 0_{1} 6_{1} 7_{1} [2]&4_{1} 2_{1} 7_{0} [3]&6_{0} 8_{0} 8_{1} [8]&0_{0} 1_{0} 4_{0} [7]&\infty _0 2_{0} 3_{1} [1]&\infty _1 5_{0} 9_{1} [4]&\infty _2 3_{0} 1_{1} [9]&\infty _3 9_{0} 5_{1} [6] \end{array} \end{aligned}$$

Example 8.2

For $n=9$:

$$\begin{aligned} \begin{array}{lllllll} 4_{0} 5_{1} 3_{1} [8] &{} 10_{0} 2_{0} 3_{0} [4] &{} 6_{0} 8_{0} 6_{1} [3] &{} 9_{1} 10_{1} 2_{1} [6] &{} \infty _0 0_{0} 4_{1} [2] &{} \infty _1 7_{0} 1_{1} [9] &{} \infty _2 9_{0} 0_{1} [1] \\ \infty _3 5_{0} 8_{1} [10] &{} \infty _4 1_{0} 7_{1} [7] &{} \end{array} \end{aligned}$$

Example 8.3

For $n=10$:

$$\begin{aligned} \begin{array}{lllllll} 3_{1} 10_{1} 11_{1} [3] &{} 3_{0} 4_{0} 6_{0} [10] &{} 1_{0} 8_{0} 4_{1} [7] &{} 7_{0} 5_{1} 8_{1} [4] &{} \infty _{0} 0_{0} 0_{1} [5] &{} \infty _{1} 11_{0} 6_{1} [1] &{} \infty _{2} 10_{0} 9_{1} [11] \\ \infty _{3} 5_{0} 2_{1} [2] &{} \infty _{4} 2_{0} 7_{1} [8] &{} \infty _{5} 9_{0} 1_{1} [9] \end{array} \end{aligned}$$

Example 8.4

For $n=11$:

$$\begin{aligned} \begin{array}{lllllll} 3_{1} 11_{0} 0_{0} [7] &{} 9_{0} 2_{0} 5_{0} [4] &{} 10_{1} 11_{1} 2_{1} [6] &{} 4_{1} 10_{0} 6_{1}[1] &{} \infty _0 3_{0} 0_{1}[0] &{} \infty _1 12_{0} 5_{1} [9] &{} \infty _2 8_{0} 9_{1} [2] \\ \infty _3 6_{0} 1_{1} [8] &{} \infty _4 4_{0} 8_{1} [11] &{} \infty _5 1_{0} 12_{1} [3] &{} \infty _6 7_{0} 7_{1} [5] \end{array} \end{aligned}$$

Example 8.5

For $n=12$:

$$\begin{aligned} \begin{array}{lllllll} 3_{1} 7_{0} 2_{0} [2] &{} 8_{1} 0_{1} 10_{1} [6] &{} 1_{1} 6_{1} 6_{0} [11] &{} 1_{0} 3_{0} 4_{0} [9] &{} \infty _{0} 11_{0} 13_{1} [5] &{} \infty _{1} 0_{0} 5_{1} [8] \\ \infty _{2} 5_{0} 9_{1} [1] &{} \infty _{3} 8_{0} 2_{1} [13] &{} \infty _{4} 10_{0} 7_{1} [4] &{} \infty _{5} 12_{0} 4_{1} [3] &{} \infty _{6} 13_{0} 11_{1} [12] &{} \infty _{7} 9_{0} 12_{1} [10] &{} \end{array} \end{aligned}$$

Example 8.6

For $n=13$:

$$\begin{aligned} \begin{array}{lllllll} 0_{0} 4_{1} 8_{1} [4] &{} 6_{0} 7_{0} 9_{0} [8] &{} 11_{0} 1_{0} 2_{1} [7] &{} 5_{1} 6_{1} 11_{1} [9] &{} \infty _0 13_{0} 7_{1} [0] &{} \infty _1 8_{0} 3_{1} [14] \\ \infty _2 4_{0} 1_{1} [5] &{} \infty _3 14_{0} 13_{1} [12] &{} \infty _4 12_{0} 0_{1} [13] &{} \infty _5 10_{0} 12_{1} [6] &{} \infty _6 5_{0} 10_{1} [1] &{} \infty _7 3_{0} 14_{1} [2] \\ \infty _8 2_{0} 9_{1} [10] \end{array} \end{aligned}$$

Example 8.7

For $n=14$:

$$\begin{aligned} \begin{array}{lllllll} 6_{0} 9_{1} 8_{1} [5] &{} 9_{0} 13_{0} 0_{0} [7] &{} 10_{0} 5_{1} 11_{0} [3] &{} 0_{1} 4_{1} 6_{1} [1] &{} \infty _0 3_{0} 12_{1} [6] &{} \infty _1 12_{0} 10_{1} [9] \\ \infty _2 8_{0} 14_{1} [14] &{} \infty _3 15_{0} 3_{1} [13] &{} \infty _4 4_{0} 11_{1} [4] &{} \infty _5 7_{0} 15_{1} [11] &{} \infty _6 1_{0} 2_{1} [2] &{} \infty _7 2_{0} 7_{1} [15] \\ \infty _8 5_{0} 1_{1} [10] &{} \infty _9 14_{0} 13_{1} [12] \end{array} \end{aligned}$$

Example 8.8

For $n=15$:

$$\begin{aligned} \begin{array}{lllllll} 10_{1} 8_{1} 16_{1} [7] &{} 16_{0} 4_{0} 13_{1} [11] &{} 7_{0} 6_{1} 5_{1} [4] &{} 2_{0} 13_{0} 0_{0} [16] &{} \infty _0 14_{0} 4_{1} [0] &{} \infty _1 15_{0} 1_{1} [15] \\ \infty _2 10_{0} 11_{1} [9] &{} \infty _3 6_{0} 0_{1} [14] &{} \infty _4 12_{0} 7_{1} [12] &{} \infty _5 9_{0} 14_{1} [8] &{} \infty _6 1_{0} 3_{1} [5] &{} \infty _7 3_{0} 9_{1} [2] \\ \infty _8 8_{0} 12_{1} [1] &{} \infty _{9} 5_{0} 15_{1} [3] &{} \infty _{10} 11_{0} 2_{1} [10] \end{array} \end{aligned}$$

Example 8.9

For $n=16$:

$$\begin{aligned} \begin{array}{lllllll} 0_{0} 16_{0} 4_{0} [10] &{} 17_{1} 2_{1} 6_{0} [14] &{} 12_{1} 2_{0} 17_{0} [5] &{} 13_{1} 0_{1} 1_{1} [7] &{} \infty _{0} 14_{0} 14_{1} [1] &{} \infty _{1} 15_{0} 5_{1} [4] \\ \infty _{2} 13_{0} 4_{1} [17] &{} \infty _{3} 11_{0} 15_{1} [13] &{} \infty _{4} 9_{0} 3_{1} [8] &{} \infty _{5} 10_{0} 16_{1} [6] &{} \infty _{6} 3_{0} 10_{1} [2] &{} \infty _{7} 12_{0} 9_{1} [15] \\ \infty _{8} 7_{0} 8_{1} [11] &{} \infty _{9} 8_{0} 11_{1} [3] &{} \infty _{10} 1_{0} 6_{1} [12] &{} \infty _{11} 5_{0} 7_{1} [16] \end{array} \end{aligned}$$

Example 8.10

For $n=17$:

$$\begin{aligned} \begin{array}{lllllll} 11_{1} 3_{1} 7_{0} [2] &{} 10_{0} 5_{1} 14_{0} [5] &{} 0_{1} 10_{1} 15_{1} [16] &{} 11_{0} 1_{0} 3_{0} [15] &{} \infty _0 2_{0} 8_{1} [0] &{} \infty _1 17_{0} 9_{1} [8] \\ \infty _2 9_{0} 12_{1} [11] &{} \infty _3 4_{0} 4_{1} [10] &{} \infty _4 0_{0} 18_{1} [12] &{} \infty _5 18_{0} 1_{1} [18] &{} \infty _6 12_{0} 2_{1} [1] &{} \infty _7 15_{0} 16_{1} [9] \\ \infty _8 16_{0} 13_{1} [7] &{} \infty _{9} 6_{0} 14_{1} [4] &{} \infty _{10} 5_{0} 17_{1} [17] &{} \infty _{11} 13_{0} 7_{1} [14] &{} \infty _{12} 8_{0} 6_{1} [3] \end{array} \end{aligned}$$

Example 8.11

For $n=18$:

$$\begin{aligned} \begin{array}{lllllll} 17_{0} 18_{1} 17_{1} [7] &{} 1_{0} 15_{0} 10_{0} [17] &{} 19_{0} 16_{0} 1_{1} [1] &{} 0_{1} 2_{1} 16_{1} [15] &{} \infty _0 9_{0} 5_{1} [4] &{} \infty _1 8_{0} 6_{1} [13] \\ \infty _2 18_{0} 11_{1} [12] &{} \infty _3 6_{0} 3_{1} [5] &{} \infty _4 5_{0} 4_{1} [3] &{} \infty _5 14_{0} 9_{1} [11] &{} \infty _6 4_{0} 13_{1} [19] &{} \infty _7 2_{0} 12_{1} [14] \\ \infty _8 13_{0} 7_{1} [9] &{} \infty _9 3_{0} 14_{1} [16] &{} \infty _{10} 7_{0} 10_{1} [8] &{} \infty _{11} 11_{0} 15_{1} [18] &{} \infty _{12} 12_{0} 19_{1} [2] &{} \infty _{13} 0_{0} 8_{1} [6] \end{array} \end{aligned}$$

Example 8.12

For $n=19$:

$$\begin{aligned} \begin{array}{llllll} 0_{0} 1_{0} 5_{0} [0] &{} 16_{1} 3_{1} 7_{0} [1] &{} 18_{0} 16_{0} 8_{1} [16] &{} 6_{1} 7_{1} 9_{1} [12] &{} \infty _0 20_{0} 20_{1} [3] &{} \infty _1 12_{0} 14_{1} [2] \\ \infty _2 17_{0} 4_{1} [10] &{} \infty _3 9_{0} 2_{1} [6] &{} \infty _4 10_{0} 11_{1} [9] &{} \infty _5 19_{0} 17_{1} [5] &{} \infty _6 14_{0} 0_{1} [11] &{} \infty _7 4_{0} 1_{1} [14] \\ \infty _8 3_{0} 18_{1} [13] &{} \infty _{9} 2_{0} 12_{1} [15] &{} \infty _{10} 13_{0} 19_{1} [7] &{} \infty _{11} 15_{0} 10_{1} [18] &{} \infty _{12} 6_{0} 5_{1} [4] &{} \infty _{13} 11_{0} 15_{1} [19] \\ \infty _{14} 8_{0} 13_{1} [20] \end{array} \end{aligned}$$

Example 8.13

For $n=20$:

$$\begin{aligned} \begin{array}{llllll} 1_{1} 2_{1} 9_{0} [15] &{} 16_{1} 5_{0} 10_{0} [21] &{} 0_{0} 20_{0} 4_{0} [3] &{} 7_{1} 17_{1} 21_{1} [5] &{} \infty _0 15_{0} 18_{1} [1] &{} \infty _1 18_{0} 15_{1} [18] \\ \infty _2 11_{0} 5_{1} [4] &{} \infty _3 2_{0} 9_{1} [9] &{} \infty _4 13_{0} 4_{1} [19] &{} \infty _5 1_{0} 10_{1} [20] &{} \infty _6 16_{0} 20_{1} [12] &{} \infty _7 14_{0} 13_{1} [16] \\ \infty _8 7_{0} 3_{1} [10] &{} \infty _9 12_{0} 0_{1} [6] &{} \infty _{10} 6_{0} 14_{1} [7] &{} \infty _{11} 3_{0} 8_{1} [17] &{} \infty _{12} 17_{0} 12_{1} [2] &{} \infty _{13} 8_{0} 6_{1} [14] \\ \infty _{14} 19_{0} 19_{1} [8] &{} \infty _{15} 21_{0} 11_{1} [13] &{} \end{array} \end{aligned}$$

Example 8.14

For $n=21$:

$$\begin{aligned} \begin{array}{llllll} 22_{0} 5_{0} 7_{0} [13] &{} 21_{0} 14_{0} 3_{1} [5] &{} 20_{1} 8_{1} 13_{0} [15] &{} 12_{1} 4_{1} 5_{1} [2] &{} \infty _0 11_{0} 9_{1} [0] &{} \infty _1 15_{0} 17_{1} [1] \\ \infty _2 0_{0} 13_{1} [4] &{} \infty _3 12_{0} 0_{1} [11] &{} \infty _4 3_{0} 22_{1} [3] &{} \infty _5 17_{0} 2_{1} [8] &{} \infty _6 6_{0} 15_{1} [9] &{} \infty _7 8_{0} 11_{1} [16] \\ \infty _8 19_{0} 18_{1} [21] &{} \infty _{9} 2_{0} 19_{1} [7] &{} \infty _{10} 4_{0} 14_{1} [6] &{} \infty _{11} 10_{0} 10_{1} [12] &{} \infty _{12} 1_{0} 16_{1} [20] &{} \infty _{13} 16_{0} 7_{1} [14] \\ \infty _{14} 20_{0} 21_{1} [17] &{} \infty _{15} 9_{0} 6_{1} [22] &{} \infty _{16} 18_{0} 1_{1} [18] \end{array} \end{aligned}$$

Example 8.15

For $n=22$:

$$\begin{aligned} \begin{array}{llllll} 13_{1} 9_{1} 20_{0} [3] &{} 10_{0} 23_{0} 7_{1} [15] &{} 17_{1} 22_{1} 19_{1} [9] &{} 19_{0} 0_{0} 1_{0} [21] &{} \infty _0 3_{0} 18_{1} [1] &{} \infty _1 4_{0} 6_{1} [2] \\ \infty _2 11_{0} 10_{1} [7] &{} \infty _3 14_{0} 4_{1} [5] &{} \infty _4 13_{0} 5_{1} [20] &{} \infty _5 17_{0} 20_{1} [14] &{} \infty _6 12_{0} 12_{1} [17] &{} \infty _7 15_{0} 16_{1} [11] \\ \infty _8 22_{0} 2_{1} [22] &{} \infty _9 9_{0} 21_{1} [23] &{} \infty _{10} 18_{0} 3_{1} [18] &{} \infty _{11} 8_{0} 14_{1} [16] &{} \infty _{12} 21_{0} 8_{1} [6] &{} \infty _{13} 16_{0} 23_{1} [19] \\ \infty _{14} 6_{0} 11_{1} [4] &{} \infty _{15} 7_{0} 1_{1} [10] &{} \infty _{16} 2_{0} 0_{1} [13] &{} \infty _{17} 5_{0} 15_{1} [8] &{} \end{array} \end{aligned}$$

Example 8.16

For $n=23$:

$$\begin{aligned} \begin{array}{llllll} 12_{0} 8_{0} 21_{1} [1] &{} 19_{1} 17_{0} 20_{0} [6] &{} 14_{1} 13_{0} 7_{0} [14] &{} 2_{1} 11_{1} 16_{1} [7] &{} \infty _0 18_{0} 5_{1} [0] &{} \infty _1 6_{0} 12_{1} [2] \\ \infty _2 3_{0} 22_{1} [9] &{} \infty _3 19_{0} 17_{1} [10] &{} \infty _4 5_{0} 20_{1} [17] &{} \infty _5 22_{0} 13_{1} [19] &{} \infty _6 24_{0} 7_{1} [8] &{} \infty _7 1_{0} 18_{1} [24] \\ \infty _8 15_{0} 1_{1} [15] &{} \infty _9 0_{0} 10_{1} [3] &{} \infty _{10} 21_{0} 24_{1} [21] &{} \infty _{11} 10_{0} 15_{1} [4] &{} \infty _{12} 9_{0} 4_{1} [22] &{} \infty _{13} 11_{0} 8_{1} [13] \\ \infty _{14} 14_{0} 3_{1} [5] &{} \infty _{15} 23_{0} 23_{1} [12] &{} \infty _{16} 2_{0} 6_{1} [18] &{} \infty _{17} 16_{0} 9_{1} [20] &{} \infty _{18} 4_{0} 0_{1} [11] &{} \end{array} \end{aligned}$$

Example 8.17

For $n=24$:

$$\begin{aligned} \begin{array}{llllll} 4_{0} 24_{0} 23_{0} [12] &{} 14_{0} 0_{0} 5_{0} [14] &{} 4_{1} 15_{1} 16_{1} [19] &{} 8_{1} 12_{1} 17_{1} [7] &{} \infty _0 21_{0} 13_{1} [1] &{} \infty _1 1_{0} 10_{1} [2] \\ \infty _2 15_{0} 25_{1} [5] &{} \infty _3 13_{0} 21_{1} [18] &{} \infty _4 16_{0} 20_{1} [11] &{} \infty _5 25_{0} 0_{1} [22] &{} \infty _6 11_{0} 14_{1} [23] &{} \infty _7 12_{0} 19_{1} [6] \\ \infty _8 20_{0} 6_{1} [10] &{} \infty _9 6_{0} 11_{1} [17] &{} \infty _{10} 17_{0} 23_{1} [21] &{} \infty _{11} 22_{0} 24_{1} [3] &{} \infty _{12} 3_{0} 3_{1} [4] &{} \infty _{13} 2_{0} 1_{1} [9] \\ \infty _{14} 8_{0} 22_{1} [24] &{} \infty _{15} 7_{0} 18_{1} [8] &{} \infty _{16} 18_{0} 7_{1} [25] &{} \infty _{17} 10_{0} 5_{1} [16] &{} \infty _{18} 9_{0} 2_{1} [15] &{} \infty _{19} 19_{0} 9_{1} [20] \end{array} \end{aligned}$$

Example 8.18

For $n=25$:

$$\begin{aligned} \begin{array}{llllll} 15_{1} 3_{1} 5_{1} [19] &{} 7_{0} 1_{0} 11_{0} [9] &{} 16_{1} 21_{1} 25_{1} [25] &{} 14_{0} 9_{0} 16_{0} [8] &{} \infty _0 13_{0} 11_{1} [0] &{} \infty _1 2_{0} 17_{1} [1] \\ \infty _2 12_{0} 7_{1} [2] &{} \infty _3 5_{0} 23_{1} [3] &{} \infty _4 21_{0} 22_{1} [5] &{} \infty _5 0_{0} 4_{1} [6] &{} \infty _6 25_{0} 9_{1} [23] &{} \infty _7 18_{0} 26_{1} [18] \\ \infty _8 6_{0} 13_{1} [26] &{} \infty _9 20_{0} 19_{1} [14] &{} \infty _{10} 23_{0} 8_{1} [22] &{} \infty _{11} 24_{0} 18_{1} [7] &{} \infty _{12} 26_{0} 1_{1} [12] &{} \infty _{13} 3_{0} 0_{1} [16] \\ \infty _{14} 22_{0} 12_{1} [17] &{} \infty _{15} 17_{0} 10_{1} [10] &{} \infty _{16} 8_{0} 14_{1} [21] &{} \infty _{17} 10_{0} 6_{1} [15] &{} \infty _{18} 15_{0} 2_{1} [13] &{} \infty _{19} 4_{0} 20_{1} [11] \\ \infty _{20} 19_{0} 24_{1} [4] &{} \end{array} \end{aligned}$$

Example 8.19

For $n=26$:

$$\begin{aligned} \begin{array}{llllll} 23_{0} 23_{1} 1_{1} [11] &{} 8_{0} 9_{0} 20_{0} [5] &{} 11_{0} 3_{0} 7_{1} [15] &{} 16_{1} 5_{1} 6_{1} [25] &{} \infty _0 1_{0} 8_{1} [1] &{} \infty _1 2_{0} 21_{1} [2] \\ \infty _2 13_{0} 2_{1} [3] &{} \infty _3 0_{0} 10_{1} [9] &{} \infty _4 27_{0} 11_{1} [4] &{} \infty _5 25_{0} 15_{1} [20] &{} \infty _6 24_{0} 17_{1} [12] &{} \infty _7 12_{0} 26_{1} [27] \\ \infty _8 16_{0} 18_{1} [24] &{} \infty _9 7_{0} 22_{1} [22] &{} \infty _{10} 19_{0} 24_{1} [8] &{} \infty _{11} 5_{0} 14_{1} [10] &{} \infty _{12} 15_{0} 3_{1} [7] &{} \infty _{13} 17_{0} 0_{1} [21] \\ \infty _{14} 26_{0} 25_{1} [23] &{} \infty _{15} 10_{0} 4_{1} [13] &{} \infty _{16} 14_{0} 27_{1} [19] &{} \infty _{17} 6_{0} 9_{1} [18] &{} \infty _{18} 4_{0} 12_{1} [16] &{} \infty _{19} 18_{0} 19_{1} [17] \\ \infty _{20} 21_{0} 13_{1} [26] &{} \infty _{21} 22_{0} 20_{1} [6] &{} \end{array} \end{aligned}$$

Example 8.20

For $n=27$:

$$\begin{aligned} \begin{array}{llllll} 4_{0} 8_{0} 22_{0} [17] &{} 11_{0} 19_{0} 22_{1} [11] &{} 5_{1} 18_{1} 6_{0} [14] &{} 2_{1} 10_{1} 4_{1} [16] &{} \infty _0 24_{0} 16_{1} [0] &{} \infty _1 14_{0} 27_{1} [1] \\ \infty _2 25_{0} 14_{1} [3] &{} \infty _3 9_{0} 26_{1} [4] &{} \infty _4 27_{0} 8_{1} [2] &{} \infty _5 10_{0} 19_{1} [6] &{} \infty _6 26_{0} 20_{1} [7] &{} \infty _7 0_{0} 24_{1} [27] \\ \infty _8 16_{0} 21_{1} [21] &{} \infty _9 18_{0} 9_{1} [5] &{} \infty _{10} 20_{0} 17_{1} [12] &{} \infty _{11} 23_{0} 25_{1} [13] &{} \infty _{12} 12_{0} 13_{1} [22] &{} \infty _{13} 2_{0} 0_{1} [15] \\ \infty _{14} 3_{0} 28_{1} [8] &{} \infty _{15} 13_{0} 6_{1} [28] &{} \infty _{16} 1_{0} 15_{1} [25] &{} \infty _{17} 15_{0} 1_{1} [23] &{} \infty _{18} 17_{0} 23_{1} [18] &{} \infty _{19} 7_{0} 7_{1} [24] \\ \infty _{20} 28_{0} 3_{1} [20] &{} \infty _{21} 21_{0} 11_{1} [26] &{} \infty _{22} 5_{0} 12_{1} [9] &{} \end{array} \end{aligned}$$

Example 8.21

For $n=28$:

$$\begin{aligned} \begin{array}{llllll} 22_{0} 3_{0} 6_{0} [26] &{} 27_{0} 2_{0} 28_{0} [24] &{} 10_{1} 18_{1} 13_{1} [23] &{} 8_{1} 14_{1} 27_{1} [17] &{} \infty _0 12_{0} 25_{1} [1] &{} \infty _1 10_{0} 22_{1} [2] \\ \infty _2 16_{0} 2_{1} [3] &{} \infty _3 1_{0} 19_{1} [4] &{} \infty _4 5_{0} 5_{1} [5] &{} \infty _5 15_{0} 24_{1} [8] &{} \infty _6 24_{0} 29_{1} [13] &{} \infty _7 4_{0} 23_{1} [29] \\ \infty _8 25_{0} 12_{1} [9] &{} \infty _9 7_{0} 17_{1} [20] &{} \infty _{10} 14_{0} 6_{1} [14] &{} \infty _{11} 26_{0} 28_{1} [18] &{} \infty _{12} 13_{0} 16_{1} [12] &{} \infty _{13} 0_{0} 7_{1} [11] \\ \infty _{14} 18_{0} 11_{1} [28] &{} \infty _{15} 29_{0} 26_{1} [21] &{} \infty _{16} 20_{0} 4_{1} [25] &{} \infty _{17} 21_{0} 20_{1} [10] &{} \infty _{18} 17_{0} 21_{1} [22] &{} \infty _{19} 9_{0} 0_{1} [27] \\ \infty _{20} 23_{0} 1_{1} [7] &{} \infty _{21} 8_{0} 3_{1} [16] &{} \infty _{22} 19_{0} 15_{1} [19] &{} \infty _{23} 11_{0} 9_{1} [6] &{} \end{array} \end{aligned}$$

Example 8.22

For $n=29$:

$$\begin{aligned} \begin{array}{llllll} 12_{0} 17_{0} 13_{1} [29] &{} 9_{0} 10_{0} 0_{0} [13] &{} 7_{1} 23_{1} 25_{1} [7] &{} 27_{1} 30_{1} 2_{0} [16] &{} \infty _{0} 27_{0} 19_{1} [0] &{} \infty _{1} 20_{0} 2_{1} [1] \\ \infty _{2} 15_{0} 22_{1} [2] &{} \infty _{3} 26_{0} 5_{1} [3] &{} \infty _{4} 3_{0} 18_{1} [4] &{} \infty _{5} 28_{0} 15_{1} [5] &{} \infty _{6} 22_{0} 26_{1} [9] &{} \infty _{7} 21_{0} 29_{1} [11] \\ \infty _{8} 14_{0} 28_{1} [10] &{} \infty _{9} 18_{0} 8_{1} [17] &{} \infty _{10} 6_{0} 17_{1} [30] &{} \infty _{11} 16_{0} 14_{1} [14] &{} \infty _{12} 24_{0} 12_{1} [19] &{} \infty _{13} 29_{0} 20_{1} [21] \\ \infty _{14} 19_{0} 0_{1} [18] &{} \infty _{15} 8_{0} 24_{1} [12] &{} \infty _{16} 25_{0} 3_{1} [20] &{} \infty _{17} 4_{0} 4_{1} [22] &{} \infty _{18} 7_{0} 10_{1} [27] &{} \infty _{19} 30_{0} 1_{1} [26] \\ \infty _{20} 23_{0} 9_{1} [24] &{} \infty _{21} 11_{0} 16_{1} [28] &{} \infty _{22} 5_{0} 11_{1} [6] &{} \infty _{23} 1_ {0} 21_{1} [8] &{} \infty _{24} 13_{0} 6_{1} [15] &{} \end{array} \end{aligned}$$

Example 8.23

For $n=31$:

$$\begin{aligned} \begin{array}{llllll} 11_{0} 19_{0} 13_{0} [28] &{} 22_{1} 17_{1} 24_{1} [26] &{} 4_{0} 24_{0} 27_{0} [9] &{} 28_{1} 19_{1} 31_{1} [11] &{} \infty _{0} 1_{0} 5_{1} [0] &{} \infty _{1} 23_{0} 30_{1} [1] \\ \infty _{2} 25_{0} 26_{1} [2] &{} \infty _{3} 17_{0} 10_{1} [3] &{} \infty _{4} 31_{0} 8_{1} [4] &{} \infty _{5} 5_{0} 18_{1} [5] &{} \infty _{6} 6_{0} 20_{1} [6] &{} \infty _{7} 12_{0} 3_{1} [13] \\ \infty _{8} 28_{0} 13_{1} [31] &{} \infty _{9} 8_{0} 25_{1} [7] &{} \infty _{10} 14_{0} 11_{1} [16] &{} \infty _{11} 3_{0} 12_{1} [8] &{} \infty _{12} 20_{0} 23_{1} [17] &{} \infty _{13} 9_{0} 14_{1} [23] \\ \infty _{14} 26_{0} 21_{1} [12] &{} \infty _{15} 21_{0} 0_{1} [29] &{} \infty _{16} 16_{0} 32_{1} [15] &{} \infty _{17} 22_{0} 4_{1} [30] &{} \infty _{18} 15_{0} 7_{1} [14] &{} \infty _{19} 0_{0} 6_{1} [18] \\ \infty _{20} 10_{0} 9_{1} [32] &{} \infty _{21} 29_{0} 27_{1} [25] &{} \infty _{22} 32_{0} 1_{1} [24] &{} \infty _{23} 30_{0} 16_{1} [19] &{} \infty _{24} 18_{0} 29_{1} [22] &{} \infty _{25} 7_{0} 15_{1} [21] \\ \infty _{26} 2_{0} 2_{1} [20] &{} \end{array} \end{aligned}$$

Example 8.24

For $n=32$:

$$\begin{aligned} \begin{array}{llllll} 0_0 3_0 11_0 [31] &{} 12_0 13_0 27_0 [3] &{} 23_1 24_1 29_1 [7] &{} 15_1 26_1 5_1 [27] &{} \infty _0 17_0 17_1 [9] &{} \infty _1 15_0 16_1 [12] \\ \infty _2 26_0 28_1 [26] &{} \infty _3 1_0 4_1 [1] &{} \infty _4 2_0 6_1 [4] &{} \infty _5 4_0 9_1 [5] &{} \infty _6 5_0 11_1 [6] &{} \infty _7 6_0 13_1 [11] \\ \infty _8 10_0 18_1 [22] &{} \infty _9 16_0 25_1 [13] &{} \infty _{10} 9_0 19_1 [30] &{} \infty _{11} 19_0 30_1 [25] &{} \infty _{12} 20_0 32_1 [2] &{} \infty _{13} 18_0 31_1 [15] \\ \infty _{14} 21_0 1_1 [32] &{} \infty _{15} 22_0 3_1 [24] &{} \infty _{16} 28_0 10_1 [19] &{} \infty _{17} 25_0 8_1 [10] &{} \infty _{18} 23_0 7_1 [18] &{} \infty _{19} 29_0 14_1 [29] \\ \infty _{20} 14_0 0_1 [23] &{} \infty _{21} 33_0 20_1 [21] &{} \infty _{22} 24_0 12_1 [33] &{} \infty _{23} 32_0 21_1 [16] &{} \infty _{24} 31_0 22_1 [28] &{} \infty _{25} 7_0 33_1 [14] \\ \infty _{26} 8_0 2_1 [20] &{} \infty _{27} 30_0 27_1 [8] \end{array} \end{aligned}$$

Example 8.25

For $n=33$:

$$\begin{aligned} \begin{array}{llllll} 0_0 11_0 32_0 [0] &{} 6_0 23_0 25_0 [ 8] &{} 16_1 17_1 21_1 [11] &{} 7_1 27_1 33_1 [32] &{} \infty _0 17_0 18_1 [22] &{} \infty _1 26_0 28_1 [14] \\ \infty _2 16_0 19_1 [26] &{} \infty _3 1_0 5_1 [ 1] &{} \infty _4 3_0 8_1 [ 3] &{} \infty _5 4_0 10_1 [ 4] &{} \infty _6 2_0 9_1 [13] &{} \infty _7 5_0 13_1 [21] \\ \infty _8 14_0 23_1 [10] &{} \infty _9 10_0 20_1 [15] &{} \infty _{10} 13_0 24_1 [34] &{} \infty _{11} 18_0 31_1 [ 5] &{} \infty _{12} 20_0 34_1 [ 9] &{} \infty _{13} 21_0 1_1 [24] \\ \infty _{14} 31_0 12_1 [17] &{} \infty _{15} 22_0 4_1 [33] &{} \infty _{16} 28_0 11_1 [ 2] &{} \infty _{17} 30_0 14_1 [ 6] &{} \infty _{18} 15_0 0_1 [19] &{} \infty _{19} 29_0 15_1 [23] \\ \infty _{20} 19_0 6_1 [25] &{} \infty _{21} 7_0 30_1 [31] &{} \infty _{22} 8_0 32_1 [20] &{} \infty _{23} 12_0 2_1 [ 7] &{} \infty _{24} 34_0 25_1 [28] &{} \infty _{25} 9_0 3_1 [12] \\ \infty _{26} 33_0 29_1 [18] &{} \infty _{27} 24_0 22_1 [29] &{} \infty _{28} 27_0 26_1 [30] \end{array} \end{aligned}$$

Example 8.26

For $n=34$:

$$\begin{aligned} \begin{array}{llllll} 0_0 1_0 11_0 [35] &{} 15_0 17_0 32_0 [1] &{} 1_1 23_1 26_1 [11] &{} 10_1 14_1 27_1 [25] &{} \infty _0 18_0 18_1 [20] &{} \infty _1 12_0 13_1 [12] \\ \infty _2 26_0 28_1 [27] &{} \infty _3 2_0 5_1 [2] &{} \infty _4 3_0 7_1 [3] &{} \infty _5 4_0 9_1 [5] &{} \infty _6 5_0 11_1 [9] &{} \infty _7 8_0 15_1 [17] \\ \infty _8 9_0 17_1 [14] &{} \infty _9 7_0 16_1 [24] &{} \infty _{10} 10_0 20_1 [16] &{} \infty _{11} 19_0 30_1 [32] &{} \infty _{12} 20_0 32_1 [28] &{} \infty _{13} 21_0 34_1 [7] \\ \infty _{14} 22_0 0_1 [15] &{} \infty _{15} 25_0 4_1 [19] &{} \infty _{16} 23_0 3_1 [6] &{} \infty _{17} 27_0 8_1 [22] &{} \infty _{18} 24_0 6_1 [23] &{} \infty _{19} 29_0 12_1 [10] \\ \infty _{20} 13_0 33_1 [21] &{} \infty _{21} 14_0 35_1 [29] &{} \infty _{22} 16_0 2_1 [4] &{} \infty _{23} 35_0 22_1 [31] &{} \infty _{24} 6_0 31_1 [13] &{} \infty _{25} 31_0 21_1 [26] \\ \infty _{26} 28_0 19_1 [30] &{} \infty _{27} 33_0 25_1 [8] &{} \infty _{28} 30_0 24_1 [33] &{} \infty _{29} 34_0 29_1 [34] \end{array} \end{aligned}$$

Example 8.27

For $n=39$:

$$\begin{aligned} \begin{array}{llllll} 0_0 11_0 18_0 [0] &{} 10_0 22_0 36_0 [17] &{} 2_1 27_1 33_1 [ 4] &{} 13_1 39_1 40_1 [13] &{} \infty _0 19_0 19_1 [ 2] &{} \infty _1 12_0 14_1 [30] \\ \infty _2 20_0 23_1 [18] &{} \infty _3 16_0 21_1 [28] &{} \infty _4 2_0 8_1 [35] &{} \infty _5 3_0 10_1 [20] &{} \infty _6 1_0 9_1 [31] &{} \infty _7 6_0 15_1 [40] \\ \infty _8 14_0 24_1 [15] &{} \infty _9 4_0 16_1 [ 6] &{} \infty _{10} 25_0 38_1 [10] &{} \infty _{11} 32_0 5_1 [11] &{} \infty _{12} 29_0 3_1 [25] &{} \infty _{13} 31_0 6_1 [29] \\ \infty _{14} 35_0 11_1 [22] &{} \infty _{15} 24_0 1_1 [26] &{} \infty _{16} 26_0 4_1 [21] &{} \infty _{17} 7_0 28_1 [23] &{} \infty _{18} 13_0 35_1 [ 7] &{} \infty _{19} 40_0 22_1 [37] \\ \infty _{20} 8_0 32_1 [14] &{} \infty _{21} 9_0 34_1 [36] &{} \infty _{22} 15_0 0_1 [19] &{} \infty _{23} 21_0 7_1 [27] &{} \infty _{24} 33_0 20_1 [16] &{} \infty _{25} 37_0 26_1 [32] \\ \infty _{26} 5_0 36_1 [ 9] &{} \infty _{27} 27_0 18_1 [38] &{} \infty _{28} 38_0 31_1 [34] &{} \infty _{29} 23_0 17_1 [33] &{} \infty _{30} 17_0 12_1 [ 8] &{} \infty _{31} 34_0 30_1 [24] \\ \infty _{32} 28_0 25_1 [39] &{} \infty _{33} 39_0 37_1 [ 1] &{} \infty _{34} 30_0 29_1 [ 3] \end{array} \end{aligned}$$

Example 8.28

For $n=44$:

$$\begin{aligned} \begin{array}{llllll} 0_0 2_0 7_0 [44] &{} 21_0 24_0 33_0 [2] &{} 0_1 11_1 17_1 [6] &{} 10_1 14_1 19_1 [40] &{} \infty _0 32_0 32_1 [28] &{} \infty _1 26_0 27_1 [29] \\ \infty _2 18_0 20_1 [45] &{} \infty _3 1_0 4_1 [18] &{} \infty _4 3_0 7_1 [36] &{} \infty _5 4_0 9_1 [16] &{} \infty _6 6_0 12_1 [9] &{} \infty _7 8_0 15_1 [5] \\ \infty _8 10_0 18_1 [30] &{} \infty _9 34_0 43_1 [37] &{} \infty _{10} 35_0 45_1 [32] &{} \infty _{11} 13_0 24_1 [15] &{} \infty _{12} 29_0 41_1 [41] &{} \infty _{13} 36_0 3_1 [43] \\ \infty _{14} 38_0 6_1 [38] &{} \infty _{15} 39_0 8_1 [3] &{} \infty _{16} 31_0 1_1 [31] &{} \infty _{17} 17_0 34_1 [1] &{} \infty _{18} 19_0 37_1 [33] &{} \infty _{19} 20_0 39_1 [25] \\ \infty _{20} 22_0 42_1 [19] &{} \infty _{21} 23_0 44_1 [14] &{} \infty _{22} 14_0 36_1 [13] &{} \infty _{23} 5_0 28_1 [17] &{} \infty _{24} 9_0 33_1 [20] &{} \infty _{25} 15_0 40_1 [34] \\ \infty _{26} 45_0 25_1 [12] &{} \infty _{27} 40_0 21_1 [8] &{} \infty _{28} 41_0 23_1 [39] &{} \infty _{29} 44_0 31_1 [10] &{} \infty _{30} 12_0 2_1 [24] &{} \infty _{31} 25_0 16_1 [22] \\ \infty _{32} 43_0 35_1 [7] &{} \infty _{33} 37_0 30_1 [21] &{} \infty _{34} 11_0 5_1 [42] &{} \infty _{35} 27_0 22_1 [11] &{} \infty _{36} 42_0 38_1 [35] &{} \infty _{37} 16_0 13_1 [27] \\ \infty _{38} 28_0 26_1 [4] &{} \infty _{39} 30_0 29_1 [26] \end{array} \end{aligned}$$

Now the intransitive starters and adders:

Example 8.29

For $n= 7$:

$$\begin{aligned} \begin{array}{lllllll} 2_{2} 3_{2} 5_{2} [0] &{} 0_{0} 1_{0} 3_{0} [6] &{} 6_{1} 0_{1} 2_{1} [1] &{} 6_{0} 3_{1} 4_{2} [ 2] &{} 5_0 1_{1} 6_{2} [5] &{} 4_{0} 5_1 0_{2} [R] &{} 5_{0} 4_{1} 0_{2} [C] \\ 2_{0} 4_{1} 1_{2} [ R] &{} 4_0 2_{1} 1_{2} [C] \\ \end{array} \end{aligned}$$

Example 8.30

For $n= 9$:

$$\begin{aligned} \begin{array}{lllllll} 0_{2} 3_{2} 5_{2} [0] &{} 6_{0} 4_{0} 8_{2} [5] &{} 2_{1} 0_{1} 4_{2} [4] &{} 1_{0} 0_{0} 7_{2} [3] &{} 4_1 3_{1} 1_{2} [6] &{} 8_{0} 3_0 7_{1} [7] &{} 5_{0} 6_{1} 1_{1} [2] \\ 2_{0} 8_{1} 2_{2} [R] &{} 8_{0} 2_{1} 2_{2} [C] &{} 7_{0} 5_{1} 6_{2} [R] &{} 5_0 7_{1} 6_{2} [C] \\ \end{array} \end{aligned}$$

Example 8.31

For $n=11$:

$$\begin{aligned} \begin{array}{lllllll} 0_{2} 1_{2} 9_{2} [0] &{} 4_{0} 8_{0} 7_{0} [5] &{} 9_{1} 2_{1} 1_{1} [6] &{} 5_{0} 4_{1} 7_{2} [ 9] &{} 2_0 3_{1} 5_{2} [2] &{} 1_{0} 6_0 6_{2} [4] &{} 5_{1} 10_{1} 10_{2} [7] \\ 9_{0} 0_{1} 4_{2} [10] &{} 10_{0} 8_{1} 3_{2} [1] &{} 0_{0} 7_{1} 8_{2} [R] &{} 7_0 0_{1} 8_{2} [C] &{} 3_{0} 6_1 2_{2} [R] &{} 6_0 3_{1} 2_{2} [C] \\ \end{array} \end{aligned}$$