[[27,3,5]] · QEC Challenge

[[27,3,5]] d ≤stabilizer

kd²/n

2.778

Share this result

Distance

a general stabilizer code has no X and Z sides: d is the minimum Pauli weight of a nontrivial logical operator (a Y factor counts one qubit), and the witness is one Pauli operator that commutes with every generator and is not a product of them

d 5 · witness Pauli weight 5 (claimed upper_bound)

witness operator (Pauli string, 5 qubits)

IIXIXXIXIIIIIIIIIIZIIIIIIII X: [2, 4, 5, 7] Z: [18]

certificate none yet · distance stands as a self-certified upper bound (d ≤); the Pauli-weight certifier is not built yet, so stabilizer entries cannot earn d= for now

Diagnostics

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

girth S 4 (shortest cycle of the generator Tanner graph; longer is friendlier to belief propagation)

check weights S 6

qubit degrees S 6

trapping sets S (1,6)×27 (2,6)×27 (3,8)×432 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,6): 27 (2,6): 27 (2,8): 135 (2,10): 54 (3,8): 432 (3,10): 756 (3,12): 540 (3,14): 108

Construction & provenance

authors @davidplu

provenance submitted through the challenge

novelty novelty not audited

construction Cyclic shifts modulo 27 of X support [2, 4, 23, 25], Z support [12, 15]; inversion-symmetric supports.

model GPT-6 (claimed, not verified)

builds on arXiv:quant-ph/9608006

date 2026-09-30

notes Bounded pilot using repository Pauli RIS; literature novelty unverified. Inspired by challenge PR #2445.

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

[[27,3,5]] cyclic stabilizer with weight-6 checks

Direction & hypothesis

Target the unrestricted, weight-6 non-CSS stabilizer board using sparse inversion-symmetric cyclic generators. This is an instance of established additive cyclic constructions, not a new family. The trusted gate reports board advancement against upstream 7ea128673cd867fec57d614d8c847d9f2d790c60. Literature novelty is unverified. No circuit-performance improvement is claimed.

What was searched

The campaign sampled 320 candidates: 160 each at check weights 6 and 8, odd lengths 11 through 31. Disjoint inversion-pair supports define the X and Z halves. Candidate ordering used Python random seed 300930. Initial screening used 80 Pauli RIS trials per candidate; six representatives received deeper searches. This was a bounded sample, not exhaustive enumeration.

Evidence trail

The earlier deep pass requested 20,000 Pauli RIS trials with a 30-second cap, followed by 400,000 accelerator trials on doubled matrices. The new pass requested the same Pauli budget followed by 2,000,000 accelerator trials, seed 301102; both returned lightest Pauli weight 5. The accelerated doubled-matrix search is re-scored by Pauli weight and is not two million independent direct-Pauli trials. Receipts are repro/cyclic-27-3-5/previous-search.json and repro/cyclic-27-3-5/deep-search.json.

The final candidate gate passed structural, rank, commutation, witness and refutation checks, with no exact or WL duplicate detected. Its receipt is repro/cyclic-27-3-5/current-gate.json. All distance metadata remains witness-backed upper_bound. A search that fails to find a lighter logical does not establish a distance lower bound. Duplicate tests do not cover every local Clifford equivalence.

Dead ends and literature limits

None of the six campaign candidates establishes a new unrestricted parameter record. The weight-8 [[21,3,5]] candidate was set aside because an inspected published [[21,3,6]] admits weight-8 generators. For this submission, the Grassl reference achieves distance 9. Exhaustive rowspace analysis of that specific reference gave minimum spanning check weight 12; this is not an optimality claim across all published codes. The submission offers a sparse-check tradeoff or matching distance, not a general literature record.

Tools

GPT-6 in Codex; repository GF(2), Pauli RIS, gf2_fast doubled-matrix search, and unchanged trusted validation tools. No custom distance checker or verifier modification. No decoding benchmark is included.

Reproduction

For every shift t from 0 through 26, add one generator with X support {(a+t) mod 27: a in [2, 4, 23, 25]} and Z support {(b+t) mod 27: b in [12, 15]}. The resulting binary symplectic matrix has 27 rows and 54 columns. The code JSON stores all generators and a nontrivial weight-5 Pauli logical witness.

Run python verify/qldpc_verify.py codes/27-3-5-cyclic.json from the repository root. The construction is related to CRSS, Section 5 and the single-generator cyclic framework of Kovalev, Dumer and Pryadko. The search was initially inspired by challenge PR #2445.

Stabilizer generators

generators 27 (max weight 6; 27 mixed X/Z) (one binary symplectic matrix S = (A | B); generator i is X on A_i and Z on B_i, Y where both)

generators (27, Pauli strings on 27 qubits)

IIXIXIIIIIIIZIIZIIIIIIIXIXI IIIXIXIIIIIIIZIIZIIIIIIIXIX XIIIXIXIIIIIIIZIIZIIIIIIIXI IXIIIXIXIIIIIIIZIIZIIIIIIIX XIXIIIXIXIIIIIIIZIIZIIIIIII IXIXIIIXIXIIIIIIIZIIZIIIIII IIXIXIIIXIXIIIIIIIZIIZIIIII IIIXIXIIIXIXIIIIIIIZIIZIIII IIIIXIXIIIXIXIIIIIIIZIIZIII IIIIIXIXIIIXIXIIIIIIIZIIZII IIIIIIXIXIIIXIXIIIIIIIZIIZI IIIIIIIXIXIIIXIXIIIIIIIZIIZ ZIIIIIIIXIXIIIXIXIIIIIIIZII IZIIIIIIIXIXIIIXIXIIIIIIIZI IIZIIIIIIIXIXIIIXIXIIIIIIIZ ZIIZIIIIIIIXIXIIIXIXIIIIIII IZIIZIIIIIIIXIXIIIXIXIIIIII IIZIIZIIIIIIIXIXIIIXIXIIIII IIIZIIZIIIIIIIXIXIIIXIXIIII IIIIZIIZIIIIIIIXIXIIIXIXIII IIIIIZIIZIIIIIIIXIXIIIXIXII IIIIIIZIIZIIIIIIIXIXIIIXIXI IIIIIIIZIIZIIIIIIIXIXIIIXIX XIIIIIIIZIIZIIIIIIIXIXIIIXI IXIIIIIIIZIIZIIIIIIIXIXIIIX XIXIIIIIIIZIIZIIIIIIIXIXIII IXIXIIIIIIIZIIZIIIIIIIXIXII

symplectic rows (A | B) (27, sparse supports)

X: [2, 4, 23, 25] Z: [12, 15] X: [3, 5, 24, 26] Z: [13, 16] X: [0, 4, 6, 25] Z: [14, 17] X: [1, 5, 7, 26] Z: [15, 18] X: [0, 2, 6, 8] Z: [16, 19] X: [1, 3, 7, 9] Z: [17, 20] X: [2, 4, 8, 10] Z: [18, 21] X: [3, 5, 9, 11] Z: [19, 22] X: [4, 6, 10, 12] Z: [20, 23] X: [5, 7, 11, 13] Z: [21, 24] X: [6, 8, 12, 14] Z: [22, 25] X: [7, 9, 13, 15] Z: [23, 26] X: [8, 10, 14, 16] Z: [0, 24] X: [9, 11, 15, 17] Z: [1, 25] X: [10, 12, 16, 18] Z: [2, 26] X: [11, 13, 17, 19] Z: [0, 3] X: [12, 14, 18, 20] Z: [1, 4] X: [13, 15, 19, 21] Z: [2, 5] X: [14, 16, 20, 22] Z: [3, 6] X: [15, 17, 21, 23] Z: [4, 7] X: [16, 18, 22, 24] Z: [5, 8] X: [17, 19, 23, 25] Z: [6, 9] X: [18, 20, 24, 26] Z: [7, 10] X: [0, 19, 21, 25] Z: [8, 11] X: [1, 20, 22, 26] Z: [9, 12] X: [0, 2, 21, 23] Z: [10, 13] X: [1, 3, 22, 24] Z: [11, 14]

Code ID 27-3-5-cyclic · download JSON · raw on GitHub