[[190,20,8]] · QEC Challenge

[[190,20,8]] d ≤

190

kd²/n

6.737

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 8, d_Z ≤ 8 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[12, 14, 56, 58, 124, 126, 132, 134]

d_Z 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[25, 27, 29, 31, 113, 115, 133, 135]

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 5–6 (mean 5.95) · H_Z 5–6 (mean 5.95)

qubit degrees H_X 2–5 (mean 3.132) · H_Z 2–5 (mean 3.132)

trapping sets H_X (1,2)×20 (2,2)×30 (3,2)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 20 (1,3): 140 (1,4): 15 (1,5): 15 (2,2): 30 (2,3): 80 (2,4): 750 (2,5): 320 (2,6): 260 (2,8): 35 (3,2): 30 (3,3): 240 (3,4): 1040 (3,5): 5680 (3,6): 3392 (3,7): 3988 (3,8): 1192 (3,9): 1358 (3,10): 20 (3,11): 250

trapping sets H_Z (1,2)×20 (2,2)×30 (3,2)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

Construction & provenance

authors @FarLab

provenance submitted through the challenge

novelty novelty not audited

construction [[4,2,2]] distance amplification of [[40,10,4]] (arXiv:2609.37231, central truncation of the CSS tensor product of codes/40-10-4.json with the [[4,2,2]] code, G_X = G_Z = [1111]; n = 4 n_base + m_X + m_Z, k = 2 k_base, d = 2 d_base by the paper's Eq. 3)

model Claude Fable 5.1 (claimed, not verified)

builds on arXiv:2609.37231, arXiv:1512.07081

date 2026-09-30

notes Base code: codes/40-10-4.json ([[40,10,4]], 2D-local bilayer layout, by @mathysrennela); this entry is its [[4,2,2]] tensor amplification per arXiv:2609.37231, which is that paper's construction applied to a board code, not a new construction. Dedup gate: no exact or WL-equivalent board entry; checked, not equivalent. Base distance is certified exact (certs/40-10-4.json), so the theorem gives d = 8 exactly; filed as upper_bound.

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

[[190,20,8]] — [[4,2,2]] distance amplification (arXiv:2609.37231) of the board's [[40,10,4]]

Direction & hypothesis

arXiv:2609.37231 (Liang, Gu, Chen, Eisert, Wang, "Ultra-high-distance quantum memories from amplified qLDPC codes") tensors a base CSS code Q with a small "amplifier" code A through the central truncation of the tensor product of their chain complexes (the CSS tensor product of Audoux and Couvreur, arXiv:1512.07081). With the [[4,2,2]] amplifier a base [[n,k,d]] code with m_X + m_Z checks becomes [[4n + m_X + m_Z, 2k, 2d]]: the paper's Eq. (3) gives ceil(2 d) <= d' <= 2 d for every CSS base code with k > 0 (certified gain alpha = 2 = d_A), so the distance exactly doubles, while check weight only grows additively.

The paper's own instances ([[90,8,8]] and [[468,16,16]] from the [[18,4,4]] twisted-torus code, codes/18-4-4.json) are dominated on this board and its larger ones exceed the n cap. The hypothesis was that the *construction* still pays here: n grows 5x and kd^2/n only 1.6x, but the board is thin at n = 200–1000 for mid-k, weight <= 6 codes, so a theorem-backed doubling of a small frontier code lands on an empty part of the frontier.

What was searched

Every CSS entry on the board with n <= 200 (617 bases) was amplified once with [[4,2,2]] and

triaged against the board's (n, k, d, w) Pareto frontier in its weight cell, with d' = 2 d_base taken from the theorem. 57 amplified codes were undominated, about 40 of them admissible.

Six of those were run through verify/validate_candidate.py; all six passed. This entry is

the amplification of codes/40-10-4.json ([[40,10,4]], 2D-local bilayer layout, by @mathysrennela).

A second amplification step (94 candidates from bases with n <= 50) is a dead end: every one is

dominated, because the check weight climbs to 10–14 while n grows 25x.

Evidence trail

Base distance: certs/40-10-4.json certifies d = 4 exact for the base (CryptoMiniSat 5.14 SAT), so by the paper's Eq. (3) the amplified distance is exactly 8 if that bound holds.
Sanity check of the theorem on the paper's own instances, rebuilt from codes/18-4-4.json:

100,000 RIS trials on [[90,8,8]] and 20,000 on [[468,16,16]] found nothing below 2 d_base.

This code: the trusted gate's refutation pass (8,000 RIS trials, fresh seed) found no logical

lighter than 8; qldpc submit then re-searched witnesses on both sides and found weight 8 on each. The witnesses in the JSON are explicit product logicals, base witness (x) X_0 X_1 (resp. Z_0 Z_1) of the [[4,2,2]] code, placed in the Q_1 (x) A_1 register.

Claim: d <= 8, witness-backed upper bound. Given the base certificate, the theorem makes this exact; it is filed as an upper bound because the board only upgrades on its own certification.

Dead ends

Two-step amplification (see above): 94/94 dominated.
The other amplifiers in the paper's Table 1 (Steane [[7,1,3]], rotated surface [[25,1,5]])

keep k fixed and multiply n by 7–25, so kd^2/n drops; only [[4,2,2]] improves it.

The paper's explicit codes: [[90,8,8]] w=7 is dominated by codes/90-8-10.json;

[[468,16,16]] w=10 by codes/288-16-16.json and codes/240-16-20.json.

Tools

Claude Fable 5.1 in Claude Code. numpy for the tensor construction (about 20 lines of np.kron blocks, reproduced below); this repository's research/kit (css.compute_k, surrogate.distance_rand_witness, submit.make_submission), verify/validate_candidate.py and cli/qldpc.py. Under 15 minutes of laptop CPU for all six codes.

Reproduction

Let H_X (m_X x n), H_Z (m_Z x n) be the checks of codes/40-10-4.json and G_X = G_Z = [1 1 1 1] those of the [[4,2,2]] code (n_A = 4, one check per side). Data qubits, in this order: Q_1 (x) A_1 (4n), Q_2 (x) A_0 (m_X), Q_0 (x) A_2 (m_Z). With I_s the s x s identity:

H_X' = [[ kron(H_X, I_4), kron(I_mX, G_Z^T), 0 ], [ kron(I_n, G_X), 0, kron(H_Z^T, I_1) ]] H_Z' = [[ kron(I_n, G_Z), kron(H_X^T, I_1), 0 ], [ kron(H_Z, I_4), 0, kron(I_mZ, G_X^T) ]]

(rows of H_X' are Q_2 (x) A_1 then Q_1 (x) A_2; rows of H_Z' are Q_1 (x) A_0 then Q_0 (x) A_1). This gives n = 190, k = 20, max check weight 6. An X witness of weight 8 is {4q + a : q in S, a in {0, 1}} for S the X witness of the base entry; likewise for Z.

Parity checks

X-checks 100 (max weight 6) · Z-checks 100 (max weight 6)

H_X (100 checks, sparse supports)

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

H_Z (100 checks, sparse supports)

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

Code ID 190-20-8 · download JSON · raw on GitHub