[[48,29,4]] · QEC Challenge

[[48,29,4]] d ≤stabilizer

kd²/n

9.667

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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 4 · witness Pauli weight 4 (claimed upper_bound)

witness operator (Pauli string, 4 qubits)

IZIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIXI X: [35, 46] Z: [1, 12]

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 14

qubit degrees S 4–7 (mean 5.542)

trapping sets S (1,4)×8 (2,0)×1 (3,1)×1 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 8 (1,5): 12 (1,6): 22 (1,7): 6 (2,0): 1 (2,1): 8 (2,2): 15 (2,3): 8 (2,4): 54 (2,5): 75 (2,6): 109 (2,7): 147 (2,8): 234 (2,9): 150 (2,10): 52 (2,11): 24 (2,12): 2 (3,1): 1 (3,2): 30 (3,3): 126 (3,4): 457 (3,5): 668 (3,6): 1297 (3,7): 1805 (3,8): 2346 (3,9): 2532 (3,10): 2333 (3,11): 1838 (3,12): 977 (3,13): 395 (3,14): 39 (3,15): 18

Construction & provenance

authors @hirobumii

provenance submitted through the challenge

novelty novelty not audited

construction A rank-19 subgroup of the span of all 24 mixed weight-14 stabilizers of the literal public FarLab [[48,10,8]] parent. Two explicit source-coordinate parity constraints and the indexed independent basis define the group. Complete general-Pauli half2 collision checking proves exact distance 4. Full enumeration of its 524287 nonidentity stabilizers proves global fixed-group w14/M19/I266. All complete cap14 bases have degree 7; unrestricted degree remains 6–7.

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

builds on H.-K. Lin and L. P. Pryadko, Quantum two-block group algebra codes, arXiv:2306.16400. https://arxiv.org/abs/2306.16400, https://github.com/unitaryfoundation/qldpc-challenge/blob/d8208c8da17e7db2285bacb62f99db9cd4863234/codes/48-10-8.json, D. Gottesman, Stabilizer Codes and Quantum Error Correction, arXiv:quant-ph/9705052., Y. Wu and Y. Lee, Self-Orthogonal Codes Constructed from Posets and Their Applications in Quantum Communication, Mathematics 8 (2020), 1495. https://doi.org/10.3390/math8091495. Its table/formula discrepancy is discussed in README; no parameter novelty claim.

date 2026-10-18

notes Displayed presentation: maximum check weight 14, maximum qubit degree 7, 19 checks and total incidence 266.

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

[[48,29,4]] - general stabilizer subgroup code

Direction & hypothesis

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

What was searched

A rank-19 subgroup of the span of all 24 mixed weight-14 stabilizers of the literal public FarLab [[48,10,8]] parent. Two explicit source-coordinate parity constraints and the indexed independent basis define the group.

The displayed presentation has maximum check weight 14, maximum qubit degree 7, 19 checks and total incidence 266.

Evidence trail

The JSON contains weight-4 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/48-29-4.json. For each checks.S row, set A[q]=1 for q in X and B[q]=1 for q in Z, using 48 columns in each half. Form S=(A|B); a column present in both halves carries Y. Preserve row and qubit order.

Coefficient annihilator: [512,131104].

Original parent rows: {"X":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],"Z":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]}.

Selected mixed-word indices: [0,1,2,3,4,6,7,8,10,11,12,13,14,15,16,18,19,21,22].

Independent mixed-source basis indices: [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,22].

Ordered mixed parent words as integer (X,Z) support masks, with bit q denoting qubit q: [["895551037953","277176435207"],["963196784898","550930268427"],["3582204151812","1108705740828"],["3852787139592","2203721073708"],["1340835238032","103381217457"],["2423166935136","155172483186"],["57315266428992","17739291853248"],["61644594233472","35259537179328"],["229261065715968","70957167412992"],["246578376933888","141038148717312"],["85813455234048","6616397917248"],["155082683848704","9931038923904"],["9002471657478","9414637547532"],["4604440879113","4810498682892"],["36009886629912","37658550190128"],["18417763516452","19241994731568"],["144036342595617","141974456438787"],["73667866656786","72636520931331"],["211144090322049","145137963892929"],["52832527975968","158346995829792"],["52836756883728","105587554651152"],["211127174694978","79166460657858"],["13209189220932","26396888662788"],["13208131993992","39586748957448"]]. Select the mixed-word indices listed above.

Public source: 48-10-8, contributed by @FarLab.

Construction references:

H.-K. Lin and L. P. Pryadko, Quantum two-block group algebra codes, arXiv:2306.16400. https://arxiv.org/abs/2306.16400
D. Gottesman, Stabilizer Codes and Quantum Error Correction, arXiv:quant-ph/9705052.
Y. Wu and Y. Lee, Self-Orthogonal Codes Constructed from Posets and Their Applications in Quantum Communication, Mathematics 8 (2020), 1495. https://doi.org/10.3390/math8091495.

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

Stabilizer generators

generators 19 (max weight 14; 19 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 (19, Pauli strings on 48 qubits)

YZZIIIIIIYIIIXYZIIIIIIIIYXIZIIIYIIIIXIYXIIIIIIII ZYIZIIIIYIIIXIZYIIIIIIIIXYZIIIYIIIIIIXXYIIIIIIII IIYZZIIIIIIYIIIXYZIIIIIIIIYXIZIIIYIIIIXIYXIIIIII IIZYIZIIIIYIIIXIZYIIIIIIIIXYZIIIYIIIIIIXXYIIIIII ZIIIYZIYIIIIYZIIIXIIIIIIIZIIYXIIIIIYYXIIXIIIIIII IIIIIIYZZIIIIIIYIIIXYZIIIIIIIIYXIZIIIYIIIIXIYXII IIIIIIZYIZIIIIYIIIXIZYIIIIIIIIXYZIIIYIIIIIIXXYII IIIIIIIIYZZIIIIIIYIIIXYZIIIIIIIIYXIZIIIYIIIIXIYX IIIIIIZIIIYZIYIIIIYZIIIXIIIIIIIZIIYXIIIIIYYXIIXI IIIIIIIZIIZYYIIIIIZYIIXIIIIIIIZIIIXYIIIIYIXYIIIX IXYZIIIIIIIIYZZIIIIIIYIIXIYXIIIIIIIIYXIZIIIYIIII XIZYIIIIIIIIZYIZIIIIYIIIIXXYIIIIIIIIXYZIIIYIIIII IIIXYZIIIIIIIIYZZIIIIIIYIIXIYXIIIIIIIIYXIZIIIYII IIXIZYIIIIIIIIZYIZIIIIYIIIIXXYIIIIIIIIXYZIIIYIII YZIIIXIIIIIIZIIIYZIYIIIIYXIIXIIIIIIIIZIIYXIIIIIY YIIIIIZYIIXIIIIIIIIZIIZYIIIIYIXYIIIXIIIIIIZIIIXY IIIIIYIIIXYZIIIIIIIIYZZIIIIYIIIIXIYXIIIIIIIIYXIZ IYIIIIYZIIIXIIIIIIZIIIYZIIIIIYYXIIXIIIIIIIIZIIYX IIYIIIXIZYIIIIIIIIZYIZIIYIIIIIIXXYIIIIIIIIXYZIII

symplectic rows (A | B) (19, sparse supports)

X: [0, 9, 13, 14, 24, 25, 31, 36, 38, 39] Z: [0, 1, 2, 9, 14, 15, 24, 27, 31, 38] X: [1, 8, 12, 15, 24, 25, 30, 37, 38, 39] Z: [0, 1, 3, 8, 14, 15, 25, 26, 30, 39] X: [2, 11, 15, 16, 26, 27, 33, 38, 40, 41] Z: [2, 3, 4, 11, 16, 17, 26, 29, 33, 40] X: [3, 10, 14, 17, 26, 27, 32, 39, 40, 41] Z: [2, 3, 5, 10, 16, 17, 27, 28, 32, 41] X: [4, 7, 12, 17, 28, 29, 35, 36, 37, 40] Z: [0, 4, 5, 7, 12, 13, 25, 28, 35, 36] X: [6, 15, 19, 20, 30, 31, 37, 42, 44, 45] Z: [6, 7, 8, 15, 20, 21, 30, 33, 37, 44] X: [7, 14, 18, 21, 30, 31, 36, 43, 44, 45] Z: [6, 7, 9, 14, 20, 21, 31, 32, 36, 45] X: [8, 17, 21, 22, 32, 33, 39, 44, 46, 47] Z: [8, 9, 10, 17, 22, 23, 32, 35, 39, 46] X: [10, 13, 18, 23, 34, 35, 41, 42, 43, 46] Z: [6, 10, 11, 13, 18, 19, 31, 34, 41, 42] X: [11, 12, 19, 22, 34, 35, 40, 42, 43, 47] Z: [7, 10, 11, 12, 18, 19, 30, 35, 40, 43] X: [1, 2, 12, 21, 24, 26, 27, 36, 37, 43] Z: [2, 3, 12, 13, 14, 21, 26, 36, 39, 43] X: [0, 3, 13, 20, 25, 26, 27, 36, 37, 42] Z: [2, 3, 12, 13, 15, 20, 27, 37, 38, 42] X: [3, 4, 14, 23, 26, 28, 29, 38, 39, 45] Z: [4, 5, 14, 15, 16, 23, 28, 38, 41, 45] X: [2, 5, 15, 22, 27, 28, 29, 38, 39, 44] Z: [4, 5, 14, 15, 17, 22, 29, 39, 40, 44] X: [0, 5, 16, 19, 24, 25, 28, 40, 41, 47] Z: [0, 1, 12, 16, 17, 19, 24, 37, 40, 47] X: [0, 7, 10, 23, 28, 30, 31, 35, 46, 47] Z: [0, 6, 7, 19, 22, 23, 28, 31, 42, 47] X: [5, 9, 10, 20, 27, 32, 34, 35, 44, 45] Z: [5, 10, 11, 20, 21, 22, 27, 34, 44, 47] X: [1, 6, 11, 22, 29, 30, 31, 34, 46, 47] Z: [1, 6, 7, 18, 22, 23, 29, 30, 43, 46] X: [2, 6, 9, 19, 24, 31, 32, 33, 42, 43] Z: [2, 8, 9, 18, 19, 21, 24, 33, 43, 44]

Code ID 48-29-4 · download JSON · raw on GitHub