CSS Codes from Dn Root Lattices: An Infinite Family with Rate 1

CSS Codes from D

Root Lattices:

An Inﬁnite Family with Rate 1 − 4/K

Raghu Kulkarni

IDrive Inc, SSMTheory and Quantum Computing Group

raghu@idrive.com

Preprint, 2026

Abstract

An old question in sphere-packing yields an explicit construction in quantum er-

ror correction. The face-centered cubic lattice—the densest sphere packing in three

dimensions—generates a CSS code with encoding rate 2

3. We demonstrate that this

construction extends to every

root lattice (

n ≥

3), yielding codes with parameters

[[

(

n−

(

n−

2)(

2 + 2

3]] and an encoding rate converging to 1

−

L → ∞

, where

= 2

(

n −

1) is the coordination number. CSS validity, the rank formula,

the asymptotic rate, and the exact distance

= 3 are proved analytically for all

n ≥

3 by

exploiting an exact graph isomorphism between the primal vertex lattice and the dual

void lattice. As

grows along the densest-packing hierarchy, the topologically protected

fraction approaches unity, establishing the ﬁrst proven inﬁnite family of topological CSS

codes whose rate converges to 1 at ﬁxed distance.

Contents

1 Introduction 1

2 The D

Root Lattice 2

3 CSS Code Construction 3

3.1 Qubits on edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.2 Z-stabilisers on vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.3 X-stabilisers on voids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4 Main Theorems 4

5 Distance 5

6 The Rate Formula 6

7 Comparison with Known CSS Codes 7

8 Open Problems 7

1 Introduction

The face-centered cubic (FCC) arrangement of spheres was conjectured by Kepler in 1611 to

be the densest packing of equal spheres in three dimensions. Gauss proved this for lattice

packings in 1831, and the full conjecture was resolved by Hales’s computer-assisted proof

in 2005 [

]. In recent work [

], it was shown that this same FCC lattice, interpreted as a

qubit graph, yields a CSS quantum code [

] with an encoding rate of 2

3—higher than

any topological code previously known at the exact same distance d = 3.

The present paper generalizes this result. The construction strictly relies on the property

that the FCC lattice is an even-parity sublattice of

, a structural characteristic that holds

for the

root lattice in every dimension. The

root lattice has

= 2

(

n −

1) nearest

neighbours; the CSS code built from it has a rate of 1

−

. For

= 3, this evaluates to

2/3; for n = 4, it evaluates to 5/6. As n → ∞, the rate strictly approaches 1.

While the asymptotic rate was expected from the graph density, the scaling of the code

distance requires formal evaluation. Initial computational sweeps of the

lattice at

= 4

indicated that all 4096 distinct triangles in the

graph operate as non-trivial logical

operators. We formalize this observation by providing a complete analytic proof that

= 3

exactly for all

n ≥

3. The proof relies on establishing an exact graph isomorphism between

the primal

graph and the adjacency graph of its topological voids, which maps the logical

operators into the strict cut space of the lattice.

Context. Topological CSS codes have been central to quantum fault-tolerance since Kitaev’s

toric code [

], which achieves

= 2 regardless of system size. Surface codes [

] simpliﬁed the

physical geometry but maintained an asymptotic rate approaching 0. Hypergraph product

codes [

] were the ﬁrst to achieve constant positive rate at constant distance, and recent

QLDPC constructions [

] achieve both constant rate and linear distance. The

family

occupies a distinct parameter regime: a rate approaching 1, small ﬁxed distance, and a purely

geometric construction.

Organisation. Section 2 reviews the structural properties of the

lattice. Section 3

formalizes the CSS code deﬁnitions. Section 4 establishes CSS validity and proves the rank

formulas. Section 5 provides the rigorous algebraic proofs establishing the exact distance

= 3. Section 6 interprets the asymptotic rate, and Section 7 compares the family with

established literature.

2 The D

Root Lattice

Deﬁnition 2.1. The D

root lattice (n ≥ 2) is deﬁned as:



x ∈ Z



+ ··· + x

≡ 0 (mod 2)



This corresponds to the

-dimensional checkerboard consisting of all sites with an even

coordinate sum. The nearest-neighbour vectors (roots) are:



±e

± e



1 ≤ i < j ≤ n



, (1)

where

is the

-th standard basis vector. Each root has a squared Euclidean length

|r|

= 2,

yielding exactly K = 4





= 2n(n − 1) nearest neighbours.

Notable instances include

∼

FCC with

= 12, proved to be the densest 3D

packing [

];

with

= 24, proved to be the densest 4D lattice packing by Korkine and

Zolotareﬀ [

]; and

⊂ E

with

= 112. (Note that

, with

= 240, is the strictly

densest 8D packing [11]).

On the torus (Z/LZ)

with L even and L ≥ 4, the coordinate sites partition into:

vertices: n

= L

/2 even-parity sites,

voids: n

= L

/2 odd-parity sites.

Each void

is geometrically surrounded by exactly 2

vertices deﬁned by the set

{o ± e

d = 1, . . . , n}.

Lemma 2.2 (Vertex- and void-transitivity). The even-parity sites and odd-parity sites each

form a single, distinct orbit under the translation group of the lattice.

Proof.

Translation by any vector

v ∈ D

(even parity) maps even-parity sites to even-parity

sites and odd-parity sites to odd-parity sites. The translation group is transitive on

establishing the vertex-transitivity of the

graph. For odd-parity sites, translation by any

odd-parity vector (e.g.,

) maps even sets to odd sets. Composing two such translations

maps odd to odd, conﬁrming the orbit of any void under the full translation group equals all

voids. Thus, the void network is strictly vertex-transitive.

3 CSS Code Construction

3.1 Qubits on edges

A single physical qubit is placed on each edge of the

graph. Since each vertex has

neighbours and every edge is shared between two vertices, the total physical qubit count is:

= n(n − 1)

. (2)

Figure 1: The CSS construction on a 2D slice of

(FCC). Left: a Z-stabiliser on a vertex

(blue, weight

). Right: an X-stabiliser on a void (red square), acting on the

non-antipodal

edges among the 2

surrounding vertices (orange). CSS validity is maintained because each

vertex-void pair shares an even number 2(n − 1) of edges.

3.2 Z-stabilisers on vertices

For each vertex

v ∈ D

, the Z-stabiliser operator applies a Pauli

to all

incident edges.

The parity-check matrix

is exactly the unoriented vertex–edge incidence matrix of the

graph, where every row has uniform weight K.

3.3 X-stabilisers on voids

For a void

, its 2

surrounding vertices are deﬁned as

s e

= 1

, . . . , n, s

}

. We

deﬁne the antipodal pair along direction

as the two vertices

, o−e

}

. The X-stabiliser

for void

applies a Pauli

to all non-antipodal edges formed among its surrounding vertices.

A non-antipodal pair of vertices (

, o

) with

i 

has a displacement vector

− s

∈ {±e

± e

}

, which identically matches the deﬁnition of a valid

root. The

number of such non-antipodal pairs is





− n

= 2

(

n −

1) =

. Therefore, both Z- and

X-stabilisers possess a uniform weight of K.

4 Main Theorems

Theorem 4.1 (CSS validity). H

= 0 over F

for all n ≥ 3 and even L ≥ 4.

Proof.

For any void

and vertex

, the matrix product evaluates the shared support:

)[o, v] = |edges(o) ∩ star(v)| (mod 2).

Case 1 :

v /∈ surrounding

(

). No X-stabiliser edge originating from

utilizes

as an endpoint.

The intersection is empty, contributing 0.

Case 2 :

∈ surrounding

(

). The vertex

connects via X-stabiliser edges to all

n −

2 non-antipodal surrounding vertices (every surrounding vertex except its antipodal

partner

o − s

). Because 2(

n −

≡

0 (

mod

2) for all integers

n ≥

2, the intersection size

is strictly even, contributing 0.

Theorem 4.2 (Edge coverage). Every edge of

is contained in exactly 2 void X-stabilisers.

Proof.

Consider an edge (

u, v

) deﬁned by the root

v −u

− e

. A void

contains (

u, v

)

in its X-stabiliser if and only if both

and

are non-antipodal surrounding vertices of

Writing

and

subject to the constraint

− s

− e

yields

exactly two geometric solutions:

u − e

and

. Because

is an even-parity

site and the basis vectors contain a single non-zero coordinate, both

and

are odd-parity

sites (valid voids). By symmetry, this applies to all root conﬁgurations. Hence, every edge is

covered by exactly 2 voids.

Corollary 4.3.

)

o,·

= 0 over F

. Hence rk

) ≤ n

− 1.

Theorem 4.4 (Rank formula). For even

L ≥

4 and

n ≥

(

) =

−

1 and

) = n

− 1.

Proof.

For

: The graph Laplacian ∆ =

has momentum-space eigenvalues

(k) =

K −

(k), where

(k) = 4

i<j

cos k

. At k = 0,

(0) = 4





, yielding

(0) = 0.

For k



= 0, the

graph on the torus is connected, leaving exactly one zero eigenvalue over

R. Over F

, the incidence matrix of any connected graph has rank n

− 1 [12].

For

: Corollary 4.3 restricts the rank to

≤ n

−

1. To establish the lower bound, we

deﬁne the void adjacency graph

: vertices are voids, and two voids are adjacent if they

share at least one X-stabiliser edge. By Lemma 2.2, the odd-parity sites form a single orbit,

making

vertex-transitive. For

n ≥

3 and

L ≥

4, Theorem 4.2 dictates that every void

shares its

edges with exactly

other voids. Since

= 2

(

n −

≥

12,

is a non-empty,

connected vertex-transitive graph. Therefore,

, acting as the incidence matrix of

, has

rank n

− 1.

Theorem 4.5 (Encoding parameters). The D

CSS code satisﬁes:

k =

(n − 2)(n + 1)

+ 2,

L→∞

−−−−→ 1 −

Proof. Applying Theorem 4.4 to the stabilizer rank-nullity equation:

k = n

− (n

− 1) − (n

− 1) = n(n − 1)

− 2



− 1



−n−2

+ 2 =

(n−2)(n+1)

+ 2.

The asymptotic rate evaluates to

k/n

→

(

n −

2)(

+ 1)

[

(

n −

1)] = 1

−

[

(

n −

1)] =

1 − 4/K.

Table 1 details the speciﬁc parameters across dimensional hierarchies.

Table 1: Analytic parameters for the

CSS code family. The exact distance

= 3 holds

analytically for all n ≥ 3, establishing the rate asymptote 1 − 4/K.

Code n

k d Rate Stab. wt.

, L = 4 192 130 3 67.71% 12

, L = 6 648 434 3 66.98% 12

, L = 4 1536 1282 3 83.46% 24

, L = 6 7776 6482 3 83.36% 24

, L = 4 10240 9218 3 90.02% 40

, large L n(n−1)L

(n−2)(n+1)

+ 2 3 1 − 4/K 2n(n−1)

5 Distance

Because Z-stabilizers are rows of the graph incidence matrix

, any vector

v ∈ ker H

deﬁnes an X-operator. Symmetrically, vectors in

ker H

deﬁne Z-operators. The distances

and

correspond to the minimum non-zero weight of vectors in

ker H

\ rowspace

(

)

and ker H

\ rowspace(H

), respectively.

Theorem 5.1 (Lower bounds:

≥

3 and

≥

3). For all

n ≥

3 and even

L ≥

4, the code

graph possesses no parallel edges.

Proof. d

≥

3: A weight-1 operator is a single edge with two distinct endpoints, making



= 0. A weight-2 operator

is in

ker H

if and only if edges

and

are parallel

(sharing identical endpoints). Two distinct

roots



yield the same translated

neighbor

≡ v

(

mod L

) only if

≡ r

(

mod L

). Since the Euclidean distance

− r

| ≤

√

8 <

≤ L

, this explicitly forces

, representing the same edge. Thus, no

parallel edges exist, giving d

≥ 3.

≥

3: By Theorem 4.2, every edge is covered by exactly 2 X-stabilisers, forbidding

weight-1 operators in

ker H

. A weight-2 operator is in

ker H

if and only if the two void-

coverage sets for edges

and

are identical. This would require two distinct edges to span the

exact same pair of voids, generating a multi-edge in the void adjacency graph

. However,

as proven in Theorem 5.2 below,

is exactly isomorphic to the

graph, inheriting its

lack of parallel edges. Thus, d

≥ 3.

Theorem 5.2 (Void Graph Isomorphism). The void adjacency graph

is strictly isomorphic

to the D

primal graph.

Proof.

The vertices of

are the odd-parity sites

o ∈ Z

. Two voids

and

share an

edge in

if they share an X-stabilizer edge. By the geometry derived in Theorem 4.2,

this overlap strictly requires their diﬀerence to be a root vector:

− o

±e

± e

∈ N

Consequently, the adjacency condition for

is mathematically identical to the adjacency

condition of the

graph. Because the odd-parity sites are merely a global translation of

the even-parity sites by e

, the graphs are exactly isomorphic: G

∼

Theorem 5.3 (Exact Distance:

= 3 and

= 3). For all

n ≥

3, every graphical triangle

forms a non-trivial logical operator.

Proof.

The matrix

is the unoriented incidence matrix of the

graph, while

acts

identically as the unoriented incidence matrix of

. Over

, the kernel of an incidence

matrix deﬁnes the graph’s cycle space, and the row space deﬁnes its cut space.

Because both the

graph and

are connected, vertex-transitive,

-regular simple

graphs, classic graph theory establishes that their edge connectivity is exactly their degree

[

]. Therefore, the minimum size of any non-empty edge cut in either graph is exactly

. This dictates that the minimum non-zero weight of any vector in

rowspace

(

) or

rowspace(H

) is strictly K = 2n(n − 1).

A triangle in the

graph constitutes a 3-cycle, represented by an indicator vector

∈ ker H

. For all

n ≥

3, the cut space bound requires

K ≥

12. Because the triangle

has weight 3

≤ K

, it is mathematically isolated from

rowspace

(

). Therefore,

is a

non-trivial X-logical operator, yielding d

≤ 3.

By symmetry via the isomorphism in Theorem 5.2, a triangle in the void graph

deﬁnes

a vector

∈ ker H

. Its weight is 3

≤ K

, isolating it from

rowspace

(

). Thus,

is a non-trivial Z-logical operator, yielding d

≤ 3.

Combined with the lower bounds from Theorem 5.1, the exact distance evaluates to

= d

= 3 for all n ≥ 3.

6 The Rate Formula

Figure 2: Asymptotic encoding rate 1

−

for the

family. The analytical formula

establishes that as the dimension and coordination number scale, the topologically protected

subset of the lattice bounds approaches 100%. All historical topological CSS codes natively

converge to a rate of 0.

Factoring the edge count as

= (

, the fractional rank utilized by the stabilizers

scales as

(

)

→ n

= 2

. The symmetric structure guarantees the same for

. Evaluating the limit:

= 1 −

rkH

−

rkH

→ 1 −

−

= 1 −

Each stabiliser type consumes exactly 2

of the available information capacity. The

remaining 1

−

fraction is fully protected by the lattice topology. Because

= 2

(

n −

expands with dimension, the rate strictly increases with

. For the lowest dimensional cases

(n = 3 and n = 4), the code rate achieves 2/3 and 5/6, respectively.

7 Comparison with Known CSS Codes

Table 2: The D

family in context.

Code family Rate Distance Stab. weight

Toric / surface codes → 0 ∼

√

n O(1)

Hypergraph product [7] const. ∼

√

n const.

Good QLDPC [8, 9] const. O(n) const.

(this work) 1 − 4/K → 1 3 2n(n − 1)

The

code architecture strategically trades distance scaling for extreme encoding rates.

They are structurally suited for fault-tolerant applications where physical noise is intrinsically

bounded below the threshold and high logical qubit density is the primary architectural

constraint. While the stabilizer weight

= 2

(

n −

1) remains physically manageable for

= 3 (

= 12) and

= 4 (

= 24), hardware implementations for

n ≥

6 will demand highly

centralized syndromic measurement networks.

8 Open Problems

O1. The

code. The

root lattice (

= 240, predicted rate 59

60) introduces half-

integer coset vectors not native to the

evaluation. A valid CSS formulation derived from

the integer Construction A realization of

would constitute the highest-rate topological

code theoretically permitted by uniform crystallographic structures.

O2. Single-shot decoding. Evaluating whether

codes natively admit single-shot

error correction requires further simulation to quantify local syndrome conﬁnement under

phenomenological noise models.

O3. Optimal rate bounds at

= 3. It remains formally unproven whether 1

−

represents the strict information-theoretic maximum encoding rate for a translationally

invariant topological CSS code operating at coordination K and distance 3.

O4. Weight reduction. For a ﬁxed parameter

, analytical methods are required to

determine if mathematically equivalent subsystem codes exist that factorize the stabilizer

operators into components strictly smaller than K = 2n(n − 1).

Acknowledgements

We acknowledge the quantum error correction community for the prior structural work that

established the necessary context for interpreting geometric stabilizers in high dimensions.

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CSS Codes from Dn Root Lattices: An Infinite Family with Rate 1 - 4/K