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Extremal diagonal-coherent binary codes: shortened Hamming cores, deep holes, and the 64 I-Ching hexagrams

Jon-Lark Kim

arXiv:2610.33676v1math.COcs.DMcs.ITmath.IT

TL;DR

The paper asks how binary error-correcting codes can fit the doubled-trigram structure of the I-Ching hexagrams, and generalizes that question to diagonal-coherent codes. It proves a sharp four-trigram bound with exactly two attaining codes, and establishes counting, distance, and covering results for the general family.

  • Problem

    The paper asks which binary codes are compatible with the doubled-trigram subspace of the 64 I-Ching hexagrams.

    Source: §1 · p. 2

  • Method

    It analyzes the intersection with the doubled-trigram subspace, then generalizes to codes containing the doubled even-weight subspace and determines their extremal structure.

    Source: §1 · p. 2

  • Results

    At most four doubled-trigram hexagrams lie in a binary code, and exactly two codes attain the bound; extremal general codes have minimum distance 3 or 4, with 4 precisely when m is a power of 2.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The shortened Hamming code and its radius-one balls, together with its deep holes, account for all 64 hexagrams, while diagonal coherence costs no redundancy in the general setting.

    Source: §1 · p. 2

    At optimal redundancy, distance 4 occurs only when m is a power of 2 and only for some extremal diagonal-coherent codes.

    Source: §6.4 · p. 17

Abstract

from arXiv · show

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 $\Delta$. We ask which binary $[6,3,3]$ codes are compatible with this trigram structure. We show that a $[6,3,3]$ code meets $\Delta$ 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.

1 Introduction

The paper treats the 64 six-line I-Ching hexagrams as binary vectors and asks how shortened Hamming codes fit their doubled-trigram structure. It proves a sharp four-hexagram bound and extends the analysis to diagonal-coherent codes of length 2m.

1 Introduction

  • The paper frames this coding-theoretic study against the I-Ching's two historically distinct layers and earlier mathematical treatments of the hexagrams.It identifies the Zhouyi as the older divination manual and the Ten Wings as later commentaries supplying the Yin-Yang cosmological vocabulary.Source: §1 · pp. 1–2
  • Exactly two binary codes attain the maximum of four doubled-trigram hexagrams: the shortened Hamming code CA and its dual.The four are forced to be Earth, Fire, Wind, and Lake.Source: §1 · p. 2
  • CA's eight codewords, their radius-one balls, and its eight deep holes together account for all 64 hexagrams.The paper also gives the extremal codes a normal form, determines their hulls, and shows each has covering radius 2 with 2^r − 1 − 2m deep-hole cosets.Source: §1 · p. 2
  • For m ≥ 2, the paper counts extremal diagonal-coherent codes with parameters [2m, 2m − r, ≥3], where r = ⌈log2(2m + 1)⌉.It also shows coherence costs no redundancy and determines that their minimum distance is 3 or 4, with 4 occurring precisely when m is a power of 2.Source: §1 · p. 2

2 Structural background of I-Ching

This section establishes the line, trigram, and hexagram conventions, then relates divinatory change and classical transformations to operations on binary vectors.

2 Structural background of I-Ching

  • The section supplies structural background for the combinatorial results and fixes terminology used throughout the paper.Readers familiar with the Book of Changes may proceed directly to Section 3.Source: §2 · p. 3

2.1 Lines, trigrams, and hexagrams

  • With lines counted from the bottom, each hexagram is an ordered pair of lower and upper trigrams, giving 64 hexagrams as 8 × 8 pairs.The binary encoding uses Yin = 0 and Yang = 1; for example, Water below Fire is encoded 010101 although its traditional name lists Fire first.Source: §2.1 · p. 3

2.2 Divination and the notion of change

  • Divination generates each line independently as Yin or Yang and stable or changing, yielding an initial hexagram and a second formed by flipping the changing lines.In binary terms, that change is translation in F2^6, and the number of flipped lines equals the Hamming distance between the hexagrams.Source: §2.2 · pp. 3–4

2.3 Classical relations between hexagrams

  • Complementation flips every line, while overturning reverses the six line positions; exactly eight hexagrams are unchanged by overturning.The two commuting involutions generate a Klein four-group, and overturning realizes the permutation that interchanges the central code with its dual.Source: §2.3 · p. 4

3 Preliminaries

The preliminaries set out binary linear-code terminology and the distance, covering, and duality notions used in the paper.

3 Preliminaries

  • The paper points readers to and for basic coding-theory facts and distinguishes self-dual, self-orthogonal, and isodual linear codes.A code is isodual when a coordinate permutation maps it to its dual; self-duality implies isoduality, but not conversely.Source: §3 · pp. 4–5
  • Hamming weight counts nonzero coordinates, Hamming distance is the weight of the difference, and a code's minimum distance is its smallest pairwise distance.A linear code is specified by its length, dimension, and minimum distance.Source: §3 · p. 4
  • The covering radius is the least radius for which balls centered at a set cover the ambient binary space.The paper follows in saying a set covers with radius R when its covering radius is at most R.Source: §3 · p. 4

4 A shortened-Hamming core with I-Ching structure

The section constructs a shortened-Hamming core and proves that its compatibility with the fixed trigram splitting uniquely determines the two possible cores.

4.1 A binary [6, 3, 3] code

  • Proposition 4.2 identifies CA as a binary linear code, with nonzero weights 3 or 4 and minimum distance 3.Since d(CA) = 3, its radius-one balls are disjoint. The remaining 8 hexagrams form a single coset.Source: §4.1 · pp. 5–6

4.2 Uniqueness of the trigram-coherent core

  • Among binary [6, 3, 3] codes, at most four doubled-trigram hexagrams can coexist, and equality forces the tetrad Earth, Fire, Wind, and Lake.These are the doubled trigrams from even-weight three-bit vectors.Source: §4.2 · pp. 7, 9
  • Exactly two binary codes contain the forced doubled-trigram subcode H: CA and its dual.Overturning interchanges them; CA is isodual but not self-dual, with overturning realizing the isoduality.Source: §4.2 · pp. 7–8
  • The two trigram-coherent codes are singled out by their position relative to the fixed trigram splitting, not by their abstract code equivalence class.The paper notes that all 30 binary codes are coordinate-equivalent, but only two contain H.Source: §4.2 · p. 9

5 Deep holes for CA

The shortened Hamming code CA has covering radius 2, and its eight deep holes form the remaining coset needed to account for all 64 hexagrams. The doubled trigrams split into a coded tetrad and a complementary deep-hole tetrad.

5 Deep holes for CA

  • CA has covering radius 2, and its eight deep holes are exactly the vectors in MA = CA + (e3 + e6).The cosets of CA and its six weight-1 translates cover the vectors at distance 0 or 1; the remaining coset consists of the deep holes.Source: §5 · pp. 9–10
  • Figure 1 places CA at the center and its eight orange-circled deep holes outside; the blue tetrad is a [6, 2, 4] code and the green tetrad its coset.The blue circled hexagrams form a diamond, and adding 000111 gives the green square.Source: Figure 1 · p. 10, §5 · p. 11
  • The deep-hole tetrad is {Heaven, Water, Mountain, Thunder}, complementary to the doubled-trigram tetrad in CA.CA contains {Earth, Fire, Wind, Lake}.Source: §5 · p. 11
  • CA alone misses exactly eight deep holes, while adjoining MA gives a [6, 4, 2] code with covering radius 1.The eight codewords, their radius-one balls, and the eight deep holes together reconstruct the 64 hexagrams.Source: §5 · p. 11
  • CA's isoduality is realized by overturning hexagrams, exchanging its nondoubled generator 000111 with 111000.The common self-orthogonal core is fixed by dualization and overturning.Source: §5 · p. 11

6 Codes coherent with a diagonal subspace

The paper extends the doubled-trigram constraint to codes of length 2m, bounding their intersection with the diagonal subspace and defining diagonal-coherent extremal codes. It shows that this constraint does not reduce the maximum dimension attainable at minimum distance at least 3.

6.1 The diagonal subspace and a general bound

  • For a code with minimum distance d, the preimage of its intersection with the diagonal subspace has minimum distance at least ceil(d/2); equality in the intersection bound occurs at the doubled even-weight code.For d ≥ 3, the largest possible intersection is attained exactly when C ∩ Δ_m = ι(E_m).Source: §6.1 · p. 12
  • A code is diagonal-coherent when it contains the doubled even-weight code, and extremal when its dimension is as large as possible.For m = 3, the diagonal subspace is the doubled-trigram subspace.Source: §6.1 · p. 12

6.2 Existence at optimal redundancy

  • For every m ≥ 2, a diagonal-coherent binary [2m, 2m − r, ≥3] code exists, so diagonal coherence entails no loss of redundancy.Here r = ceil(log2(2m + 1)), the optimal redundancy for length 2m codes with minimum distance at least 3.Source: §6.2 · p. 13
  • The existence proof constructs parity-check columns that are pairwise distinct and nonzero while retaining the doubled even-weight subcode.A code contains that subcode and has minimum distance at least 3 precisely when its dual lies in D⊥ and its parity-check columns are pairwise distinct and nonzero.Source: §6.2 · pp. 13–14

6.3 Enumeration

The enumeration section counts extremal diagonal-coherent codes by representing them through subspaces of D⊥. It counts suitable ordered tuples and divides by the number of basis changes, whose action is free.

6.3 Enumeration

  • Extremal codes correspond bijectively to dimension-r subspaces R = C⊥ of D⊥ satisfying the column condition from the existence proof.A basis of R corresponds to a tuple satisfying condition (6).Source: §6.3 · p. 14
  • The count starts from tuples whose projected columns are ordered, distinct, nonzero elements, with two preimages for each projected element.The nonzero vector e may be chosen freely.Source: §6.3 · p. 14
  • Changing the basis of R gives a free GL_r(F2) action on those tuples, so the tuple count is divided by |GL_r(F2)| to obtain the number of codes.The action is free because the vectors in each tuple span F2^r.Source: §6.3 · p. 14

6.4 The minimum distances of the extremal diagonal-coherent codes

At minimal redundancy, extremal diagonal-coherent codes have minimum distance 3, except that distance 4 is possible when m is a power of 2. The section characterizes those distance-4 cases and counts them in several examples.

Lemma 6.8

  • A sum-free subset of F_2^r \ {0} has size at most 2^(r−1), with equality exactly for a hyperplane complement {x : λ(x) = 1}.For a parity-check column set, distance at least 4 is equivalent to being sum-free.Source: §6.4 · p. 15

Theorem 6.9

  • At minimal redundancy, a binary [n, n − r] code has maximum distance 4 when n is a power of 2 and 3 otherwise; the distance-4 case is an extended Hamming code.The extended Hamming code is unique up to coordinate permutation when n = 2^r−1.Source: §6.4 · p. 15

Remark 6.10

  • For n = 10, redundancy 5 permits a [10, 5, 4] code although 10 is not a power of 2; this exception disappears at minimal redundancy.The minimal redundancy for n = 10 is 4.Source: §6.4 · pp. 15–16

Corollary 6.11

  • For extremal diagonal-coherent codes, d = 4 is possible only when m is a power of 2; otherwise d = 3, and not every code has d = 4 when m is a power of 2.When m is a power of 2, at least one extremal code has d = 4; for m = 4, two of ten do, while the other eight have d = 3.Source: §6.4 · p. 16

Corollary 6.12

  • The best-distance extremal codes are both distance-optimal and dimension-optimal among binary codes with the same length and at least that distance.For m = 3, this recovers the distance-optimality of C_A.Source: §6.4 · p. 16

Proposition 6.14

  • The number of distance-4 extremal codes is N4(2) = 1, N4(4) = 2, and N4(8) = 480; these are respectively fewer than the total at m = 4 and 8.The corresponding totals are N(2) = 1, N(4) = 10, and N(8) = 205,920.Source: §6.4 · p. 17

6.5 Normal forms for the extremal diagonal-coherent codes

The paper represents every extremal diagonal-coherent code as the kernel of a matrix determined by distinct nonzero columns and a binary vector. In this form, the minimum distance is controlled by whether the columns form a sum-free set.

Proposition 6.15

  • Every extremal diagonal-coherent code is ker M(A, ε) for some A and ε, and distinct nonzero columns of A give such a code.The construction has dimension 2m − r, minimum distance at least 3, and contains ι(E_m).Source: §6.5 · p. 18
  • The code has d = 4 exactly when the columns of A form a sum-free set; otherwise d = 3, independently of ε.The parity-check column set is the product of the columns of A with F_2, and its sum-freeness is equivalent to that of A's column set.Source: §6.5 · p. 18

Corollary 6.17

  • For every m ≥ 3, the binary-expansion column choice yields an extremal diagonal-coherent [2m, 2m − r, 3] code.In that construction, v1 + v2 + v3 = 0, so the columns are not sum-free.Source: §6.5 · p. 18

Remark 6.18

  • When m = 3, the eight choices of ε yield only C_A and its dual, consistent with N(3) = 2.The all-zero choice gives C_A’s dual as the kernel, while the all-one choice gives C_A.Source: §6.5 · pp. 18–19

6.6 Hulls of the extremal diagonal-coherent codes

The normal form determines the hull of every extremal diagonal-coherent code. Its form depends on the column sum σ and the parity of m, and no such code is linear complementary dual.

6.6 Hulls of the extremal diagonal-coherent codes

  • Every extremal diagonal-coherent code has a nonzero-dimensional hull, so none is linear complementary dual.The hull has dimension at least r−2 ≥ 1.Source: §6.6 · p. 19
  • The hull is ι(Row(A) ∩ E_m) with dimension r−2 when σ ≠ 0; when σ = 0, it is ι(Row(A)) of dimension r−1 for odd m and C⊥ of dimension r for even m.In the even case, C⊥ is contained in C.Source: §6.6 · p. 19
  • When m = 2^(r−1)−1, the hull is the doubled simplex code; for r = 3 this is ι(E_3), and for r = 4 it is the doubled simplex code.In this case Row(A) is the simplex code and σ = 0 with m odd.Source: §6.6 · p. 20
  • At m = 8, codes with d = 3 can have hull dimensions 3 or 5, depending on the column sum σ.The dimension-5 example has C⊥ ⊆ C.Source: §6.6 · p. 20

6.7 Deep holes for an extremal diagonal-coherent code

Every extremal diagonal-coherent code has covering radius 2, with a number of deep-hole cosets determined by m rather than by the code chosen. The single-coset hexagram picture occurs only at particular lengths.

6.7 Deep holes for an extremal diagonal-coherent code

  • Each code has covering radius 2 and exactly 2^r−1−2m deep-hole cosets, each containing 2^(2m−r) words.The number of deep-hole cosets is independent of the choice of extremal code.Source: §6.7 · p. 20
  • The parity-check syndromes have coset-leader weights 0 once, 1 for exactly 2m syndromes, and 2 for all remaining syndromes.The proof shows every syndrome outside the zero syndrome and parity-check columns is a sum of two columns.Source: §6.7 · p. 20
  • The deep holes form a single coset exactly when m = 2^(r−1)−1, including the hexagram case m = 3.For m = 3, this yields the eight deep holes in the hexagram decomposition; the single-coset pattern is not typical for general m.Source: §6.7 · pp. 20–21
  • The table reports parameters, N(m), and deep-hole-coset counts for the extremal codes, with stars marking optimal minimum distances.Each deep-hole coset has 2^(2m−r) elements.Source: Table 1 · p. 21

6.8 Specialisations

At the special lengths m = 2^(r−1)−1, extremal diagonal-coherent codes are shortened Hamming codes, and the hexagram case has exactly two such codes. Other specializations distinguish when diagonal coherence is restrictive and when minimum distance 4 or self-duality occurs.

6.8 Specialisations

  • When m = 2^(r−1)−1, every extremal code is a shortened Hamming code with parameters [2^r−2, 2^r−2−r, 3] and minimum distance exactly 3.Its parity-check matrix is obtained by deleting one coordinate from a Hamming code of length 2^r−1.Source: §6.8 · p. 21
  • At these lengths every optimal [2^r−2, 2^r−2−r, 3] code is equivalent to a diagonal-coherent code; for smaller m, optimal codes not equivalent to any such code exist.For m = 6, omitting three linearly independent columns gives an optimal code that is not equivalent to a diagonal-coherent code.Source: §6.8 · p. 22
  • For m = 3, the two extremal codes are exchanged by coordinate reversal τ.This recovers the shortened-Hamming-code pair for the hexagram case.Source: §6.8 · p. 22
  • Minimum distance 4 occurs exactly when m is a power of 2, and those codes are extended Hamming codes.The family includes m = 2, 4, 8, 16, …Source: §6.8 · p. 22
  • An extremal code has dimension half its length exactly when m = r = ⌈log2(2m + 1)⌉; self-duality is possible only when both this and distance 4 hold.For m = 4, exactly two of the ten extremal codes are self-dual extended Hamming codes.Source: §6.8 · p. 22
  • The diagonal-coherent part of the doubled-trigram subspace is precisely its even-weight half; for m = 3 this is the four doubled trigrams Earth, Fire, Wind, and Lake.This is the maximum diagonal intersection described for a single-error-correcting code.Source: §6.8 · p. 22

6.9 Open problems

The paper settles the minimum-distance question for extremal diagonal-coherent codes at the single-error-correcting redundancy, while posing whether analogous optimality holds for larger minimum distances. It also asks for corresponding extensions to repeated diagonals and nonbinary codes.

6.9 Open problems

  • At redundancy r0(2m), extremal diagonal-coherent codes have minimum distance 3 or 4, with 4 occurring only for some codes when m is a power of 2; no distance at least 5 occurs.The remaining question concerns the optimal dimension for each other minimum distance.Source: §6.9 · p. 22
  • The open problem asks whether, for every m and d, a d-diagonal-coherent code can attain the largest dimension κ(2m, d) among length-2m binary codes with minimum distance at least d.The paper answers this affirmatively for d = 3.Source: §6.9 · p. 23
  • For d = 5 and m = 7, a computer search finds a length-14 code containing ι(H_3) with dimension κ(14, 5) = 6; whether this no-loss behavior persists is open.Here H_3 is the Hamming code and ι(H_3) has dimension 4 with weights 6 and 8.Source: §6.9 · p. 23
  • The paper also asks for analogues over F_q and for codes coherent with s-fold repeated diagonals.The binary sum-free argument does not immediately extend to F_q.Source: §6.9 · p. 23
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