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C-16

proof verified via machine
There is a (85,24,3) covering design with 76 blocks, so C(85,24,3) ≤ 76.

Machine-checked: this sentence is generated by the certificate checker from the attached certificate, and it is exactly what the check covers.

The author's statement — not covered by the machine check:

C(85,24,3) <= 76. A 76-block (85,24,3) covering design, beating the prior best of 77.

by caleb with Claude Opus 5.5 (Anthropic) · 2026-11-07 16:14 UTC · via web · hash a74036c692a6… · AI-assisted

Evidence

Explicit block file attached (one block per line, 1-indexed, sorted). Coverage verified exhaustively (0 uncovered t-subsets).

Prior best 77 (LJCR v1.2 and coveringrepository.com).
Method: weighted tabu local search with single-point swaps from random blocks.
All 98,770 triples covered.
Author: Caleb Ashton.
SHA-256 of the block file: d111b0feb0cd74f5af84f8bbf660d65979caca2eb005885191dbd3fb63c14ea53.
LJCR entry: https://ljcr.dmgordon.org/cover/show_cover.php?v=85&k=24&t=3

All blocks (one per line):
11 12 13 14 16 20 21 22 23 24 27 30 31 33 40 41 44 52 54 57 58 59 68 84
3 4 7 16 19 21 22 23 29 32 36 40 41 47 48 55 58 61 65 69 70 78 81 82
5 9 14 15 16 20 24 28 29 32 39 45 46 54 55 56 63 66 72 74 77 78 79 83
2 3 6 11 13 19 20 23 28 30 40 42 44 45 46 56 58 63 70 72 74 80 83 84
6 10 11 18 19 25 27 28 29 41 45 49 50 53 55 66 71 73 74 75 77 78 79 83
1 2 5 6 9 10 13 20 24 27 30 34 39 41 42 51 58 65 66 71 77 79 82 84
1 5 6 7 9 11 14 17 21 22 28 29 34 39 46 47 53 54 57 65 67 73 76 85
1 4 11 13 15 17 18 23 29 30 34 40 44 46 49 50 62 66 69 73 75 77 82 84
3 5 9 12 13 15 17 18 21 22 30 32 39 51 52 53 56 58 59 63 70 72 75 84
1 12 13 19 21 22 25 28 30 34 35 36 46 55 57 59 60 61 66 77 78 82 84 85
2 7 8 12 18 20 21 22 24 29 31 38 43 47 49 50 52 56 63 65 68 69 72 75
10 12 16 18 26 29 32 37 44 49 50 51 55 57 58 59 64 67 71 75 76 78 80 85
2 3 10 15 17 19 23 25 29 31 33 40 51 52 57 58 66 67 68 70 71 76 77 85
3 13 20 24 25 27 29 30 36 41 48 56 57 61 62 63 67 70 72 73 76 81 84 85
5 7 9 11 12 20 24 26 33 36 39 44 47 49 50 52 59 61 64 66 67 76 77 79
1 6 8 11 16 17 25 27 29 31 32 34 38 41 42 43 44 46 52 55 57 68 73 78
11 14 18 19 21 22 26 36 42 44 52 54 55 56 61 63 64 72 73 75 78 79 80 82
1 2 4 7 12 18 19 31 32 34 35 36 41 47 51 58 59 60 61 68 69 73 75 79
12 17 25 26 28 31 41 45 46 49 50 56 59 62 63 64 65 67 68 72 74 76 82 83
1 3 7 8 13 16 26 30 33 34 37 38 43 44 47 56 63 64 66 70 72 77 82 84
4 6 7 10 14 17 19 26 32 33 41 44 45 47 54 57 64 66 69 71 74 77 83 85
4 5 9 12 17 20 21 22 24 25 31 35 39 44 48 53 60 66 68 69 77 79 80 81
4 5 7 8 9 11 15 19 25 32 38 39 41 42 43 47 56 62 63 67 69 72 76 80
2 3 4 11 12 15 20 24 33 44 45 53 55 57 59 62 65 69 70 73 74 78 83 85
1 5 6 9 15 23 31 32 33 34 36 37 39 40 44 49 50 56 61 63 65 68 72 73
1 7 10 12 13 21 22 27 29 30 36 37 45 47 59 61 62 65 71 74 79 80 83 84
4 8 20 23 29 31 33 35 38 40 42 43 45 57 60 67 68 69 74 76 79 82 83 85
1 5 6 8 9 12 16 19 34 35 38 39 43 44 45 48 52 59 60 62 65 74 81 83
2 3 8 11 14 21 22 25 32 35 37 38 43 44 45 49 50 54 60 66 70 74 77 83
2 5 6 9 12 13 16 29 30 32 33 35 39 41 42 52 53 55 59 60 62 78 84 85
2 3 5 7 9 10 11 16 18 27 31 39 44 45 47 67 68 70 71 74 75 76 82 83
2 4 5 9 10 12 14 23 25 27 28 29 37 39 40 46 52 54 59 69 71 73 80 82
3 7 12 15 16 17 25 28 32 37 42 46 47 48 49 50 53 59 62 70 73 79 81 82
7 15 16 19 21 22 25 26 27 29 33 35 42 44 47 51 53 58 60 64 65 66 73 77
2 4 10 11 14 15 16 19 20 21 22 24 25 26 37 48 51 54 62 64 65 69 71 81
2 4 6 8 12 13 21 22 26 30 38 42 43 48 49 50 59 64 67 69 73 76 81 84
11 12 16 19 27 31 42 48 49 50 51 53 56 57 58 63 65 66 68 72 77 80 81 85
1 3 10 20 21 22 24 26 28 31 32 34 42 44 46 57 58 62 64 68 70 71 73 85
6 10 16 18 21 22 23 25 35 40 41 46 52 56 60 62 63 66 67 71 72 75 76 77
2 5 8 9 10 17 19 21 22 27 33 36 38 39 43 49 50 53 55 61 62 71 78 82
6 15 19 20 21 22 24 31 36 37 44 45 51 55 58 61 67 68 74 76 78 82 83 85
3 5 8 9 12 18 19 20 23 24 37 38 39 40 43 57 62 66 70 73 75 77 79 85
3 5 9 14 18 28 29 33 35 36 39 41 42 44 46 48 49 50 54 60 61 70 75 81
1 3 12 14 15 19 23 26 27 34 35 40 48 49 50 53 54 58 59 60 64 67 70 76
2 5 9 10 15 20 26 27 32 35 39 44 48 56 57 58 60 63 64 71 72 81 82 85
2 4 6 16 17 23 24 26 29 36 40 51 53 56 59 61 63 64 66 69 72 73 77 82
2 6 7 12 13 14 15 17 18 25 30 31 36 37 42 47 54 57 59 61 68 75 84 85
3 4 8 17 27 28 37 38 42 43 44 46 51 52 55 58 65 66 67 69 70 76 77 78
2 7 12 18 23 26 28 31 33 40 46 47 48 53 55 62 64 66 68 75 77 78 80 81
12 14 18 20 23 27 32 34 36 40 42 45 51 54 58 59 61 62 65 66 74 75 77 83
1 4 10 11 12 17 19 20 24 29 33 34 35 37 42 44 56 58 59 60 63 71 72 80
1 4 18 21 22 27 28 33 34 37 45 46 48 52 56 57 63 69 72 74 75 81 83 85
5 8 9 13 14 19 26 28 29 30 31 38 39 43 46 48 51 54 58 62 64 68 81 84
5 9 16 17 20 21 22 23 26 31 37 39 40 41 42 45 53 57 64 68 73 74 83 85
7 8 10 12 28 35 36 38 43 45 46 47 51 53 56 58 60 61 63 71 72 73 74 83
4 5 9 12 13 16 25 30 33 36 39 45 49 50 51 57 58 61 69 74 80 83 84 85
6 8 11 12 18 20 24 25 28 29 32 33 38 43 44 46 48 51 53 58 59 75 81 82
5 7 9 11 13 23 30 35 37 39 40 44 47 48 51 55 58 60 67 73 76 78 81 84
1 13 14 30 31 32 34 45 48 52 53 54 66 67 68 73 74 76 77 80 81 82 83 84
3 4 10 13 14 30 31 35 49 50 54 55 56 60 63 65 68 69 70 71 72 78 79 84
1 4 8 10 14 16 18 20 24 34 36 38 42 43 53 54 58 61 67 69 71 75 76 80
10 11 13 15 16 17 18 19 20 28 30 32 33 46 48 65 67 71 73 75 76 79 81 84
8 11 14 15 18 20 33 35 37 38 41 43 44 52 53 54 57 60 65 73 75 80 82 85
4 6 11 15 16 20 26 27 28 31 35 36 37 44 46 52 58 60 61 62 64 68 69 79
1 7 10 15 20 23 24 25 34 40 42 45 46 47 48 49 50 52 55 71 74 78 81 83
2 3 6 17 21 22 23 24 27 28 32 33 35 36 40 46 60 61 65 67 70 76 77 80
3 6 7 14 16 17 20 24 35 47 49 50 51 52 54 57 58 60 62 70 73 80 82 85
8 13 15 17 18 20 24 26 27 30 35 38 41 43 45 55 60 64 74 75 78 80 83 84
4 7 13 19 20 24 27 28 30 32 34 37 41 44 46 47 49 50 52 53 62 69 80 84
1 3 4 5 6 9 18 25 26 31 34 39 52 55 57 59 64 65 68 69 70 75 78 80
8 10 11 13 23 25 26 30 32 36 38 40 43 44 52 53 57 61 64 65 71 79 84 85
1 2 3 11 17 20 26 29 34 36 41 44 45 48 51 52 58 61 64 70 74 79 81 83
1 2 14 20 25 33 34 37 41 44 51 53 54 55 56 58 62 63 67 69 72 76 78 79
1 2 6 8 15 16 21 22 23 28 34 38 40 41 43 46 49 50 51 57 58 79 80 85
6 7 8 12 14 17 23 27 38 40 43 47 48 54 55 56 57 59 63 72 78 79 81 85
3 6 8 10 12 15 29 31 36 37 38 41 43 48 53 59 61 66 68 70 71 77 80 81

Citations

Machine verification

  • Certificate check passed score 76 2026-11-07 20:07 UTC
    Valid (85,24,3) covering with 76 blocks.

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