convert_hamming-bridge

The ternary Hamming code

\mathcal{H}_3 = [13,10,3]_3

is the unique perfect single-error-correcting code over

\mathbf{F}_3

of length

13

. Its parity-check matrix has

3

rows and

13

columns, one for each point of the projective plane

\mathrm{PG}(2,\mathbf{F}_3)

. The code, the plane, and the

2\text{-}(13,4,1)

block design are three faces of the same combinatorial object.

In this note we show that this object arises canonically from the Hecke algebra at level

N=163

, the largest class-number-one Heegner prime.

No choice is made at any step. Step (iii) is canonical because the

x^2

-primary component is determined by the Hecke action, not by a basis. Step (vi) is forced by uniqueness: there is exactly one

2\text{-}(13,4,1)

design up to isomorphism .

Main results

The canonical summand and its projective plane

Let

V_\mathbf{Z}\subset S_2(\Gamma_0(163))

be the integral modular-symbols lattice, and set

V_{\mathbf{F}_3} = V_\mathbf{Z}/ 3V_\mathbf{Z}

The operator $T_2$ on $V_{\mathbf{F}_3}$ has characteristic polynomial $\chi_{T_2}(x) \equiv x^3(x^4+2x^3+2)(x^6+x^4+x^3+x+1) \pmod{3}.$ The three factors are pairwise coprime, giving a Hecke-stable decomposition $V_{\mathbf{F}_3} = Q \oplus W_4 \oplus W_6,$ where $Q = \operatorname{ker}(T_2^2)$ has dimension $3$ , $W_4$ has dimension $4$ , and $W_6$ has dimension $6$ .

The Hecke projective plane at level $163$ is $\mathbf{P}(Q) = \{\text{one-dimensional $\mathbf{F}_3$-subspaces of $Q$}\}.$ Since $\dim Q = 3$ , the set $\mathbf{P}(Q)$ has $(3^3-1)/(3-1) = 13$ points.

The canonical Hamming geometry

The incidence structure $(\mathbf{P}(Q), \mathcal{L})$ , where $\mathcal{L}$ is the set of lines, satisfies:

there are $13$ points and $13$ lines;
each line contains exactly $4$ points;
each point lies on exactly $4$ lines;
any two distinct points determine exactly one line.

This is a $2\text{-}(13,4,1)$ balanced incomplete block design. By the uniqueness theorem for projective planes of order $3$ , $(\mathbf{P}(Q), \mathcal{L}) \cong \mathrm{PG}(2,\mathbf{F}_3).$

Proof. Properties (a)–(d) are the standard properties of $\mathbf{P}^2$ over any field with $3$ elements. Since $Q$ is a $3$ -dimensional $\mathbf{F}_3$ -vector space, $\mathbf{P}(Q)$ is abstractly isomorphic to $\mathbf{P}^2(\mathbf{F}_3)$ , and the incidence of points and lines is that of the standard projective plane. The $2\text{-}(13,4,1)$ design is unique up to isomorphism , so the isomorphism class of the incidence structure is independent of any basis choice. ◻

Let $E = \mathbf{F}_3^{\mathbf{P}(Q)}$ be the free module on the $13$ points. The parity-check map $H_Q : E \to Q$ sending each basis vector to a nonzero representative of the corresponding projective point has $\mathrm{rank}\; H_Q = 3$ , and $\mathcal{C}_Q := \operatorname{ker}(H_Q) \subset E$ is a $[13,10,3]_3$ ternary code. This is the ternary Hamming code, determined up to $\operatorname{GL}(Q)$ -equivalence (i.e., column permutation and scaling).

Proof. After choosing a basis of $Q$ , the $13$ projective points give $13$ distinct nonzero column vectors in $\mathbf{F}_3^3$ , forming a standard Hamming parity-check matrix . The resulting code has parameters $[13,10,3]_3$ . Changing the basis of $Q$ acts by $\operatorname{GL}(3,\mathbf{F}_3)$ on all columns simultaneously and therefore preserves the code up to equivalence. ◻

The canonicity chain is: $S_2(\Gamma_0(163)) \;\xrightarrow{\bmod\; 3}\; V_{\mathbf{F}_3} \;\xrightarrow{\operatorname{ker}(T_2^2)}\; Q \;\xrightarrow{\mathbf{P}}\; \mathbf{P}(Q) \;\xrightarrow{\text{incidence}}\; \mathrm{PG}(2,\mathbf{F}_3) \;\xrightarrow{\operatorname{ker}(H_Q)}\; [13,10,3]_3.$ Each arrow is either functorial (mod- $3$ reduction, primary decomposition, projectivization, parity-check kernel) or forced by a uniqueness theorem (the $2\text{-}(13,4,1)$ design is unique).

The Hecke symmetry of the plane

The seven Heegner operators

T_d

for

d \in \{3,7,11,19,43,67,163\}

act on

Q

through the syndrome algebra

A_H = \operatorname{Span}_{\mathbf{F}_3}\{I, E, N\} \cong \mathbf{F}_3 \times \mathbf{F}_3[\varepsilon]/(\varepsilon^2),

with

E^2 = E

N^2 = EN = NE = 0

(see ). The invertible elements of

A_H

act on

\mathbf{P}(Q)

by projectivization.

The projective unit group $A_H^\times / \mathbf{F}_3^\times \cong C_6$ acts on $\mathbf{P}(Q)$ with orbit decomposition $1 + 1 + 2 + 3 + 6 = 13.$ The cycle types of the invertible Heegner operators are:

Operator	Projective order	Cycle type on $13$ points
$T_7$	$3$	$3{+}3{+}3{+}1{+}1{+}1{+}1$
$T_{43}$	$6$	$6{+}3{+}2{+}1{+}1$
$T_{67}$	$6$	$6{+}3{+}2{+}1{+}1$
$T_{163}$	$2$	$2{+}2{+}2{+}2{+}1{+}1{+}1{+}1{+}1$

These four operators generate the full projective unit group. The two fixed points are the eigenspaces of the idempotent $E = T_3|_Q$ .

Proof. The unit group computation follows from the algebra structure: $|A_H^\times| = |\mathbf{F}_3^\times| \cdot |(\mathbf{F}_3[\varepsilon]/(\varepsilon^2))^\times| = 2 \cdot 6 = 12$ , so $|A_H^\times / \mathbf{F}_3^\times| = 6$ . The cycle types and orbit sizes are computed by explicit matrix action on the $13$ points of $\mathbf{P}(Q)$ , using the restricted Hecke matrices from the appendix of . The two fixed points correspond to the $E$ -eigenspace $\langle e_1 \rangle$ (where $E$ acts as $1$ ) and the kernel of $E$ restricted to the pair block. ◻

The group $C_6$ embeds in $\operatorname{PGL}(3,\mathbf{F}_3)$ , which is the full automorphism group of $\mathrm{PG}(2,\mathbf{F}_3)$ . Since $|\operatorname{PGL}(3,\mathbf{F}_3)| = 11{,}232$ , the Hecke symmetry is a small but arithmetically distinguished subgroup. The orbit decomposition $1+1+2+3+6$ is a partition of the $13$ code positions into Hecke-equivalent classes.

The syndrome algebra as decoder

The classical role of the syndrome in coding theory is error detection: given a received word

r \in \mathbf{F}_3^{13}

, the syndrome is

s = H_Q(r) \in Q

. If

s = 0

, no error is detected. If

s \ne 0

, the syndrome identifies the error position as the projective point

[s] \in \mathbf{P}(Q)

The Heegner syndrome algebra $A_H$ acts on the syndrome space $Q$ . In particular:

The idempotent $E = T_3|_Q$ splits the syndrome into a semisimple component (the $E$ -eigenspace) and a pair component (the $\operatorname{ker}(E)$ -subspace).
The nilpotent $N = T_{11}|_Q$ acts within the pair component. It maps one syndrome to a neighboring syndrome within the nilpotent fiber, without changing the semisimple part.
The collision $T_{43}|_Q = T_{67}|_Q$ means that the syndromes of these two Heegner operators are indistinguishable at the residue level. They separate only after $3$ -adic lifting.

The name “syndrome algebra” is thus doubly justified: it is both the residue algebra of the singular $3$ -adic Hecke node and the algebra that acts on the syndrome space of the canonically associated Hamming code. The two meanings coincide because the $3$ -dimensional space $Q$ plays both roles simultaneously: it is the mod- $3$ Hecke summand and the syndrome space of the code.

Census of singular

3

-adic Hecke factors

We apply the same method — mod-

3

primary decomposition of

T_2

, identification of

3

-dimensional summands, and

3

-adic lifting — to all prime levels

N \le 500

Among the $91$ prime levels $N \le 500$ (respectively $164$ prime levels $N \le 1000$ ):

$56$ (resp. $95$ ) levels have a smooth $3$ -adic Hecke algebra;
$20$ (resp. $39$ ) levels have a cuspidal singularity;
$8$ (resp. $17$ ) levels have a split-nodal singularity;
$7$ (resp. $13$ ) levels have a non-split-nodal singularity.

The split-nodal levels up to $1000$ and their syndrome dimensions are:

$N$	127	163	199	307	337	449	487	499	$\cdots$
$\dim Q_N$	3	3	4	2	3	5	7	6

Continuing to $1000$ : $N \in \{541, 577, 631, 773, 811, 829, 857, 883, 919\}$ .

The non-split-nodal levels up to $1000$ are: $N \in \{71, 269, 271, 359, 421, 439, 443, 509, 523, 571, 601, 947, 997\}$ .

Proof. Systematic computation in SageMath. For each prime $N$ , we compute $T_2$ on $V_{\mathbf{F}_3}$ , extract $Q_N = \operatorname{ker}(T_2^2)$ , and classify the $3$ -adic local factor by testing for a mod- $3$ idempotent and its $\mathbf{Z}_3$ -lift. Full data and scripts are available in the accompanying repository. ◻

The canonical Hamming geometry $\mathrm{PG}(2,\mathbf{F}_3)$ arises exactly when $\dim Q_N = 3$ . Among the split-nodal levels $\le 500$ , this occurs at $N \in \{127, 163, 337\}$ . Of these, only $N = 163$ is a class-number-one Heegner prime.

Among the four class-number-one prime levels $N \in \{19, 43, 67, 163\}$ , the dimensions of $Q_N$ are $1, \quad 0, \quad 0, \quad 3,$ respectively. Level $163$ is the unique class-number-one prime with a split-nodal singularity, a $3$ -dimensional syndrome summand, and a canonical $\mathrm{PG}(2,\mathbf{F}_3)$ Hamming geometry.

Higher-dimensional syndrome summands ( $\dim Q_N > 3$ ) produce higher-order projective geometries: $\mathbf{P}(Q_{199})$ has $40$ points ( $\dim Q_{199} = 4$ ), $\mathbf{P}(Q_{449})$ has $121$ points ( $\dim Q_{449} = 5$ ), and so on. The Hamming codes at these levels have longer block lengths and different parameters. The $[13,10,3]_3$ ternary Hamming code is specific to $\dim Q_N = 3$ .

The classification problem — characterize which prime levels $N$ produce singular $3$ -adic Hecke factors, determine their local type and syndrome dimension — is a natural next step. The data suggest that split-nodal singularities become more frequent and higher-dimensional as $N$ grows. We leave the asymptotic analysis to future work.

Summary

The singular

3

-adic Hecke factor at level

163

produces a canonical chain of objects:

\begin{aligned} &\text{Hecke algebra } \mathbb{T}_{163} \;\longrightarrow\; \text{mod-}3\text{ summand } Q \\ &\;\longrightarrow\; \text{projective plane } \mathbf{P}(Q) \cong \mathrm{PG}(2,\mathbf{F}_3) \\ &\;\longrightarrow\; \text{Hamming code } [13,10,3]_3. \end{aligned}

The syndrome algebra

A_H \cong \mathbf{F}_3 \times \mathbf{F}_3[\varepsilon]/(\varepsilon^2)

is simultaneously:

Among all prime levels

\le 1000

, seventeen exhibit split-nodal

3

-adic singularities. Of these, only three (

127

163

337

) have

\dim Q_N = 3

and therefore carry the

\mathrm{PG}(2,\mathbf{F}_3)

Hamming geometry. Of those three, only

163

is a class-number-one Heegner prime with the full seven-operator syndrome algebra.

Computational verification. All results were computed in SageMath. The census data and verification scripts are available in the accompanying repository.

M. Hall Jr., Uniqueness of the projective plane with $57$ points, Proc. Amer. Math. Soc. 4 (1953), 912–916. Note: The uniqueness of the projective plane of order $3$ ( $13$ points) is classical; see also R. H. Bruck and H. J. Ryser, The nonexistence of certain finite projective planes, Canad. J. Math. 1 (1949), 88–93.

J. H. van Lint, Introduction to Coding Theory, third ed., Springer, 1999.

R. A. Hoekstra, The Heegner syndrome algebra on the canonical mod- $3$ Hecke quotient at level $163$ , preprint, 2026.

R. A. Hoekstra, The singular $3$ -adic Hecke node at level $163$ : presentation, ideal theory, and module classification, preprint, 2026.

The Sage Developers, SageMath, the Sage Mathematics Software System, https://www.sagemath.org.

Introduction