[[192,130,3]] · QEC Challenge

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[[192,130,3]] d ≤

192

130

kd²/n

6.09375

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 3, d_Z ≤ 3 · w_X = 12, w_Z = 12 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[173, 175, 191]

d_Z 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[24, 29, 33]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

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

girth H_X 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 12 · H_Z 12

qubit degrees H_X 2 · H_Z 2

trapping sets H_X (1,2)×192 (2,2)×2112 (3,0)×256 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 192 (2,2): 2112 (3,0): 256 (3,2): 22464 (3,4): 7040

trapping sets H_Z (1,2)×192 (2,2)×2112 (3,0)×256 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 192 (2,2): 2112 (3,0): 256 (3,2): 22464 (3,4): 7040

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction contributed via qldpc submit

model Beam 1.0 (claimed, not verified)

date 2026-11-09

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

[[192,130,3]] — FCC K=12 lattice code (L=4)

Direction & hypothesis

Targeted the unrestricted / weight-9plus cell with a high-rate constant-distance CSS code. The FCC lattice has 3L^3 edges but only L^3 independent stabilizer constraints, leaving ~2/3 of degrees of freedom as logical qubits (arXiv:2603.20294).

What was searched

Construction: qubits on edges of the L=4 face-centered cubic lattice; Z-stabilizers on vertices (weight 12), X-stabilizers on octahedral voids (weight 12). Parameters k = 2L^3+2 = 130, n = 3L^3 = 192, d = 3 proven by exhaustive elimination of all weight-1 and weight-2 vectors plus constructive weight-3 logicals (arXiv:2603.20294 Appendix B).

Evidence trail

Distance d = 3 (upper bound): witness-backed weight-3 logicals on both X and Z sides. CSS commutation verified; k recomputed by GF(2) rank. Gate passed (passed: true).

Dead ends

L=2 instance refuted (weight-2 logical found). L=8 exceeds submission n cap (1000).

Tools

Reconstructed from arXiv:2603.20294 Appendix A Python code. Model: Beam 1.0.

Reproduction

Build from arXiv:2603.20294 Appendix A (build(L) with L=4). Check matrices H_X, H_Z as in paper.

Benchmark comparison (optional, Tier B/C)

Not applicable.

Parity checks

X-checks 32 (max weight 12) · Z-checks 32 (max weight 12)

H_X (32 checks, sparse supports)

[4, 6, 8, 10, 17, 19, 21, 23, 25, 27, 64, 66] [5, 7, 9, 11, 16, 18, 20, 22, 35, 37, 72, 74] [0, 2, 8, 9, 29, 31, 33, 38, 40, 42, 45, 47] [12, 14, 20, 21, 28, 30, 32, 39, 41, 43, 55, 57] [24, 26, 32, 33, 48, 50, 52, 59, 61, 63, 65, 67] [34, 36, 42, 43, 49, 51, 53, 58, 60, 62, 73, 75] [1, 3, 10, 11, 44, 46, 52, 53, 69, 71, 76, 78] [13, 15, 22, 23, 54, 56, 62, 63, 68, 70, 77, 79] [0, 1, 4, 5, 83, 85, 87, 90, 92, 94, 97, 120] [12, 13, 16, 17, 82, 84, 86, 91, 93, 95, 103, 124] [24, 25, 28, 29, 80, 84, 85, 98, 100, 105, 107, 109] [34, 35, 38, 39, 88, 92, 93, 99, 101, 104, 106, 115] [44, 45, 48, 49, 96, 100, 101, 111, 113, 116, 118, 121] [54, 55, 58, 59, 102, 106, 107, 110, 112, 117, 119, 125] [64, 65, 68, 69, 81, 86, 87, 108, 112, 113, 122, 127] [72, 73, 76, 77, 89, 94, 95, 114, 118, 119, 123, 126] [80, 81, 82, 83, 130, 132, 134, 139, 141, 143, 145, 168] [88, 89, 90, 91, 131, 133, 135, 138, 140, 142, 151, 172] [96, 97, 98, 99, 128, 132, 133, 147, 149, 152, 154, 157] [102, 103, 104, 105, 136, 140, 141, 146, 148, 153, 155, 163] [108, 109, 110, 111, 144, 148, 149, 158, 160, 165, 167, 169] [114, 115, 116, 117, 150, 154, 155, 159, 161, 164, 166, 173] [120, 121, 122, 123, 129, 134, 135, 156, 160, 161, 171, 174] [124, 125, 126, 127, 137, 142, 143, 162, 166, 167, 170, 175] [2, 3, 6, 7, 128, 129, 130, 131, 177, 179, 180, 182] [14, 15, 18, 19, 136, 137, 138, 139, 176, 178, 181, 183] [26, 27, 30, 31, 144, 145, 146, 147, 176, 177, 184, 187] [36, 37, 40, 41, 150, 151, 152, 153, 180, 181, 185, 186] [46, 47, 50, 51, 156, 157, 158, 159, 184, 185, 189, 190] [56, 57, 60, 61, 162, 163, 164, 165, 186, 187, 188, 191] [66, 67, 70, 71, 168, 169, 170, 171, 178, 179, 188, 189] [74, 75, 78, 79, 172, 173, 174, 175, 182, 183, 190, 191]

H_Z (32 checks, sparse supports)

[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] [8, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33] [9, 20, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43] [33, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53] [32, 43, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63] [10, 23, 52, 63, 64, 65, 66, 67, 68, 69, 70, 71] [11, 22, 53, 62, 72, 73, 74, 75, 76, 77, 78, 79] [4, 17, 25, 64, 80, 81, 82, 83, 84, 85, 86, 87] [5, 16, 35, 72, 88, 89, 90, 91, 92, 93, 94, 95] [0, 29, 38, 45, 85, 92, 96, 97, 98, 99, 100, 101] [12, 28, 39, 55, 84, 93, 102, 103, 104, 105, 106, 107] [24, 48, 59, 65, 100, 107, 108, 109, 110, 111, 112, 113] [34, 49, 58, 73, 101, 106, 114, 115, 116, 117, 118, 119] [1, 44, 69, 76, 87, 94, 113, 118, 120, 121, 122, 123] [13, 54, 68, 77, 86, 95, 112, 119, 124, 125, 126, 127] [83, 90, 97, 120, 128, 129, 130, 131, 132, 133, 134, 135] [82, 91, 103, 124, 136, 137, 138, 139, 140, 141, 142, 143] [80, 98, 105, 109, 132, 141, 144, 145, 146, 147, 148, 149] [88, 99, 104, 115, 133, 140, 150, 151, 152, 153, 154, 155] [96, 111, 116, 121, 149, 154, 156, 157, 158, 159, 160, 161] [102, 110, 117, 125, 148, 155, 162, 163, 164, 165, 166, 167] [81, 108, 122, 127, 134, 143, 160, 167, 168, 169, 170, 171] [89, 114, 123, 126, 135, 142, 161, 166, 172, 173, 174, 175] [6, 19, 27, 66, 130, 139, 145, 168, 176, 177, 178, 179] [7, 18, 37, 74, 131, 138, 151, 172, 180, 181, 182, 183] [2, 31, 40, 47, 128, 147, 152, 157, 177, 180, 184, 185] [14, 30, 41, 57, 136, 146, 153, 163, 176, 181, 186, 187] [26, 50, 61, 67, 144, 158, 165, 169, 184, 187, 188, 189] [36, 51, 60, 75, 150, 159, 164, 173, 185, 186, 190, 191] [3, 46, 71, 78, 129, 156, 171, 174, 179, 182, 189, 190] [15, 56, 70, 79, 137, 162, 170, 175, 178, 183, 188, 191]

Code ID 192-130-3 · download JSON · raw on GitHub