Extremal diagonal-coherent binary codes: shortened Hamming cores, deep holes, and the 64 I-Ching hexagrams
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 𝒞A and its dual. Since 𝒞A 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 ≥ 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≥2 and r=⌈log2(2m+1)⌉, we determine the number N(m) of extremal diagonal-coherent codes with parameters [ 2m, 2m−r, ≥3 ], 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 2r−1−2m deep-hole cosets. Finally we propose three open problems.
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BibTeX
@misc{kim2026extremal,
title = {{Extremal diagonal-coherent binary codes: shortened Hamming cores, deep holes, and the 64 I-Ching hexagrams}},
author = {Jon-Lark Kim},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33676},
eprint = {2610.33676},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33676},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Extremal diagonal-coherent binary codes: shortened Hamming cores, deep holes, and the 64 I-Ching hexagrams
AU - Jon-Lark Kim
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33676
UR - https://arxiv.org/abs/2610.33676
AB - 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 𝒞_A and its dual. Since 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 ≥ 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≥2 and r=⌈log_2(2m+1)⌉, we determine the number N(m) of extremal diagonal-coherent codes with parameters [ 2m, 2m−r, ≥3 ], 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.
ER -