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Canonical Hamming geometry from the Hecke singularity at level 163

Paper IV in the singular Hecke node series

Richard Hoekstra · March 2026 · PDF

The construction

The ternary Hamming code [13,10,3]₃ is the unique perfect single-error-correcting code over F₃ of length 13. Its parity-check geometry is the projective plane PG(2,F₃) — 13 points, 13 lines, each line containing 4 points, each pair of points on exactly one line.

I show that this geometry arises canonically from the Hecke algebra at level 163. The construction requires no choices:

S₂(Γ₀(163)) → mod 3 → ker(T₂²) = Q, dim 3 P(Q): 13 points → incidence = 2-(13,4,1) design unique design → PG(2,F₃) → [13,10,3]₃ Hamming

Theorem (Canonical Hamming geometry) The incidence structure on P(Q) — points and lines of a 3-dimensional F₃-projective plane — is a 2-(13,4,1) balanced incomplete block design. By the uniqueness theorem for projective planes of order 3, this is PG(2,F₃). The parity-check kernel is the [13,10,3]₃ ternary Hamming code.

Each step is either functorial (mod-3 reduction, primary decomposition, projectivisation) or forced by a uniqueness theorem (there is exactly one 2-(13,4,1) design up to isomorphism).

The Hamming code was not constructed. It was found.

Hecke symmetry of the plane

The syndrome algebra A_H ≅ F₃ × F₃[ε]/(ε²) acts on PG(2,F₃) through its projective unit group C₆ ⊂ PGL(3,F₃), with orbit decomposition:

1 + 1 + 2 + 3 + 6 = 13

The cycle types of the invertible Heegner operators: T₇ has order 3 with cycles 3+3+3+1+1+1+1. T₄₃ and T₆₇ have order 6 with cycles 6+3+2+1+1. T₁₆₃ has order 2 with cycles 2+2+2+2+1+1+1+1+1. The two fixed points are the eigenspaces of the idempotent E = T₃.

The quantum code

The parity-check matrix H of the canonical Hamming code satisfies a remarkable property:

Result HH^T = 0 over F₃. The row space C^⊥ is self-orthogonal with C^⊥ ⊂ C. This yields a CSS quantum stabilizer code with parameters:

[[13, 7, 3]]₃

A quantum error correcting code on 13 qutrits, encoding 7 logical qutrits, correcting 1 error.

The Hecke symmetry group C₆ acts as code automorphisms of the CSS pair (C, C^⊥), preserving the quantum code structure. The self-orthogonality is not a generic property of Hamming codes — it is a specific property of this particular parity-check representation, produced canonically by the Hecke construction.

Census of singular Hecke factors

A systematic scan of all prime levels N ≤ 1000 classifies the 3-adic Hecke factors:

Type	N ≤ 500	N ≤ 1000
Smooth	56	95
Cuspidal	20	39
Split-nodal	8	17
Non-split-nodal	7	13

The split-nodal levels up to 500 and their syndrome dimensions:

N	127	163	199	307	337	449	487	499
dim Q	3	3	4	2	3	5	7	6

The PG(2,F₃) Hamming geometry arises exactly when dim Q = 3. Among the split-nodal levels up to 500, this occurs at N ∈ {127, 163, 337}. Of these, only 163 is a class-number-one Heegner prime with the full seven-operator syndrome algebra.

Higher-dimensional summands produce higher-order projective geometries: P(Q₁₉₉) has 40 points (dim 4), P(Q₄₄₉) has 121 points (dim 5). The [13,10,3]₃ Hamming code is specific to dim Q = 3.

The complete chain

From one relation in the Hecke algebra to a quantum error correcting code:

η² = 3η → O_sing ≅ Z₃ × Z₃[η]/(η²−3η) → A_H ≅ F₃ × F₃[ε]/(ε²) → P(Q) = PG(2,F₃), 13 points → [13,10,3]₃ Hamming code → HHᵀ = 0, C⊥ ⊂ C → [[13,7,3]]₃ quantum stabilizer → D_sg: Σ² = Id, K₀ ≅ Z/2Z

Every step is a computation. Every arrow is canonical or forced by uniqueness. The syndrome algebra is simultaneously the residue of the 3-adic node, the algebra acting on the syndrome space of the Hamming code, and the source of a C₆ symmetry of the Hamming plane.

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