[[44,2,9]] · QEC Challenge

[[44,2,9]] d ≤stabilizer

kd²/n

3.682

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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 9 · witness Pauli weight 9 (3 Y factors; Hamming weight over 2n bits 12) (claimed upper_bound)

witness operator (Pauli string, 9 qubits)

XIIIIIIIIIIIZXZZYYZIIIIIIIIIIIYIIIIIIIIIIIII X: [0, 13, 16, 17, 30] Z: [12, 14, 15, 16, 17, 18, 30]

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)×44 (2,6)×44 (3,8)×484 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,6): 44 (2,6): 44 (2,8): 220 (2,10): 88 (3,8): 484 (3,10): 1452 (3,12): 1364 (3,14): 132

Construction & provenance

authors @davidplu

provenance submitted through the challenge

novelty novelty not audited

construction Cyclic shifts modulo 44 of X support [1, 43], Z support [2, 15, 29, 42]; inversion-symmetric supports.

model GPT-6 (claimed, not verified)

builds on arXiv:1108.5490

date 2026-10-03

notes Known one-block additive cyclic construction; specific-instance literature novelty unverified. Witness-backed board metadata; supplementary exhaustive evidence in accompanying note.

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

[[44,2,9]] — weight-6 one-block cyclic stabilizer

Direction & hypothesis

Target the unrestricted weight-6 general-stabilizer cell by varying inversion-symmetric cyclic supports. At upstream ff48155b8897e6796950c64d860e9102d59c36e9 this entry extends the Pareto frontier: no existing entry dominates it. This is a board-relative contribution within a known construction (arXiv:1108.5490), not a claim of a new family or literature record.

What was searched

The completed campaign examined 4,151 unique seed instances across 4,526 runs: complete normalized two-color odd supports through n=53, sampled odd n=55..101 and even n=24..80, and a connected k=3 branch. The normalized seed has X support {1,-1} and Z support {u,-u,v,-v}; screening used randomized information-set search, followed by exhaustive checks for selected survivors. Initial odd screening used seed 100301 and 50 RIS trials; refinement used seed 100400 plus the candidate index and 100 trials. Search success alone is not a lower bound. The exact submitted supports are given below.

Evidence trail

PROVED/VERIFIED externally: stabilizer rank 42, k=2, commuting cyclic generators of Pauli weight 6. All 11,362,066 Pauli errors of weight at most 4 were keyed by syndrome and four logical commutation bits. No equal-syndrome pair had different logical labels, excluding every nontrivial logical of weight at most 8. The stored weight-9 witness supplies the opposite bound. The supplementary calculation is reproducible from research/evidence/44-2-9/scripts/ and its outputs are committed under research/evidence/44-2-9/. The trusted gate receipt is research/evidence/44-2-9/gate.json: passed=true, fresh refutation seed 100344, no lighter logical found. Repository distance metadata remains upper_bound with a checked Pauli witness; the supplementary exhaustive result is not the board's exact-tier certificate.

Uniqueness is relative to the checked board snapshot. There is no general-stabilizer entry with the same n and k; n and k are invariant under LC+permutation. The included algebraic certificate excludes equivalence to ANY CSS code: necessary trace-one local-projector preservation equations sum to 0=1 over GF(2). Both exact/WL duplicate checks are negative. No matching submission appeared among the seven open PRs checked on 2026-10-03. Global literature novelty remains UNKNOWN.

Dead ends

The campaign's [[31,1,9]] candidate is a duplicate: multiplying the existing entry's qubit labels by 2 modulo 31 gives the same stabilizer rowspace, so it was withheld. The [[33,1,9]] and [[39,3,7]] candidates are dominated on the current board. The [[53,1,11]] candidate is dominated by this campaign's [[47,1,11]]. None is used as evidence of a new record.

Tools

GPT-6 via Codex; Python, NumPy, exact binary symplectic routines, and the unchanged challenge validator. Exhaustive verification takes seconds to tens of seconds on the development machine. No circuit-tier files: the tier does not support general stabilizer codes.

Reproduction

Generate all 44 cyclic shifts modulo 44 of X=[1, 43], Z=[2, 15, 29, 42]. This specifies the complete submitted check matrix without search randomness. Install NumPy, then run from the repository root:

python research/evidence/44-2-9/scripts/k2_certify.py
python research/evidence/44-2-9/scripts/verify_css.py
python verify/qldpc_verify.py codes/44-2-9.json

The supplementary scripts replay the completed investigation; they do not replace or modify the trusted verifier.

Stabilizer generators

generators 44 (max weight 6; 44 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 (44, Pauli strings on 44 qubits)

IXZIIIIIIIIIIIIZIIIIIIIIIIIIIZIIIIIIIIIIIIZX XIXZIIIIIIIIIIIIZIIIIIIIIIIIIIZIIIIIIIIIIIIZ ZXIXZIIIIIIIIIIIIZIIIIIIIIIIIIIZIIIIIIIIIIII IZXIXZIIIIIIIIIIIIZIIIIIIIIIIIIIZIIIIIIIIIII IIZXIXZIIIIIIIIIIIIZIIIIIIIIIIIIIZIIIIIIIIII IIIZXIXZIIIIIIIIIIIIZIIIIIIIIIIIIIZIIIIIIIII IIIIZXIXZIIIIIIIIIIIIZIIIIIIIIIIIIIZIIIIIIII IIIIIZXIXZIIIIIIIIIIIIZIIIIIIIIIIIIIZIIIIIII IIIIIIZXIXZIIIIIIIIIIIIZIIIIIIIIIIIIIZIIIIII IIIIIIIZXIXZIIIIIIIIIIIIZIIIIIIIIIIIIIZIIIII IIIIIIIIZXIXZIIIIIIIIIIIIZIIIIIIIIIIIIIZIIII IIIIIIIIIZXIXZIIIIIIIIIIIIZIIIIIIIIIIIIIZIII IIIIIIIIIIZXIXZIIIIIIIIIIIIZIIIIIIIIIIIIIZII IIIIIIIIIIIZXIXZIIIIIIIIIIIIZIIIIIIIIIIIIIZI IIIIIIIIIIIIZXIXZIIIIIIIIIIIIZIIIIIIIIIIIIIZ ZIIIIIIIIIIIIZXIXZIIIIIIIIIIIIZIIIIIIIIIIIII IZIIIIIIIIIIIIZXIXZIIIIIIIIIIIIZIIIIIIIIIIII IIZIIIIIIIIIIIIZXIXZIIIIIIIIIIIIZIIIIIIIIIII IIIZIIIIIIIIIIIIZXIXZIIIIIIIIIIIIZIIIIIIIIII IIIIZIIIIIIIIIIIIZXIXZIIIIIIIIIIIIZIIIIIIIII IIIIIZIIIIIIIIIIIIZXIXZIIIIIIIIIIIIZIIIIIIII IIIIIIZIIIIIIIIIIIIZXIXZIIIIIIIIIIIIZIIIIIII IIIIIIIZIIIIIIIIIIIIZXIXZIIIIIIIIIIIIZIIIIII IIIIIIIIZIIIIIIIIIIIIZXIXZIIIIIIIIIIIIZIIIII IIIIIIIIIZIIIIIIIIIIIIZXIXZIIIIIIIIIIIIZIIII IIIIIIIIIIZIIIIIIIIIIIIZXIXZIIIIIIIIIIIIZIII IIIIIIIIIIIZIIIIIIIIIIIIZXIXZIIIIIIIIIIIIZII IIIIIIIIIIIIZIIIIIIIIIIIIZXIXZIIIIIIIIIIIIZI IIIIIIIIIIIIIZIIIIIIIIIIIIZXIXZIIIIIIIIIIIIZ ZIIIIIIIIIIIIIZIIIIIIIIIIIIZXIXZIIIIIIIIIIII IZIIIIIIIIIIIIIZIIIIIIIIIIIIZXIXZIIIIIIIIIII IIZIIIIIIIIIIIIIZIIIIIIIIIIIIZXIXZIIIIIIIIII IIIZIIIIIIIIIIIIIZIIIIIIIIIIIIZXIXZIIIIIIIII IIIIZIIIIIIIIIIIIIZIIIIIIIIIIIIZXIXZIIIIIIII IIIIIZIIIIIIIIIIIIIZIIIIIIIIIIIIZXIXZIIIIIII IIIIIIZIIIIIIIIIIIIIZIIIIIIIIIIIIZXIXZIIIIII IIIIIIIZIIIIIIIIIIIIIZIIIIIIIIIIIIZXIXZIIIII IIIIIIIIZIIIIIIIIIIIIIZIIIIIIIIIIIIZXIXZIIII IIIIIIIIIZIIIIIIIIIIIIIZIIIIIIIIIIIIZXIXZIII IIIIIIIIIIZIIIIIIIIIIIIIZIIIIIIIIIIIIZXIXZII IIIIIIIIIIIZIIIIIIIIIIIIIZIIIIIIIIIIIIZXIXZI IIIIIIIIIIIIZIIIIIIIIIIIIIZIIIIIIIIIIIIZXIXZ ZIIIIIIIIIIIIZIIIIIIIIIIIIIZIIIIIIIIIIIIZXIX XZIIIIIIIIIIIIZIIIIIIIIIIIIIZIIIIIIIIIIIIZXI

symplectic rows (A | B) (44, sparse supports)

X: [1, 43] Z: [2, 15, 29, 42] X: [0, 2] Z: [3, 16, 30, 43] X: [1, 3] Z: [0, 4, 17, 31] X: [2, 4] Z: [1, 5, 18, 32] X: [3, 5] Z: [2, 6, 19, 33] X: [4, 6] Z: [3, 7, 20, 34] X: [5, 7] Z: [4, 8, 21, 35] X: [6, 8] Z: [5, 9, 22, 36] X: [7, 9] Z: [6, 10, 23, 37] X: [8, 10] Z: [7, 11, 24, 38] X: [9, 11] Z: [8, 12, 25, 39] X: [10, 12] Z: [9, 13, 26, 40] X: [11, 13] Z: [10, 14, 27, 41] X: [12, 14] Z: [11, 15, 28, 42] X: [13, 15] Z: [12, 16, 29, 43] X: [14, 16] Z: [0, 13, 17, 30] X: [15, 17] Z: [1, 14, 18, 31] X: [16, 18] Z: [2, 15, 19, 32] X: [17, 19] Z: [3, 16, 20, 33] X: [18, 20] Z: [4, 17, 21, 34] X: [19, 21] Z: [5, 18, 22, 35] X: [20, 22] Z: [6, 19, 23, 36] X: [21, 23] Z: [7, 20, 24, 37] X: [22, 24] Z: [8, 21, 25, 38] X: [23, 25] Z: [9, 22, 26, 39] X: [24, 26] Z: [10, 23, 27, 40] X: [25, 27] Z: [11, 24, 28, 41] X: [26, 28] Z: [12, 25, 29, 42] X: [27, 29] Z: [13, 26, 30, 43] X: [28, 30] Z: [0, 14, 27, 31] X: [29, 31] Z: [1, 15, 28, 32] X: [30, 32] Z: [2, 16, 29, 33] X: [31, 33] Z: [3, 17, 30, 34] X: [32, 34] Z: [4, 18, 31, 35] X: [33, 35] Z: [5, 19, 32, 36] X: [34, 36] Z: [6, 20, 33, 37] X: [35, 37] Z: [7, 21, 34, 38] X: [36, 38] Z: [8, 22, 35, 39] X: [37, 39] Z: [9, 23, 36, 40] X: [38, 40] Z: [10, 24, 37, 41] X: [39, 41] Z: [11, 25, 38, 42] X: [40, 42] Z: [12, 26, 39, 43] X: [41, 43] Z: [0, 13, 27, 40] X: [0, 42] Z: [1, 14, 28, 41]

Code ID 44-2-9 · download JSON · raw on GitHub