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The quantum stabilizer code from the Hecke singularity

Paper XII in the singular Hecke node series

Richard Hoekstra · 2026

From modular forms to code

The construction proceeds through four canonical steps, each forced by the Hecke algebra at level 163.

Step 1: The syndrome space

Let Q = ker(T₂²) in S₂(Γ₀(163)) ⊗ F₃. This is the kernel of the square of the Hecke operator T₂ acting modulo 3 on the 13-dimensional space of weight-2 cusp forms. The syndrome space Q is 3-dimensional over F₃ — this is the q_dim = 3 that the census (Paper VI) identifies as the signature of a split node at level 163.

Step 2: The projective points

The projective space P(Q) over F₃ has |P(F₃³)| = (3³ − 1)/(3 − 1) = 13 points. These 13 points are the columns of the parity matrix H ∈ M(3×13, F₃).

Step 3: Self-orthogonality

The matrix H satisfies HH^T = 0 over F₃. This means the rows of H are mutually orthogonal and self-orthogonal. The row space C^⊥ = rowspan(H) is contained in C = ker(H). This self-orthogonality is the condition needed for the CSS construction.

Step 4: The codes

Code	Parameters	Description
C = ker(H)	[13, 10, 3]₃	Ternary Hamming code
C^⊥ = rowspan(H)	[13, 3, 9]₃	Ternary simplex code (dual of Hamming)

The Hamming code has minimum distance d(C) = 3. Its dual, the simplex code, has minimum distance d(C^⊥) = 3³⁻¹ = 9. Since C^⊥ ⊆ C (self-orthogonality), the CSS construction applies.

The quantum code

CSS stabilizer code The CSS construction on (C, C^⊥) yields a quantum stabilizer code with parameters [[13, 7, 3]]₃.

The parameters are determined by:

Quantity	Value	Derivation
n (physical qutrits)	13	Length of both codes
k (logical qutrits)	7	dim C − dim C^⊥ = 10 − 3
d (distance)	3	min weight of C \ C^⊥

The CSS distance is the minimum weight of a codeword in C that is not in C^⊥. Since C has minimum distance 3 and C^⊥ has minimum distance 9, there exist weight-3 codewords in C that are not in C^⊥. So d = 3.

Hecke automorphisms

The Hecke operators act on the 13 projective coordinates of P(Q). The action splits into two classes:

Generators	Primes p	Action on Q
Invertible	7, 43, 67, 163	Monomial matrices S_p = diag(scales) · P (scaling + permutation)
Non-invertible	3, 11, 19	det = 0 on Q (singular action)

For each invertible generator p, H · S_p has the same row space as H. Since S_p is monomial (invertible), ker(H · S_p) = ker(H), so S_p preserves both C and C^⊥. The Hecke-induced group acts by code automorphisms of the CSS pair (C, C^⊥).

No choices at any step. The syndrome space Q is determined by the Hecke algebra. The 13 projective points are determined by Q. The self-orthogonality HH^T = 0 is a property of the Hecke structure, not a design choice. The quantum code [[13, 7, 3]]₃ is as canonical as the level-163 singularity itself.

This extends Paper IV (Hamming bridge) to the quantum setting. The classical Hamming code [13,10,3]₃ corrects single errors. The CSS code [[13,7,3]]₃ corrects single qutrit errors in a quantum register. Both arise from the same Hecke singularity.