{
  "claim": "For every abelian group G of order 16 acting regularly on itself, the minimum size of a G-translation-invariant (16,5,3) covering is 80.",
  "method": "Exhaust all C(273,4) choices of four 5-set orbits; validate a five-orbit witness for each group.",
  "combination_formula": "226387980",
  "results": [
    {
      "group_invariant_factors": [
        16
      ],
      "block_orbits": 273,
      "triple_orbits": 35,
      "all_block_orbit_sizes": [
        16
      ],
      "four_orbit_combinations_tested": 226387980,
      "maximum_triple_orbits_covered_by_four": 33,
      "number_of_maximizing_four_tuples": 296,
      "one_maximizing_four_tuple": [
        [
          0,
          1,
          2,
          3,
          6
        ],
        [
          0,
          1,
          2,
          7,
          12
        ],
        [
          0,
          1,
          6,
          8,
          10
        ],
        [
          0,
          1,
          6,
          9,
          13
        ]
      ],
      "no_four_orbit_cover": true,
      "witness_base_blocks": [
        [
          0,
          1,
          2,
          13,
          14
        ],
        [
          0,
          1,
          6,
          11,
          14
        ],
        [
          0,
          1,
          7,
          9,
          13
        ],
        [
          0,
          1,
          8,
          10,
          12
        ],
        [
          0,
          2,
          5,
          9,
          11
        ]
      ],
      "witness_block_count": 80,
      "witness_uncovered_triples": 0,
      "witness_coverage_histogram": {
        "1": 384,
        "2": 128,
        "3": 32,
        "4": 16
      },
      "witness_sha256": "e1bf66b60a5ca34dbe1221150aff412a3a23fb0bbcd34b3c87783774c3519429",
      "enumeration_seconds": 1.074812889099121
    },
    {
      "group_invariant_factors": [
        8,
        2
      ],
      "block_orbits": 273,
      "triple_orbits": 35,
      "all_block_orbit_sizes": [
        16
      ],
      "four_orbit_combinations_tested": 226387980,
      "maximum_triple_orbits_covered_by_four": 33,
      "number_of_maximizing_four_tuples": 336,
      "one_maximizing_four_tuple": [
        [
          0,
          1,
          2,
          3,
          8
        ],
        [
          0,
          1,
          4,
          10,
          12
        ],
        [
          0,
          2,
          4,
          11,
          15
        ],
        [
          0,
          2,
          9,
          12,
          15
        ]
      ],
      "no_four_orbit_cover": true,
      "witness_base_blocks": [
        [
          0,
          1,
          2,
          13,
          15
        ],
        [
          0,
          1,
          4,
          9,
          10
        ],
        [
          0,
          1,
          6,
          8,
          14
        ],
        [
          0,
          2,
          4,
          8,
          15
        ],
        [
          0,
          2,
          9,
          12,
          15
        ]
      ],
      "witness_block_count": 80,
      "witness_uncovered_triples": 0,
      "witness_coverage_histogram": {
        "1": 368,
        "2": 144,
        "3": 48
      },
      "witness_sha256": "c818a2e1c43a140a1e8947671df738681e72ceb8716e30cd21e13b9387dde621",
      "enumeration_seconds": 0.6934852600097656
    },
    {
      "group_invariant_factors": [
        4,
        4
      ],
      "block_orbits": 273,
      "triple_orbits": 35,
      "all_block_orbit_sizes": [
        16
      ],
      "four_orbit_combinations_tested": 226387980,
      "maximum_triple_orbits_covered_by_four": 34,
      "number_of_maximizing_four_tuples": 96,
      "one_maximizing_four_tuple": [
        [
          0,
          1,
          2,
          4,
          5
        ],
        [
          0,
          1,
          6,
          8,
          11
        ],
        [
          0,
          2,
          4,
          8,
          15
        ],
        [
          0,
          2,
          5,
          8,
          12
        ]
      ],
      "no_four_orbit_cover": true,
      "witness_base_blocks": [
        [
          0,
          1,
          2,
          14,
          15
        ],
        [
          0,
          1,
          5,
          14,
          15
        ],
        [
          0,
          1,
          7,
          8,
          12
        ],
        [
          0,
          1,
          8,
          10,
          14
        ],
        [
          0,
          1,
          9,
          11,
          14
        ]
      ],
      "witness_block_count": 80,
      "witness_uncovered_triples": 0,
      "witness_coverage_histogram": {
        "1": 384,
        "2": 144,
        "3": 16,
        "5": 16
      },
      "witness_sha256": "5461d0bbbe21b01085fa3a91310053709fc7f29138181b03c6da6120bb42ae59",
      "enumeration_seconds": 0.6586389541625977
    },
    {
      "group_invariant_factors": [
        4,
        2,
        2
      ],
      "block_orbits": 273,
      "triple_orbits": 35,
      "all_block_orbit_sizes": [
        16
      ],
      "four_orbit_combinations_tested": 226387980,
      "maximum_triple_orbits_covered_by_four": 33,
      "number_of_maximizing_four_tuples": 384,
      "one_maximizing_four_tuple": [
        [
          0,
          1,
          2,
          4,
          8
        ],
        [
          0,
          1,
          2,
          5,
          12
        ],
        [
          0,
          1,
          6,
          11,
          14
        ],
        [
          0,
          2,
          9,
          13,
          14
        ]
      ],
      "no_four_orbit_cover": true,
      "witness_base_blocks": [
        [
          0,
          1,
          2,
          14,
          15
        ],
        [
          0,
          1,
          8,
          11,
          12
        ],
        [
          0,
          2,
          4,
          13,
          14
        ],
        [
          0,
          2,
          5,
          10,
          14
        ],
        [
          0,
          2,
          5,
          11,
          13
        ]
      ],
      "witness_block_count": 80,
      "witness_uncovered_triples": 0,
      "witness_coverage_histogram": {
        "1": 352,
        "2": 176,
        "3": 32
      },
      "witness_sha256": "6c969bb38a309954336b47388bdf0f0a23168f9af06b05c5a01e6bca8593e592",
      "enumeration_seconds": 0.6734654903411865
    },
    {
      "group_invariant_factors": [
        2,
        2,
        2,
        2
      ],
      "block_orbits": 273,
      "triple_orbits": 35,
      "all_block_orbit_sizes": [
        16
      ],
      "four_orbit_combinations_tested": 226387980,
      "maximum_triple_orbits_covered_by_four": 32,
      "number_of_maximizing_four_tuples": 34440,
      "one_maximizing_four_tuple": [
        [
          0,
          1,
          2,
          4,
          8
        ],
        [
          0,
          1,
          2,
          4,
          9
        ],
        [
          0,
          1,
          10,
          13,
          14
        ],
        [
          0,
          2,
          5,
          10,
          12
        ]
      ],
      "no_four_orbit_cover": true,
      "witness_base_blocks": [
        [
          0,
          1,
          2,
          14,
          15
        ],
        [
          0,
          1,
          4,
          11,
          12
        ],
        [
          0,
          1,
          6,
          9,
          12
        ],
        [
          0,
          2,
          4,
          7,
          9
        ],
        [
          0,
          2,
          4,
          9,
          10
        ]
      ],
      "witness_block_count": 80,
      "witness_uncovered_triples": 0,
      "witness_coverage_histogram": {
        "1": 384,
        "2": 128,
        "3": 32,
        "4": 16
      },
      "witness_sha256": "a8b6f32d2f7a3777a5fa16189cbc05d9ffa67b921e24aa8f5f54839f5658460a",
      "enumeration_seconds": 0.6799335479736328
    }
  ]
}