Abstract
Let \(v \equiv 4 \pmod 6\) and \(v\ge 4\). A canonical Kirkman packing design of order v, denoted by \(\hbox {CKPD}(v)\), is a resolvable packing with \(r=(v-4)/2\) parallel classes such that (i) each parallel class consists of a size 4 block and \((v- 4)/3\) triples; (ii) the leave consists of the union of \((v- 4)/2\) vertex-disjoint edges and a \(K_4\) with no vertices in common with those edges. A canonical Kirkman packing design is said to be doubly resolvable if there exist a pair of orthogonal resolutions. A doubly resolvable packing design is the generalization of the Kirkman square, which is inextricably bound up with the existence of some constant weight codes such as constant composition codes and permutation codes, etc. In this paper, we establish the spectra of doubly resolvable \(\hbox {CKPD}(v)\hbox {s}\) with 36 possible exceptions for v. As its direct application, a class of permutation codes are obtained.
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Acknowledgements
The author is grateful to the referees for their careful reading of the original version of this paper, their detailed comments and the suggestions that much improved the quality of this paper. This research was carried out while the author was visiting the University of Tsukuba. She wishes to express her gratitude to Prof. Miao and the Faculty of Engineering, Information and Systems for their hospitality.
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Research supported by the National Natural Science Foundation of China under Grant no. 11371207.
Appendix
Appendix
Here, using the method in Lemma 8, we provide starter blocks and adders for DRCKPDs of orders \(6u+4\) with \(5 \le u \le 14\).
u | Starter | Adder | \(\hbox {Starter}+ \hbox {adder}\) | Starter | Adder | \(\hbox {Starter}+ \hbox {adder}\) |
|---|---|---|---|---|---|---|
5 | \(\{ 4, 5, 10,15\}\) | 0 | \(\{ 4 , 5 , 10 , 15\}\) | \(\{ 8 , 21 ,25\}\) | 8 | \(\{ 16 , 29 , 3\}\) |
\(\{ 0 , 9 , 27 \}\) | 28 | \(\{28 , 7 ,25\}\) | \(\{11 , 18, \infty _1\}\) | 20 | \(\{ 1 , 8, \infty _1\}\) | |
\(\{ 1 , 7 , 29\}\) | 12 | \(\{13 , 19 , 11\}\) | \(\{12 , 19 , \infty _2\}\) | 2 | \(\{ 14 , 21, \infty _2\}\) | |
\(\{ 2 , 16 , 20\}\) | 10 | \(\{12, 26 , 0\}\) | \(\{ 13 , 22, \infty _3\}\) | 14 | \(\{ 27 , 6, \infty _3\}\) | |
\(\{ 3 , 14 , 17\}\) | 6 | \(\{9 , 20 , 23\}\) | \(\{23 , 24, \infty _4\}\) | 24 | \(\{17 , 18, \infty _4\}\) | |
\(\{ 6 , 26 , 28 \}\) | 26 | \(\{2 , 22 , 24\}\) | ||||
6 | \(\{ 5 , 8 , 12 , 35\}\) | 0 | \(\{5 , 8 , 12 ,35\}\) | \(\{ 11 , 25 , 30\}\) | 34 | \(\{9 , 23 , 28\}\) |
\(\{0 , 1 , 2 \}\) | 30 | \(\{30 , 31 , 32\}\) | \(\{13 , 15 , 23\}\) | 24 | \(\{1 , 3 , 11\}\) | |
\(\{3 , 7 ,19\}\) | 14 | \(\{17 , 21 , 33\}\) | \(\{17 , 34 , \infty _1\}\) | 26 | \(\{7 , 24, \infty _1\}\) | |
\(\{4 , 14 , 28 \}\) | 12 | \(\{16 , 26 , 4\}\) | \(\{18 , 33, \infty _2\}\) | 18 | \(\{0 , 15, \infty _2\}\) | |
\(\{ 6 , 22 , 31\}\) | 32 | \(\{2 , 18 , 27\}\) | \(\{20 , 27, \infty _3\}\) | 2 | \(\{22 , 29, \infty _3\}\) | |
\(\{ 9 , 24 , 32\}\) | 10 | \(\{19 , 34 , 6\}\) | \(\{26 , 29 , \infty _4\}\) | 20 | \(\{10 , 13, \infty _4\}\) | |
\(\{ 10 , 16 , 21\}\) | 4 | \(\{14 , 20 ,25\}\) | ||||
7 | \(\{1 , 18 , 33 , 34\}\) | 0 | \(\{1 , 18 , 33, 34\}\) | \(\{15 , 27 , 40\}\) | 32 | \(\{ 5 , 17 , 30\}\) |
\(\{0 , 2 , 3\}\) | 38 | \(\{38 , 40 , 41\}\) | \(\{16 , 23 , 30\}\) | 12 | \(\{ 28 , 35 , 0\}\) | |
\(\{4 , 8 , 14\}\) | 8 | \(\{12 , 16 , 22\}\) | \(\{17 , 25 , 31\}\) | 36 | \(\{ 11 , 19 , 25\}\) | |
\(\{5 , 7 , 10\}\) | 22 | \(\{27 , 29 , 32\}\) | \(\{19 , 28, \infty _1\}\) | 18 | \(\{ 37 , 4, \infty _1\}\) | |
\(\{6 , 11 , 29\}\) | 2 | \(\{8 , 13 , 31\}\) | \(\{21 , 36, \infty _2\}\) | 30 | \(\{9 , 24, \infty _2\}\) | |
\(\{9 , 20 , 38\}\) | 6 | \(\{15 , 26 , 2\}\) | \(\{22 , 41, \infty _3\}\) | 40 | \(\{20 , 39, \infty _3\}\) | |
\(\{12 , 24 , 32\}\) | 24 | \(\{36 , 6 , 14\}\) | \(\{26 , 37, \infty _4\}\) | 26 | \(\{10 , 21, \infty _4\}\) | |
\(\{13 , 35 , 39\}\) | 10 | \(\{23 , 3 , 7\}\) | ||||
8 | \(\{15 , 24 , 30 , 43\}\) | 0 | \(\{15 , 24 , 30 , 43\}\) | \(\{9 , 12 , 22\}\) | 4 | \(\{13 , 16 , 26\}\) |
\(\{20 , 29 , 40\}\) | 26 | \(\{46 , 7 ,18\}\) | \(\{10 , 32 , 39\}\) | 30 | \(\{40 , 14 , 21\}\) | |
\(\{19 , 31 ,37\}\) | 10 | \(\{ 29 , 41 , 47\}\) | \(\{13 , 35 , 45\}\) | 36 | \(\{ 1 , 23 , 33\}\) | |
\(\{17 , 25 , 42\}\) | 34 | \(\{3 , 11 , 28\}\) | \(\{14 , 28 ,46\}\) | 24 | \(\{ 38 , 4 , 22\}\) | |
\(\{0 , 1 , 2 \}\) | 8 | \(\{8 , 9 , 10\}\) | \(\{16 , 27, \infty _1\}\) | 18 | \(\{ 34 , 45, \infty _1\}\) | |
\(\{3 , 5 , 36\}\) | 14 | \(\{17 , 19 , 2\}\) | \(\{18 , 23, \infty _2\}\) | 2 | \(\{20 , 25, \infty _2\}\) | |
\(\{4 , 8 , 44\}\) | 40 | \(\{44 , 0 ,36\}\) | \(\{21 , 26, \infty _3\}\) | 16 | \(\{37 , 42, \infty _3\}\) | |
\(\{6 ,33 , 47\}\) | 6 | \(\{12 , 39 , 5\}\) | \(\{38 , 41, \infty _4\}\) | 42 | \(\{32 , 35, \infty _4\}\) | |
\(\{7 , 11 , 34\}\) | 20 | \(\{27 , 31 , 6\}\) | ||||
9 | \(\{6 , 19 , 35 , 36\}\) | 0 | \(\{6 , 19 , 35 , 36\}\) | \(\{2 , 17 , 23\}\) | 30 | \(\{32 , 47, 53\}\) |
\(\{7 , 9 , 27\}\) | 2 | \(\{9 , 11 , 29\}\) | \(\{4 , 8 , 43\}\) | 38 | \(\{42 , 46 , 27\}\) | |
\(\{5 , 47 , 50\}\) | 32 | \(\{37 , 25 , 28\}\) | \(\{11 , 37 , 42\}\) | 22 | \(\{33 , 5 , 10\}\) | |
\(\{3 , 10 , 53\}\) | 40 | \(\{43 , 50 , 39\}\) | \(\{12 , 13, 34\}\) | 28 | \(\{40 , 41 , 8\}\) | |
\(\{21 , 31 , 44\}\) | 36 | \(\{3 ,13, 26\}\) | \(\{16 , 22 , 41\}\) | 8 | \(\{24 , 30 , 49\}\) | |
\(\{18 , 26 , 38\}\) | 50 | \(\{14 , 22, 34\}\) | \(\{28 , 45, \infty _1\}\) | 26 | \(\{ 0 , 17, \infty _1\}\) | |
\(\{15 , 24 , 29\}\) | 46 | \(\{7 , 16 , 21\}\) | \(\{32 , 39, \infty _2\}\) | 12 | \(\{44 , 51, \infty _2\}\) | |
\(\{20 , 30 , 48\}\) | 18 | \(\{38 , 48 , 12\}\) | \(\{40 , 51, \infty _3\}\) | 34 | \(\{20 , 31, \infty _3\}\) | |
\(\{0 , 14 , 52\}\) | 4 | \(\{4 , 18 , 2\}\) | \(\{46 , 49, \infty _4\}\) | 6 | \(\{ 52 , 1, \infty _4\}\) | |
\(\{1 , 25 , 33\}\) | 44 | \(\{45 , 15 , 23\}\) | ||||
10 | \(\{0 , 11 , 49 , 50\}\) | 0 | \(\{0 , 11 , 49 , 50\}\) | \(\{3 , 32, 51\}\) | 6 | \(\{9 ,38 , 57\}\) |
\(\{20 , 25 , 40\}\) | 2 | \(\{22 , 27 , 42\}\) | \(\{4 , 39 , 58\}\) | 14 | \(\{18 , 53, 12\}\) | |
\(\{16 , 18 , 44\}\) | 8 | \(\{24 , 26 , 52\}\) | \(\{6 , 21 , 48 \}\) | 26 | \(\{32 , 47, 14\}\) | |
\(\{13 , 53 , 56\}\) | 12 | \(\{25 , 5 , 8\}\) | \(\{7 , 41 , 43\}\) | 34 | \(\{41 ,15 ,17\}\) | |
\(\{12 , 24 , 28\}\) | 16 | \(\{28 , 40 , 44\}\) | \(\{9 , 27 , 59\}\) | 52 | \(\{1 , 19 ,51\}\) | |
\(\{8 , 33 , 37\}\) | 58 | \(\{6 , 31 , 35\}\) | \(\{ 15 , 23 , 36\}\) | 40 | \(\{55 , 3 , 16\}\) | |
\(\{2 , 5 , 10\}\) | 54 | \(\{56 , 59 , 4\}\) | \(\{17 , 26, \infty _1\}\) | 28 | \(\{45 , 54, \infty _1\}\) | |
\(\{22 , 31 , 45\}\) | 36 | \(\{58 , 7 , 21\}\) | \(\{29 , 46, \infty _2\}\) | 44 | \(\{13 , 30, \infty _2\}\) | |
\(\{19 , 35 ,42 \}\) | 4 | \(\{23, 39 , 46\}\) | \(\{30 , 57, \infty _3\}\) | 32 | \(\{2 , 29, \infty _3\}\) | |
\(\{14 , 38 , 52 \}\) | 56 | \(\{10 ,34, 48\}\) | \(\{34 ,47, \infty _4\}\) | 46 | \(\{20 , 33, \infty _4\}\) | |
\(\{1 , 54 , 55\}\) | 42 | \(\{43, 36, 37\}\) |
u | Starter | Adder | \(\hbox {Starter}+ \hbox {adder}\) | Starter | Adder | \(\hbox {Starter}+ \hbox {adder}\) |
|---|---|---|---|---|---|---|
11 | \(\{0 , 9 , 49 , 50\}\) | 0 | \(\{0 , 9 , 49 ,50\}\) | \(\{4 , 31 , 42\}\) | 56 | \(\{60, 21 , 32\}\) |
\(\{28, 33 , 57\}\) | 2 | \(\{30 ,35 , 59\}\) | \(\{14, 21 , 40\}\) | 24 | \(\{38, 45 , 64\}\) | |
\(\{15 , 36 ,47\}\) | 52 | \(\{1 , 22 , 33\}\) | \(\{8 , 16 , 38\}\) | 36 | \(\{44 , 52 , 8\}\) | |
\(\{19 ,46 , 63\}\) | 54 | \(\{7 , 34, 51\}\) | \(\{11 , 39 , 54\}\) | 14 | \(\{25 , 53, 2\}\) | |
\(\{32 , 51 , 64\}\) | 38 | \(\{4 ,23 ,36\}\) | \(\{2, 58 , 61\}\) | 4 | \(\{6 , 62 , 65\}\) | |
\(\{13 , 29, 65\}\) | 64 | \(\{11 , 27, 63\}\) | \(\{20 ,22, 34 \}\) | 58 | \(\{12, 14, 26\}\) | |
\(\{30 , 43 , 48\}\) | 28 | \(\{58 , 5 , 10\}\) | \(\{23 , 26 , 41\}\) | 16 | \(\{39, 42 , 57\}\) | |
\(\{12 , 18 , 55\}\) | 6 | \(\{18 ,24 , 61\}\) | \(\{24 , 59, \infty _1\}\) | 22 | \(\{46 , 15, \infty _1\}\) | |
\(\{5 , 17 , 25\}\) | 12 | \(\{17, 29 , 37\}\) | \(\{35 , 60, \infty _2\}\) | 62 | \(\{31 , 56, \infty _2\}\) | |
\(\{1 , 3 , 7\}\) | 40 | \(\{41, 43, 47\}\) | \(\{44 , 45, \infty _3\}\) | 10 | \(\{54 , 55, \infty _3\}\) | |
\(\{6 , 27 , 37\}\) | 42 | \(\{48 , 3 , 13\}\) | \(\{53 , 62, \infty _4\}\) | 32 | \(\{19 ,28, \infty _4\}\) | |
\(\{10 , 52 , 56\}\) | 30 | \(\{40 ,16 , 20\}\) | ||||
12 | \(\{0 , 9 , 49 , 50\}\) | 0 | \(\{0 , 9 , 49 ,50\}\) | \(\{14 , 26, 43\}\) | 10 | \(\{24, 36 , 53\}\) |
\(\{2 , 35 ,39\}\) | 66 | \(\{68 , 29 , 33\}\) | \(\{11 , 41 , 65\}\) | 26 | \(\{37 ,67, 19\}\) | |
\(\{18 , 24, 45\}\) | 40 | \(\{58, 64 ,13\}\) | \(\{8 , 40 , 63\}\) | 44 | \(\{52 , 12, 35\}\) | |
\(\{7 ,34 , 68\}\) | 64 | \(\{71 , 26 ,60\}\) | \(\{21 , 37 , 47\}\) | 4 | \(\{25 , 41 , 51\}\) | |
\(\{1 , 16 ,29\}\) | 30 | \(\{31 , 46 ,59\}\) | \(\{23, 25 , 36\}\) | 20 | \(\{43 ,45 , 56\}\) | |
\(\{10 , 61 , 66\}\) | 68 | \(\{6 , 57 , 62\}\) | \(\{22 , 30 ,48\}\) | 52 | \(\{2 , 10, 28\}\) | |
\(\{12 , 17 , 54\}\) | 22 | \(\{34 , 39 , 4\}\) | \(\{32 , 42 , 57\}\) | 6 | \(\{38 , 48, 63\}\) | |
\(\{13, 27, 33\}\) | 42 | \(\{55 ,69 , 3\}\) | \(\{38 , 58 , 62\}\) | 54 | \(\{20, 40 , 44\}\) | |
\(\{15 , 44, 46\}\) | 58 | \(\{1 , 30 , 32\}\) | \(\{51 ,60, \infty _1\}\) | 28 | \(\{7 , 16, \infty _1\}\) | |
\(\{4 , 5 ,69\}\) | 18 | \(\{22 ,23 ,15\}\) | \(\{52 , 55, \infty _2\}\) | 38 | \(\{18 , 21, \infty _2\}\) | |
\(\{3 ,28 , 56\}\) | 14 | \(\{17, 42 ,70\}\) | \(\{64 , 71, \infty _3\}\) | 62 | \(\{54 , 61, \infty _3\}\) | |
\(\{19 , 31 , 53\}\) | 46 | \(\{65 , 5 , 27\}\) | \(\{67 ,70, \infty _4\}\) | 16 | \(\{11 , 14, \infty _4\}\) | |
\(\{6 , 20 , 59\}\) | 60 | \(\{66 , 8 ,47\}\) | ||||
13 | \(\{4 ,17 ,55 , 56\}\) | 0 | \(\{4 , 17 ,55 , 56\}\) | \(\{15 , 47, 70\}\) | 56 | \(\{71 , 25 , 48\}\) |
\(\{5 , 8 , 49\}\) | 36 | \(\{41, 44 , 7\}\) | \(\{19 , 45 , 60\}\) | 48 | \(\{67 , 15 ,30\}\) | |
\(\{3 ,65 , 75\}\) | 72 | \(\{75 , 59 ,69\}\) | \(\{25 , 54 , 58\}\) | 8 | \(\{33 ,62 , 66\}\) | |
\(\{28, 34 , 43\}\) | 66 | \(\{16 , 22 , 31\}\) | \(\{30 , 35 , 59\}\) | 46 | \(\{76 , 3 , 27\}\) | |
\(\{26, 27 , 57\}\) | 20 | \(\{46, 47, 77\}\) | \(\{12 , 50 ,71\}\) | 30 | \(\{42 , 2 , 23\}\) | |
\(\{23 , 37 , 44\}\) | 60 | \(\{5 , 19 , 26\}\) | \(\{16 , 36, 46\}\) | 42 | \(\{58 , 0 , 10\}\) | |
\(\{21 , 41 , 72\}\) | 58 | \(\{1 , 21 , 52\}\) | \(\{18, 40 , 64\}\) | 10 | \(\{28 ,50 , 74\}\) | |
\(\{6 , 31 , 76\}\) | 34 | \(\{40, 65 , 32\}\) | \(\{22, 38 , 66\}\) | 76 | \(\{20 , 36 , 64\}\) | |
\(\{7 , 9 , 20\}\) | 4 | \(\{11 , 13 ,24\}\) | \(\{32 , 39 , 74\}\) | 22 | \(\{54 , 61 , 18\}\) | |
\(\{0 , 2 , 14\}\) | 70 | \(\{70, 72, 6\}\) | \(\{48 , 67, \infty _1\}\) | 64 | \(\{34 , 53, \infty _1\}\) | |
\(\{1 ,10 , 13\}\) | 50 | \(\{51 , 60 , 63\}\) | \(\{52, 69, \infty _2\}\) | 38 | \(\{12 , 29, \infty _2\}\) | |
\(\{29 , 33 , 51\}\) | 6 | \(\{35 , 39 ,57\}\) | \(\{62 , 73, \infty _3\}\) | 54 | \(\{38 , 49, \infty _3\}\) | |
\(\{24 , 42 , 77\}\) | 44 | \(\{68 , 8 , 43\}\) | \(\{63 , 68, \infty _4\}\) | 24 | \(\{9 , 14, \infty _4\}\) | |
\(\{11 , 53 ,61\}\) | 62 | \(\{73 , 37, 45\}\) | ||||
14 | \(\{4 , 17, 67 ,68\}\) | 0 | \(\{4 , 17, 67 ,68\}\) | \(\{28 , 32, 35\}\) | 2 | \(\{30, 34 , 37\}\) |
\(\{9 ,57 ,72\}\) | 74 | \(\{83 , 47 , 62\}\) | \(\{25 , 37 ,55\}\) | 18 | \(\{43 , 55 ,73\}\) | |
\(\{34 , 40, 50\}\) | 56 | \(\{6 ,12 , 22\}\) | \(\{3, 20 , 58\}\) | 6 | \(\{9 , 26 ,64\}\) | |
\(\{16 , 53, 59\}\) | 32 | \(\{48 , 1 , 7\}\) | \(\{13 , 29 , 82\}\) | 22 | \(\{35, 51 , 20\}\) | |
\(\{5, 33 ,43\}\) | 54 | \(\{59 , 3 , 13\}\) | \(\{6 , 11 , 24\}\) | 4 | \(\{10 , 15 ,28\}\) | |
\(\{12, 42 , 51\}\) | 44 | \(\{56 , 2, 11\}\) | \(\{10 , 61 , 75\}\) | 14 | \(\{24 ,75 , 5\}\) | |
\(\{26, 54 , 66\}\) | 16 | \(\{42 , 70, 82\}\) | \(\{22 , 77 , 79\}\) | 36 | \(\{58 , 29, 31\}\) | |
\(\{1 , 23 , 81\}\) | 26 | \(\{27 , 49 , 23\}\) | \(\{15 , 39, 64\}\) | 66 | \(\{81 ,21 , 46\}\) | |
\(\{7 , 14 , 74\}\) | 70 | \(\{77 , 0 , 60\}\) | \(\{38 ,60 , 83\}\) | 62 | \(\{16 , 38 , 61\}\) | |
\(\{8 , 49 , 69\}\) | 68 | \(\{76 , 33 , 53\}\) | \(\{41 , 44 , 46\}\) | 28 | \(\{69 , 72, 74\}\) | |
\(\{19, 27, 71\}\) | 52 | \(\{71 , 79 ,39\}\) | \(\{45 , 80, \infty _1\}\) | 12 | \(\{57 , 8, \infty _1\}\) | |
\(\{18 , 65, 76\}\) | 60 | \(\{78 , 41 , 52\}\) | \(\{47, 56, \infty _2\}\) | 82 | \(\{45 , 54, \infty _2\}\) | |
\(\{2 , 21, 78\}\) | 42 | \(\{44 , 63 , 36\}\) | \(\{48 , 73, \infty _3\}\) | 76 | \(\{40 , 65, \infty _3\}\) | |
\(\{0 , 36 , 70\}\) | 80 | \(\{80 , 32 , 66\}\) | \(\{52 , 63, \infty _4\}\) | 46 | \(\{14 , 25, \infty _4\}\) | |
\(\{30 , 31 , 62\}\) | 72 | \(\{18 , 19 ,50\}\) |
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Wang, S. Doubly Resolvable Canonical Kirkman Packing Designs and its Applications. Graphs and Combinatorics 35, 1239–1251 (2019). https://doi.org/10.1007/s00373-019-02072-9
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DOI: https://doi.org/10.1007/s00373-019-02072-9