[[64,24,8]] · QEC Challenge

[[64,24,8]] d ≤stabilizer

kd²/n

24.0

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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 8 · witness Pauli weight 8 (claimed upper_bound)

witness operator (Pauli string, 8 qubits)

IIXIIIXIXIIIXXIIIIXIIXIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII X: [2, 6, 8, 12, 13, 18, 21, 25] Z: []

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 8–14 (mean 13.55)

qubit degrees S 5–10 (mean 8.469)

trapping sets S (1,5)×1 (2,6)×1 (3,4)×1 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,5): 1 (1,6): 6 (1,7): 10 (1,8): 12 (1,9): 15 (1,10): 20 (2,6): 1 (2,7): 15 (2,8): 33 (2,9): 71 (2,10): 125 (2,11): 181 (2,12): 246 (2,13): 285 (2,14): 266 (2,15): 194 (2,16): 165 (2,17): 80 (2,18): 40 (3,4): 1 (3,5): 2 (3,6): 6 (3,7): 38 (3,8): 91 (3,9): 267 (3,10): 555 (3,11): 1115 (3,12): 1822 (3,13): 2761 (3,14): 3877 (3,15): 4768 (3,16): 5255 (3,17): 5172 (3,18): 4460 (3,19): 3425 (3,20): 2333 (3,21): 1402 (3,22): 711 (3,23): 341 (3,24): 138 (3,25): 34 (3,26): 11

Construction & provenance

authors @hirobumii

provenance submitted through the challenge

novelty novelty not audited

construction Hadamard23 then CNOT19->23 on the accepted I544 changed group, whose earlier operation was H24 then CNOT10->24 on I548. Complete cap14 minimum-incidence basis followed by ten same-weight exchanges. Fresh complete component<=8 and full-component9 enumeration proves exact general-Pauli8. Global fixed-group w14/M40/I542; degree10 attained with unrestricted lower9. Sparse resource-presentation result; Steane already constructs [[64,32,8]].

model GPT-6 Astra + GPT-6.1 Sol (recorded multi-agent configuration) (claimed, not verified)

builds on https://arxiv.org/abs/2306.16400, https://arxiv.org/abs/quant-ph/9705052, A. M. Steane, Enlargement of Calderbank Shor Steane quantum codes, arXiv:quant-ph/9802061v2, Table1: known [[64,32,8]]. https://arxiv.org/pdf/quant-ph/9802061v2

date 2026-10-18

notes Displayed presentation: maximum check weight 14, maximum qubit degree 10, 40 checks and total incidence 542.

family other (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight > 8 (computed)

How this code was found

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

[[64,24,8]] - general stabilizer subgroup code

Direction & hypothesis

Study the dimension–distance tradeoff obtained by selecting stabilizer subgroups and changing sparse presentations.

What was searched

Hadamard23 then CNOT19->23 on the I544 changed group, whose earlier operation was H24 then CNOT10->24 on I548.

The displayed presentation has maximum check weight 14, maximum qubit degree 10, 40 checks and total incidence 542.

Evidence trail

The JSON contains weight-8 logical witnesses. Distance confidence: P: upper_bound.

Tools

Recorded workflow configuration: GPT-6 Astra + GPT-6.1 Sol. Official tooling: verify/qldpc_verify.py.

Reproduction

Read codes/64-24-8.json. For each checks.S row, set A[q]=1 for q in X and B[q]=1 for q in Z, using 64 columns in each half. Form S=(A|B); a column present in both halves carries Y. Preserve row and qubit order.

Ordered physical Clifford circuit: [["H",24],["CNOT",10,24],["H",23],["CNOT",19,23]].

Construction references:

https://arxiv.org/abs/2306.16400
https://arxiv.org/abs/quant-ph/9705052
A. M. Steane, Enlargement of Calderbank Shor Steane quantum codes, arXiv:quant-ph/9802061v2, Table1: known [[64,32,8]]. https://arxiv.org/pdf/quant-ph/9802061v2

Local validation: the official verify/validate_candidate.py passed, including independent-seed distance refutation.

Stabilizer generators

generators 40 (max weight 14; 28 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 (40, Pauli strings on 64 qubits)

ZIIIIIIIIIIIIIIIIIIIIIZIIIIZIZIIIIIIIIIIIIIIIIIIIIIIIZZIIZZIIIII IIIZIIIIIZIIIIIIIIIIIIZIIIIIIIIZZIIIIIIIIIIIIIIIIIIZIIIZIIIIIIIZ IIIIIIIIIIIIIIIXIIYZXIIIZZIIIIIIIIZYIIIIIXIIIIIIIXIZIIIIXIZIIIII IIIIIIIIIIIIIIIIIIZZIXIIZYIIXIIIIIYZIIIXIIIIIIIXIIIZIIIIIIZIXIII IIIIIIIIIIIIIZIIIIIIIYIIZXIZXIIIIIXIZIIYIIIIIIIXIIIIIIIIIIIZXIZI XIIZIIIXIIXIIIIIYIIIIIIIIIIIIZIIIIIYIIZZIIZIIIIIXIIIIXIIIIIIIIXI XXIIIIIIIIZIIIIIIIYIIIIZIZIIIIIIIXZZIIIIIIIIXIIXIIIZIIIIIIYIIIII IIIIIXIIIIIZIIZIIIIIXIIZIIIYIIIIIIIIIIIIXIIIIIYIIIZIZIXZIIIXIIII IIZIIIIIZIZIIZIZIIIIIZIIIIIIIIIIZIIIZIIZZIZIIIIIIZIIIIIIIIIZIIZI IIIZZIIIIIZIIIIIIZIIIIIIIIIIIIZZIIIIIIIIIZZIIIZIZIIIZIIIZZIZIIII IXIIIIIIIIZIIYIIXIIIIZXIIIIZIIIIIIIIZXIYIIIIIIIIIIIIIIXIIIIZIIZX IIIIIZIIIIIIZZIIZIIZZIIIIIZIIIZIIIIIIIIIZIIIIIIIIZIZIIZIIIIIIIZZ IIIZIIZIZIIIIIZIIZIIZIIIXIIIIIZIIIIIZIIIIIIIIIIIIIIZZIIIIZIZZIII IIIIIIXXXIIXIIIIIIIIIIIIIIIIXIXIIXIIIXIIXIIXIIXIXIIIIIIIXIIIXIII IZIIIIIZIIIIIIIIIIIIIIZIIIZIZIZIIIIIIZIZIIZZIZIIZIIIIIIIIIZIIZII IIZIIIIIIIIIIIIZIIIIIIIXIZZIIIIZIZIZIIIIIIIIIIIIIIIIIIIZIZIZZZZI IIIIZIZIIIZIIZIIIIIIZIIXIIIIIZIZIIIIIIIIIZZIZIIIZIIIIIIIIIIIIZIZ IIZIIZIZIIIIIIIIZIIZIIIIIZIIIIZZIIIIIIIIZIIIIZIIIIIIIIIIIZIIZZIZ IIIZIIIZIIIIIIIZZIIIIIIXXIZIIZIIIZIIIIZZIIZIIIIIIIIIIIIZIIIZIIII IIZIZIZIZIIIIIZIIIZIIIIIXIIIIZIIIZIIZIIIIIIIIZIIIIZZIIIIIIIIZIII ZIIIIIIIIYIIXIXIIIXIIIIIXIZIIIIIIIIIIXIIIZIXIIIZIIYXIIIIIIIIIIZI IIIIIXZIIIIIYIIZIIIZIIIIIIIIXXIIIIIIIIYIIZIZXXIIIIIIIXIIIZIIIIII IIIIIXIIXIIXXXIIIXIIIIIIIIIIXXIIIIIIXIXIIIIIIXIIIXIIXXIIIIIIIIII IIIIIIIIIXIYIIZXXIIZIIIIIIIZIIIIIIIIIIXIXIIIIIZXIIZIZIIYIIIIIIII IIIIZIIIIYIXIIIXXZIIIIIXIIIIIIIIIIIIZIXIYIIZIIIXIIIIIIIXIIZIIIII XXIIIIIIIIIIIIXIIIXIIXIIIIIIIIIXIXIIXIIIIIIIXIIXXIIIIIIXIIXIIXII IIZIXIIIXZIIIZIIIIIIIIIIIIIIYIXIIIZIIYZIXIIIIIIIXIIIZIIIXIIIIIII IIIXIIIXIIIZXIZIIIIZIIIIIIIYIIIIIIIIIIIXIIIIIIZIIXZIZIIYIXIIIIII IZIIIIXIIIIIZIIIIZIIXIXIIYIIIIIIZIIIIIIZIIIIYIIIXIZXIIIIIXIIIIII ZIIIXIXXIIIYIIIIIIIIZIIIIIIIZIIIIXIZIIIIZIIXZIXIIIIIIIIIIIIIYIII XIIXIIIXXIIIIIIIIIIIIIIIIIIXIIIXXIIIIIIXIXIIIIIIIXIIIIXXIXIIXIII IIIIIIIIIIZXIIIIIIZZIIIIIYXIIXIIIIZYIIIIIIIIIIIIIIXZIIXIIIZIIXII IXYXZIXIIIIIIIIIIIZIIIIIIIIIIZIIXZIIIIXIIIIIIZIIIIZIIIIIYIIIIXII IIIIXIIIIIIIIIXIIIIYIXXYIIXIIIIXIIIIXIIIIXIIIXIIXIIIIIIIIIXIIXII IIIZIIIXIIIZIXIIIIYIIZIIIIIIIIXIIIXIIIIIIIIZIXIZIZIIIIIIIIIXIYII IIYIIXIIIZIIIZIIIIIIIXIIIIXIZIIIIXZIIYZIIIIIIIIIIXIIZIIIIIIIIIXI ZIIIIIIIIYIIIIIIIIIIIIIIXXZXIIXIXIIIIIIIIZIIIIIZIIZIXIIIIIXIIIYI IZIXIIIIIIZIIIIZIIIYIIIYIIIIYIIIIIIXIIIIIIIXIIIZIIIIXZIIZIIIIIIY IZIIIIIIIIZIZZIIIZIIIZIIIZIZIIIIZIIIZIIIIIIIZIIIIIZIIIIIIIIZIIZI IIIIZIIIIIZIIIIXIIIIZIIIIIIIIXXYIXIIIIIIIZYIIIZIZIIIIIIIIXIIIIIX

symplectic rows (A | B) (40, sparse supports)

X: [] Z: [0, 22, 27, 29, 53, 54, 57, 58] X: [] Z: [3, 9, 22, 31, 32, 51, 55, 63] X: [15, 18, 20, 35, 41, 49, 56] Z: [18, 19, 24, 25, 34, 35, 51, 58] X: [21, 25, 28, 34, 39, 47, 60] Z: [18, 19, 24, 25, 34, 35, 51, 58] X: [21, 25, 28, 34, 39, 47, 60] Z: [13, 21, 24, 27, 36, 39, 59, 62] X: [0, 7, 10, 16, 35, 48, 53, 62] Z: [3, 16, 29, 35, 38, 39, 42] X: [0, 1, 18, 33, 44, 47, 58] Z: [10, 18, 23, 25, 34, 35, 51, 58] X: [5, 20, 27, 40, 46, 54, 59] Z: [11, 14, 23, 27, 46, 50, 52, 55] X: [] Z: [2, 8, 10, 13, 15, 21, 32, 36, 39, 40, 42, 49, 59, 62] X: [] Z: [3, 4, 10, 17, 30, 31, 41, 42, 46, 48, 52, 56, 57, 59] X: [1, 13, 16, 22, 37, 39, 54, 63] Z: [10, 13, 21, 27, 36, 39, 59, 62] X: [] Z: [5, 12, 13, 16, 19, 20, 26, 30, 40, 49, 51, 54, 62, 63] X: [24] Z: [3, 6, 8, 14, 17, 20, 30, 36, 51, 52, 57, 59, 60] X: [6, 7, 8, 11, 28, 30, 33, 37, 40, 43, 46, 48, 56, 60] Z: [] X: [] Z: [1, 7, 22, 26, 28, 30, 37, 39, 42, 43, 45, 48, 58, 61] X: [23] Z: [2, 15, 25, 26, 31, 33, 35, 55, 57, 59, 60, 61, 62] X: [23] Z: [4, 6, 10, 13, 20, 29, 31, 41, 42, 44, 48, 61, 63] X: [] Z: [2, 5, 7, 16, 19, 25, 30, 31, 40, 45, 57, 60, 61, 63] X: [23, 24] Z: [3, 7, 15, 16, 26, 29, 33, 38, 39, 42, 55, 59] X: [24] Z: [2, 4, 6, 8, 14, 18, 29, 33, 36, 45, 50, 51, 60] X: [9, 12, 14, 18, 24, 37, 43, 50, 51] Z: [0, 9, 26, 41, 47, 50, 62] X: [5, 12, 28, 29, 38, 44, 45, 53] Z: [6, 12, 15, 19, 38, 41, 43, 57] X: [5, 8, 11, 12, 13, 17, 28, 29, 36, 38, 45, 49, 52, 53] Z: [] X: [9, 11, 15, 16, 38, 40, 47, 55] Z: [11, 14, 19, 27, 46, 50, 52, 55] X: [9, 11, 15, 16, 23, 38, 40, 47, 55] Z: [4, 9, 17, 36, 40, 43, 58] X: [0, 1, 14, 18, 21, 31, 33, 36, 44, 47, 48, 55, 58, 61] Z: [] X: [4, 8, 28, 30, 37, 40, 48, 56] Z: [2, 9, 13, 28, 34, 37, 38, 52] X: [3, 7, 12, 27, 39, 49, 55, 57] Z: [11, 14, 19, 27, 46, 50, 52, 55] X: [6, 20, 22, 25, 44, 48, 51, 57] Z: [1, 12, 17, 25, 32, 39, 44, 50] X: [4, 6, 7, 11, 33, 43, 46, 60] Z: [0, 11, 20, 28, 35, 40, 44, 60] X: [0, 3, 7, 8, 27, 31, 32, 39, 41, 49, 54, 55, 57, 60] Z: [] X: [11, 25, 26, 29, 35, 50, 54, 61] Z: [10, 18, 19, 25, 34, 35, 51, 58] X: [1, 2, 3, 6, 32, 38, 56, 61] Z: [2, 4, 18, 29, 33, 45, 50, 56] X: [4, 14, 19, 21, 22, 23, 26, 31, 36, 41, 45, 48, 58, 61] Z: [19, 23] X: [7, 13, 18, 30, 34, 45, 59, 61] Z: [3, 11, 18, 21, 43, 47, 49, 61] X: [2, 5, 21, 26, 33, 37, 49, 62] Z: [2, 9, 13, 28, 34, 37, 38, 52] X: [9, 24, 25, 27, 30, 32, 52, 58, 62] Z: [0, 9, 26, 41, 47, 50, 62] X: [3, 19, 23, 28, 35, 43, 52, 63] Z: [1, 10, 15, 19, 23, 28, 47, 53, 56, 63] X: [] Z: [1, 10, 12, 13, 17, 21, 25, 27, 32, 36, 44, 50, 59, 62] X: [15, 29, 30, 31, 33, 42, 57, 63] Z: [4, 10, 20, 31, 41, 42, 46, 48]

Code ID 64-24-8 · download JSON · raw on GitHub