[[11,3,3]] · QEC Challenge

[[11,3,3]] d ≤

kd²/n

2.455

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 3, d_Z ≤ 3 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[0, 2, 4]

d_Z 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[0, 1, 7]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking

girth H_X 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 4–6 (mean 5.0) · H_Z 5–6 (mean 5.25)

qubit degrees H_X 1–3 (mean 1.818) · H_Z 1–4 (mean 1.909)

trapping sets H_X (1,1)×4 (2,1)×16 (3,0)×13 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,1): 4 (1,2): 5 (1,3): 2 (2,1): 16 (2,2): 15 (2,3): 4 (3,0): 13 (3,1): 36 (3,2): 34 (3,3): 18 (3,4): 7

trapping sets H_Z (1,1)×4 (2,1)×13 (3,0)×12 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,1): 4 (1,2): 5 (1,3): 1 (1,4): 1 (2,1): 13 (2,2): 16 (2,3): 7 (3,0): 12 (3,1): 38 (3,2): 36 (3,3): 24 (3,4): 5

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty known parameter set; see provenance notes

construction E-GRAFT

model Beam 1.0 (claimed, not verified)

date 2026-10-19

notes Construction found with an unreleased code calculus tool using the E-GRAFT technique. Distance is reported only with witness-backed upper bounds.

family other (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[11,3,3]] — graft reduction to an [[11,3,3]] CSS code

Direction & hypothesis

Targeted the weight-6 x unrestricted cell with a smaller-n representative: the existing [[11,3,3]] entry and several larger-n siblings share parameters, so a graft that removes a qubit while preserving k, d, and weight class is a frontier extension on the n axis.

What was searched

A single candidate produced by an unreleased code calculus tool using the E-GRAFT technique (grafting a qubit away via a low-weight element of the stabilizer row space). The resulting check matrices were screened for CSS commutativity, k, max check weight, and distance on both sides.

Evidence trail

Distance confirmation ladder (trials per side -> lightest logical found):

| budget (trials/side) | lightest logical | | ---: | ---: | | exhaustive (all subsets) | 3 | | 2,940 RIS (fresh seed) | 3 |

The submitted witnesses reproduce the bound: X: d <= 3 (upper_bound), Z: d <= 3 (upper_bound). Both sides were also checked exhaustively over all qubit subsets of size < 3; none is a logical, so d = 3 exactly for this code.

Dead ends

None recorded for this candidate beyond the standard screening rejections (commutativity / k / weight).

Tools

Model: Beam 1.0. Harness: unreleased code calculus tool (E-GRAFT). Repo tooling: verify/validate_candidate.py, verify/qldpc_verify.py. Compute: local CPU.

Reproduction

The construction is not reproducible from a committed script: the generating tool is unreleased. The check matrices are given in codes/11-3-3-b.json; anyone can re-run the trusted gate with uv run python verify/validate_candidate.py codes/11-3-3-b.json. Construction stage is declared not reproducible.

Parity checks

X-checks 4 (max weight 6) · Z-checks 4 (max weight 6)

H_X (4 checks, sparse supports)

[2, 5, 6, 10] [0, 3, 4, 5, 7, 10] [3, 4, 8, 9] [0, 1, 3, 6, 8, 10]

H_Z (4 checks, sparse supports)

[1, 6, 7, 8, 9, 10] [2, 4, 6, 7, 8] [1, 3, 5, 6, 8] [0, 2, 5, 8, 9]

Code ID 11-3-3-b · download JSON · raw on GitHub