{
  "abstract": "Each of the 64 hexagrams of the I-Ching, or Book of Changes, is a stack of six Yin-Yang lines and is therefore a binary vector of length 6; the eight hexagrams formed by repeating a single trigram span a doubled-trigram subspace $Δ$. We ask which binary $[6,3,3]$ codes are compatible with this trigram structure. We show that a $[6,3,3]$ code meets $Δ$ in at most four codewords, that the four are then forced to be the doubled trigrams Earth, Fire, Wind and Lake, and that exactly two codes attain this bound: the shortened Hamming code $\\mathcal C_A$ and its dual. Since $\\mathcal C_A$ has covering radius 2, its eight codewords, their radius-one balls, and its eight deep holes together reconstruct all 64 hexagrams. In general, we call a binary code of length $2m$ with $d \\ge 3$ diagonal-coherent if it contains the half of the diagonal subspace $\\{(u,u)\\}$ coming from even-weight vectors $u$ -- the largest part of that subspace a single-error-correcting code can contain -- and extremal if its dimension is as large as possible. For $m\\ge2$ and $r=\\lceil\\log_2(2m+1)\\rceil$, we determine the number $N(m)$ of extremal diagonal-coherent codes with parameters $[\\,2m,\\ 2m-r,\\ \\ge3\\,]$, show that diagonal coherence entails no loss of redundancy, and determine their minimum distances. We also put these codes in a normal form, determine their hulls, and show that each has covering radius $2$ with exactly $2^r-1-2m$ deep-hole cosets. Finally we propose three open problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33676",
  "author_total": 1,
  "authors": [
    {
      "name": "Jon-Lark Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cg6p6wg92mp61.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33676",
  "fulltext_url": null,
  "id": "cg6p6wg92mp61",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33676",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7223820836",
  "pdf_url": "https://arxiv.org/pdf/2610.33676",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cg6p6wg92mp61.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33676",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33676"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Extremal diagonal-coherent binary codes: shortened Hamming cores, deep holes, and the 64 I-Ching hexagrams",
  "title_text": "Extremal diagonal-coherent binary codes: shortened Hamming cores, deep holes, and the 64 I-Ching hexagrams",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/cg6p6wg92mp61",
  "venue": "arXiv",
  "versions": []
}