[[48,11,8]] · QEC Challenge

[[48,11,8]] d ≤stabilizer

kd²/n

14.667

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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 8 · witness Pauli weight 8 (claimed upper_bound)

witness operator (Pauli string, 8 qubits)

IIIIIIIIIIIIXIXXIXIIIIIIIXIIIXXIXIIIIIIIIIIIIIII X: [12, 14, 15, 17, 25, 29, 30, 32] 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 10–14 (mean 10.108)

qubit degrees S 5–9 (mean 7.792)

trapping sets S (1,5)×1 (2,5)×7 (3,5)×2 (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): 4 (1,7): 13 (1,8): 16 (1,9): 14 (2,5): 7 (2,6): 12 (2,7): 16 (2,8): 35 (2,9): 66 (2,10): 87 (2,11): 127 (2,12): 182 (2,13): 164 (2,14): 99 (2,15): 52 (2,16): 13 (3,5): 2 (3,6): 9 (3,7): 41 (3,8): 117 (3,9): 277 (3,10): 511 (3,11): 815 (3,12): 1231 (3,13): 1560 (3,14): 1962 (3,15): 2121 (3,16): 2020 (3,17): 1668 (3,18): 1186 (3,19): 751 (3,20): 355 (3,21): 89 (3,22): 17 (3,23): 1

Construction & provenance

authors @hirobumii

provenance submitted through the challenge

novelty novelty not audited

construction A rank37 hyperplane of the literal FarLab public [[48,10,8]] 2BGA stabilizer, obtained by identifying selected X-index2 and Z-index16 coefficient parities. A minimum-incidence same-group basis has w14,D9,M37,I374. Exact general-Pauli distance follows from fresh complete parent and singleton syndrome enumeration through7. Sparse resource-presentation study; no parameter or literature-wide novelty claim.

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/pdf/2306.16400, https://github.com/unitaryfoundation/qldpc-challenge/blob/d8208c8da17e7db2285bacb62f99db9cd4863234/codes/48-10-8.json, A. M. Steane, Enlargement of Calderbank-Shor-Steane quantum codes, arXiv:quant-ph/9802061v2.

date 2026-10-18

notes Displayed presentation: maximum check weight 14, maximum qubit degree 9, 37 checks and total incidence 374.

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

[[48,11,8]] - general stabilizer subgroup code

Direction & hypothesis

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

What was searched

A rank37 hyperplane of the literal FarLab public [[48,10,8]] 2BGA stabilizer, obtained by identifying selected X-index2 and Z-index16 coefficient parities. Exact general-Pauli distance follows from fresh complete parent and singleton syndrome enumeration through7.

The displayed presentation has maximum check weight 14, maximum qubit degree 9, 37 checks and total incidence 374.

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/48-11-8.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: "34359738372".

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 parent coefficient masks: ["2","16","32","8","4096","262144","274627","32768","16384","495993","64","128","131072","1024","65536","2048","256","512","2147483648","4294967296","8589934592","17179869184","4194304","137438953472","2097152","16777216","268435456","8388608","134217728","234669735936","1048576","33554432","1073741824","524288","67108864","536870912","34359738372"].

Merged parent basis indices: [2,16].

Coefficient masks use the selected X parent rows first, then selected Z parent rows; bit 19 starts the Z sector. Sum selected supports over GF(2).

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/pdf/2306.16400
A. M. Steane, Enlargement of Calderbank-Shor-Steane quantum codes, arXiv:quant-ph/9802061v2.

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

Stabilizer generators

generators 37 (max weight 14; 1 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 (37, Pauli strings on 48 qubits)

IXIIIIIIXIIIXIIXIIIIIIIIXXIIIIXIIIIIIXXXIIIIIIII IIIIXIIXIIIIXIIIIXIIIIIIIIIIXXIIIIIXXXIIXIIIIIII IIIIIXXIIIIIIXIIXIIIIIIIIIIIXXIIIIXIXXIIIXIIIIII IIIXIIIIIIXIIIXIIXIIIIIIIIXXIIIIXIIIIIIXXXIIIIII IXXIIIIIIIIIXIIIIIIIIXIIXIXXIIIIIIIIXXIIIIIXIIII IIIXIIIXXIIIIIIIIIXIIIIIIXIIIIXIXXIIIIIIIIXXIIII IIXIIIXIIXIIIIIIIIIXIIIIXIIIIIIXXXIIIIIIIIXXIIII IIXIIXIIIIIIIIIXIIIIIIXIIIIXXXIIIIIIIIXXIIIIXIII IIIXXIIIIIIIIIXIIIIIIIIXIIXIXXIIIIIIIIXXIIIIIXII IIIIIXIIIXXIIIIIIIIIXIIIIIIXIIIIXIXXIIIIIIIIXXII IIIIIIXIIIIIIIIXIIIXXIIIIIIIIIXXIIIIIXIIIIXIXXII IIIIIIIXIIIIIIXIIIXIIXIIIIIIIIXXIIIIXIIIIIIXXXII IXIIXIIIIIIIIIIIIXXIIIIIXXIIIXIIIIIIIIIIXXIIIIXI IIIIIIIIIIXIIXIIIIXIIIIXIIIIIIIIIIXXIIIIIXXXIIXI XIIIIXIIIIIIIIIIXIIXIIIIXXIIXIIIIIIIIIIIXXIIIIIX IIIIIIIIIIIXXIIIIIIXIIXIIIIIIIIIIIXXIIIIXIXXIIIX IIIIIIIIXIIIIIIIIXIIIXXIIIIIIIIIXXIIIIIXIIIIXIXX IIIIIIIIIXIIIIIIXIIIXIIXIIIIIIIIXXIIIIXIIIIIIXXX ZIIIZZIZIIIIZZIIIIIIIIIIIZIIZIIIIIIZZIIIIIIIIIII IZIIZZZIIIIIZZIIIIIIIIIIZIIIIZIIIIZIIZIIIIIIIIII ZZZIIIIIIZIIIIZZIIIIIIIIZIIZIIIZIIIIIIZIIIIIIIII ZZIZIIIIZIIIIIZZIIIIIIIIIZZIIIZIIIIIIIIZIIIIIIII IIZZIIIIIIIIZZIZIIIIZIIIIIIZIIIIIIIIIZZIIIZIIIII IIIIIIZIIIZZIZIIIIZZIIIIIIIIIIIZIIZIIIIIIZZIIIII IIZZIIIIIIIIZZZIIIIIIZIIIIZIIIIIIIIIZIIZIIIZIIII IIIIZZIIIIIIIIZZIZIIIIZIIIIIIZIIIIIIIIIZZIIIZIII IIZIIIIIZZIIIIIIIIZZIZIIZIIIIIIIIZIIIIIIIIIZZIII IIIIZZIIIIIIIIZZZIIIIIIZIIIIZIIIIIIIIIZIIZIIIZII IIIZIIIIZZIIIIIIIIZZZIIIIZIIIIIIZIIIIIIIIIZIIZII IIIIIIIIZZZIIIIIIZIIIIZZIIIIIIIIZIIZIIIZIIIIIIZI ZZIIIIIIIIIIIZIIZZZIIIIIIZIIIIIIIIIIZIIIIZIIIIZI IZIIIIZZIIIIIIIIIIZIIIZZIIIIIZZIIIIIIIIIIIIZIIZI IIIIZIIIIIZZIIIIIIIIZZIZIIZIIIIIIIIZIIIIIIIIIZZI ZZIIIIIIIIIIZIIIZZIZIIIIZIIIIIIIIIIIIZIIZIIIIIIZ ZIIIIIZZIIIIIIIIIIIZIIZZIIIIZIIZIIIIIIIIIIZIIIIZ IIIIIZIIIIZZIIIIIIIIZZZIIIIZIIIIIIZIIIIIIIIIZIIZ IIYZZIIIIIIYIIIXYZIIIIIIIIYXIZIIIYIIIIXIYXIIIIII

symplectic rows (A | B) (37, sparse supports)

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

Code ID 48-11-8 · download JSON · raw on GitHub