files / problems / covering / compute / odd_r_residue_2026-08-20.json

1.8 KB · on GitHub

{
  "format": "odd-r-table-5.1-residue-v1",
  "status": "search residue, no dent",
  "warning": "Positive hole counts are search residues or local exclusions, not lower bounds. The paper lengths ell_2(11,2)<=79 and ell_2(13,2)<=159 still stand.",
  "paper": {
    "r11": 79,
    "r13": 159,
    "family": "f(r)=5*2^{(r-3)/2}-1 Gabidulin-Davydov-Tombak"
  },
  "exclusions": [
    {
      "claim": "Gabidulin m=6 w1=1 w2=2 is an explicit 11x79 covering",
      "file": "compute/odd_r11_n79_gabidulin.cols",
      "covered": "2048/2048",
      "note": "Reconstructs the paper upper bound. Not a dent."
    },
    {
      "claim": "Every 1-column deletion of that 79-set leaves 16, 29, 30, 36, or 64 uncovered syndromes",
      "best_deletion_holes": 16,
      "best_deleted_columns": [1, 1025, 513, 1537],
      "files": [
        "compute/odd_r11_n78_16hole_a.cols",
        "compute/odd_r11_n78_16hole_b.cols",
        "compute/odd_r11_n78_16hole_c.cols",
        "compute/odd_r11_n78_16hole_d.cols"
      ]
    },
    {
      "claim": "Exhaustive 1-swap on the 16-hole puncture (delete column 1) finds no improvement",
      "start_holes": 16,
      "end_holes": 16,
      "file": "compute/odd_r11_n78_repair_a.json"
    },
    {
      "claim": "No 2-out-1-in replacement from the same 79-set reaches 0 holes",
      "best_2_deletion_holes": 34
    }
  ],
  "search_residues": [
    {
      "r": 11,
      "n": 78,
      "best_uncovered": 16,
      "seed": "Gabidulin 79 minus one",
      "file": "compute/odd_r11_n78_checkpoint.json"
    },
    {
      "r": 11,
      "n": 78,
      "best_uncovered": 37,
      "seed": "78-subset of a longer union-greedy covering",
      "file": "compute/odd_r11_n78_from85.json"
    },
    {
      "r": 13,
      "n": 158,
      "note": "Gabidulin m=7 1-deletions leave 32 to 128 holes; no n=158 search completed before wrap"
    }
  ]
}