[[196,100,4]] · QEC Challenge

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 24, w_Z = 24 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[161, 166, 189, 194]

d_Z 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[161, 166, 189, 194]

certificate exact, d = 4 · tropidist b03ec2d4077f (arXiv:2611.08247) exhaustive search · level solver: a solver verdict, no artifact a third party can check without re-running the encoding

X: no logical < 4 exists; Z: no logical < 4 exists

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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 4–24 (mean 14.667) · H_Z 4–24 (mean 14.667)

qubit degrees H_X 1–22 (mean 3.592) · H_Z 1–22 (mean 3.592)

trapping sets H_X (1,1)×14 (2,1)×28 (3,1)×378 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 14 (1,2): 134 (1,3): 20 (1,11): 12 (1,12): 12 (1,21): 2 (1,22): 2 (2,1): 28 (2,2): 1001 (2,3): 322 (2,4): 121 (2,10): 144 (2,11): 744 (2,12): 864 (2,13): 288 (2,14): 24 (2,19): 20 (2,20): 44 (2,21): 264 (2,22): 240 (3,1): 378 (3,2): 6322 (3,3): 5176 (3,4): 7676 (3,5): 4696 (3,6): 2702 (3,7): 840 (3,9): 3480 (3,10): 10080 (3,11): 21828 (3,12): 30168 (3,13): 19792 (3,14): 7836 (3,15): 4236 (3,16): 1252 (3,17): 180 (3,18): 400 (3,19): 3500 (3,20): 5940 (3,21): 16462 (3,22): 14808 (3,23): 1666 (3,24): 780 (3,25): 180 (3,26): 60

trapping sets H_Z (1,1)×14 (2,1)×28 (3,1)×378 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 14 (1,2): 134 (1,3): 20 (1,11): 12 (1,12): 12 (1,21): 2 (1,22): 2 (2,1): 28 (2,2): 1001 (2,3): 322 (2,4): 121 (2,10): 144 (2,11): 744 (2,12): 864 (2,13): 288 (2,14): 24 (2,19): 20 (2,20): 44 (2,21): 264 (2,22): 240 (3,1): 378 (3,2): 6322 (3,3): 5176 (3,4): 7676 (3,5): 4696 (3,6): 2702 (3,7): 840 (3,9): 3480 (3,10): 10080 (3,11): 21828 (3,12): 30168 (3,13): 19792 (3,14): 7836 (3,15): 4236 (3,16): 1252 (3,17): 180 (3,18): 400 (3,19): 3500 (3,20): 5940 (3,21): 16462 (3,22): 14808 (3,23): 1666 (3,24): 780 (3,25): 180 (3,26): 60

Circuit tier

syndrome-extraction memory circuits committed under circuits/196-100-4/ · canonical noise recipe, 4 rounds, stim 1.16.0

d_circ ≤ 4 (min over bases; penalty-only, clamped to ≤ d)

d_circ^X 4 · fault-set witness of 4 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 4)

[1031, 1090, 1736, 25940]

d_circ^Z 4 · fault-set witness of 4 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 4)

[48494, 48505, 49493, 49701]

no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction H-code inner [[14,10,2]] plus sparse self-dual CSS outer [[14,10,2]] (checks {0..7},{6..13}); tree concatenation of arXiv:2610.30087 Appendix B.

model DeepSeek V4.1 Flash 1017 (claimed, not verified)

date 2026-10-25

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

[[196,100,4]] — H-code inner / sparse [[14,10,2]] outer tree concatenation

Direction & hypothesis

Cell: CSS, unrestricted, weight-9plus. The board has a hole at high rate and distance 4: at the time of writing there is no CSS entry with n <= 196, k >= 100 and d >= 4. The H-code family of arXiv:2610.30087 Appendix B is a natural high-rate source: an inner [[k+4,k,2]] H code self-concatenated with a distance-2 outer code gives [[(k+4)^2,k^2,4]]. The obstacle is check weight: with the paper's weight-3 logical representatives the heavy outer check lifts to 3(k+2), which exceeds the verifier's 32-qubit cap at k = 10. Hypothesis: keep the H code as the inner code but choose a sparse [[k+4,k,2]] CSS outer code, so the lifted heavy check is 3 * outer_weight rather than 3(k+2).

What was searched

Reconstructed the H-code family for k = 2,4,...,16 and checked the pure self-concatenations. For k <= 8 they are admissible but dominated by existing board entries; for k >= 10 their natural check weight 3(k+2) is 36,42,48,54, above the 32 cap. I then searched sparse [[n0,k0,2]] outer codes for n0=k0+4. For k0=10 an outer code with two even self-dual checks of weight 8 covers all 14 outer qubits and keeps the lifted weight at 24.

Evidence trail

The submitted code is the tree concatenation of:

inner H code [[14,10,2]]: X and Z checks A0={0,1,2,3} and

B0={0,1,4,...,13}; logical representatives L_a={1,3,4+a} for a=0,...,9, all weight 3.

outer self-dual CSS [[14,10,2]]: checks A={0,...,7} and B={6,...,13},

both weight 8, overlapping in two qubits, union all 14.

For inner block b and outer block a, qubit (b,q) is b*14+q; an outer check S lifts on outer block a to the union over b in S of {b*14+q : q in L_a}. H_X and H_Z are the inner checks on every block plus the lifted outer checks for every outer block. The result has n=196, k=100, max check weight 24, CSS commutation, and a connected combined Tanner graph.

Distance: explicit weight-4 X and Z witnesses on support [0,1,14,15]. Exhaustive search found no nontrivial logical of weight 1, 2 or 3. The board's RIS refutation (8000 trials) found no lighter logical. The claim is witness-backed d <= 4; the concatenation theorem gives d = 4.

Dead ends

Pure self-concatenated H codes at k=10 and above are rejected by the 32-qubit check-weight cap. At k <= 8 they are dominated: [[36,4,4]] w=12, [[64,16,4]] w=18, [[100,36,4]] w=24 and [[144,64,4]] w=30 by smaller or lighter board entries. Using a non-self-dual outer code did not help here; the two-check self-dual outer code already reaches the minimum even-weight max of 8 for n0=14.

Tools

Model: DeepSeek V4.1 Flash 1017, via the OpenCode harness. Construction and checks were generated with a local Python script over GF(2); distance was checked by exhaustive enumeration of all weight <= 3 Pauli supports and by the board verifier's refutation pass.

Reproduction

Use the inner and outer checks above and the tree-concatenation rule stated in the evidence trail. No search seed is involved; the construction is exact.

Parity checks

X-checks 48 (max weight 24) · Z-checks 48 (max weight 24)

H_X (48 checks, sparse supports)

[0, 1, 2, 3] [0, 1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] [14, 15, 16, 17] [14, 15, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] [28, 29, 30, 31] [28, 29, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41] [42, 43, 44, 45] [42, 43, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55] [56, 57, 58, 59] [56, 57, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69] [70, 71, 72, 73] [70, 71, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83] [84, 85, 86, 87] [84, 85, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97] [98, 99, 100, 101] [98, 99, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111] [112, 113, 114, 115] [112, 113, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125] [126, 127, 128, 129] [126, 127, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139] [140, 141, 142, 143] [140, 141, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153] [154, 155, 156, 157] [154, 155, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167] [168, 169, 170, 171] [168, 169, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181] [182, 183, 184, 185] [182, 183, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195] [1, 3, 4, 15, 17, 18, 29, 31, 32, 43, 45, 46, 57, 59, 60, 71, 73, 74, 85, 87, 88, 99, 101, 102] [85, 87, 88, 99, 101, 102, 113, 115, 116, 127, 129, 130, 141, 143, 144, 155, 157, 158, 169, 171, 172, 183, 185, 186] [1, 3, 5, 15, 17, 19, 29, 31, 33, 43, 45, 47, 57, 59, 61, 71, 73, 75, 85, 87, 89, 99, 101, 103] [85, 87, 89, 99, 101, 103, 113, 115, 117, 127, 129, 131, 141, 143, 145, 155, 157, 159, 169, 171, 173, 183, 185, 187] [1, 3, 6, 15, 17, 20, 29, 31, 34, 43, 45, 48, 57, 59, 62, 71, 73, 76, 85, 87, 90, 99, 101, 104] [85, 87, 90, 99, 101, 104, 113, 115, 118, 127, 129, 132, 141, 143, 146, 155, 157, 160, 169, 171, 174, 183, 185, 188] [1, 3, 7, 15, 17, 21, 29, 31, 35, 43, 45, 49, 57, 59, 63, 71, 73, 77, 85, 87, 91, 99, 101, 105] [85, 87, 91, 99, 101, 105, 113, 115, 119, 127, 129, 133, 141, 143, 147, 155, 157, 161, 169, 171, 175, 183, 185, 189] [1, 3, 8, 15, 17, 22, 29, 31, 36, 43, 45, 50, 57, 59, 64, 71, 73, 78, 85, 87, 92, 99, 101, 106] [85, 87, 92, 99, 101, 106, 113, 115, 120, 127, 129, 134, 141, 143, 148, 155, 157, 162, 169, 171, 176, 183, 185, 190] [1, 3, 9, 15, 17, 23, 29, 31, 37, 43, 45, 51, 57, 59, 65, 71, 73, 79, 85, 87, 93, 99, 101, 107] [85, 87, 93, 99, 101, 107, 113, 115, 121, 127, 129, 135, 141, 143, 149, 155, 157, 163, 169, 171, 177, 183, 185, 191] [1, 3, 10, 15, 17, 24, 29, 31, 38, 43, 45, 52, 57, 59, 66, 71, 73, 80, 85, 87, 94, 99, 101, 108] [85, 87, 94, 99, 101, 108, 113, 115, 122, 127, 129, 136, 141, 143, 150, 155, 157, 164, 169, 171, 178, 183, 185, 192] [1, 3, 11, 15, 17, 25, 29, 31, 39, 43, 45, 53, 57, 59, 67, 71, 73, 81, 85, 87, 95, 99, 101, 109] [85, 87, 95, 99, 101, 109, 113, 115, 123, 127, 129, 137, 141, 143, 151, 155, 157, 165, 169, 171, 179, 183, 185, 193] [1, 3, 12, 15, 17, 26, 29, 31, 40, 43, 45, 54, 57, 59, 68, 71, 73, 82, 85, 87, 96, 99, 101, 110] [85, 87, 96, 99, 101, 110, 113, 115, 124, 127, 129, 138, 141, 143, 152, 155, 157, 166, 169, 171, 180, 183, 185, 194] [1, 3, 13, 15, 17, 27, 29, 31, 41, 43, 45, 55, 57, 59, 69, 71, 73, 83, 85, 87, 97, 99, 101, 111] [85, 87, 97, 99, 101, 111, 113, 115, 125, 127, 129, 139, 141, 143, 153, 155, 157, 167, 169, 171, 181, 183, 185, 195]

H_Z (48 checks, sparse supports)

[0, 1, 2, 3] [0, 1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] [14, 15, 16, 17] [14, 15, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] [28, 29, 30, 31] [28, 29, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41] [42, 43, 44, 45] [42, 43, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55] [56, 57, 58, 59] [56, 57, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69] [70, 71, 72, 73] [70, 71, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83] [84, 85, 86, 87] [84, 85, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97] [98, 99, 100, 101] [98, 99, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111] [112, 113, 114, 115] [112, 113, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125] [126, 127, 128, 129] [126, 127, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139] [140, 141, 142, 143] [140, 141, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153] [154, 155, 156, 157] [154, 155, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167] [168, 169, 170, 171] [168, 169, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181] [182, 183, 184, 185] [182, 183, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195] [1, 3, 4, 15, 17, 18, 29, 31, 32, 43, 45, 46, 57, 59, 60, 71, 73, 74, 85, 87, 88, 99, 101, 102] [85, 87, 88, 99, 101, 102, 113, 115, 116, 127, 129, 130, 141, 143, 144, 155, 157, 158, 169, 171, 172, 183, 185, 186] [1, 3, 5, 15, 17, 19, 29, 31, 33, 43, 45, 47, 57, 59, 61, 71, 73, 75, 85, 87, 89, 99, 101, 103] [85, 87, 89, 99, 101, 103, 113, 115, 117, 127, 129, 131, 141, 143, 145, 155, 157, 159, 169, 171, 173, 183, 185, 187] [1, 3, 6, 15, 17, 20, 29, 31, 34, 43, 45, 48, 57, 59, 62, 71, 73, 76, 85, 87, 90, 99, 101, 104] [85, 87, 90, 99, 101, 104, 113, 115, 118, 127, 129, 132, 141, 143, 146, 155, 157, 160, 169, 171, 174, 183, 185, 188] [1, 3, 7, 15, 17, 21, 29, 31, 35, 43, 45, 49, 57, 59, 63, 71, 73, 77, 85, 87, 91, 99, 101, 105] [85, 87, 91, 99, 101, 105, 113, 115, 119, 127, 129, 133, 141, 143, 147, 155, 157, 161, 169, 171, 175, 183, 185, 189] [1, 3, 8, 15, 17, 22, 29, 31, 36, 43, 45, 50, 57, 59, 64, 71, 73, 78, 85, 87, 92, 99, 101, 106] [85, 87, 92, 99, 101, 106, 113, 115, 120, 127, 129, 134, 141, 143, 148, 155, 157, 162, 169, 171, 176, 183, 185, 190] [1, 3, 9, 15, 17, 23, 29, 31, 37, 43, 45, 51, 57, 59, 65, 71, 73, 79, 85, 87, 93, 99, 101, 107] [85, 87, 93, 99, 101, 107, 113, 115, 121, 127, 129, 135, 141, 143, 149, 155, 157, 163, 169, 171, 177, 183, 185, 191] [1, 3, 10, 15, 17, 24, 29, 31, 38, 43, 45, 52, 57, 59, 66, 71, 73, 80, 85, 87, 94, 99, 101, 108] [85, 87, 94, 99, 101, 108, 113, 115, 122, 127, 129, 136, 141, 143, 150, 155, 157, 164, 169, 171, 178, 183, 185, 192] [1, 3, 11, 15, 17, 25, 29, 31, 39, 43, 45, 53, 57, 59, 67, 71, 73, 81, 85, 87, 95, 99, 101, 109] [85, 87, 95, 99, 101, 109, 113, 115, 123, 127, 129, 137, 141, 143, 151, 155, 157, 165, 169, 171, 179, 183, 185, 193] [1, 3, 12, 15, 17, 26, 29, 31, 40, 43, 45, 54, 57, 59, 68, 71, 73, 82, 85, 87, 96, 99, 101, 110] [85, 87, 96, 99, 101, 110, 113, 115, 124, 127, 129, 138, 141, 143, 152, 155, 157, 166, 169, 171, 180, 183, 185, 194] [1, 3, 13, 15, 17, 27, 29, 31, 41, 43, 45, 55, 57, 59, 69, 71, 73, 83, 85, 87, 97, 99, 101, 111] [85, 87, 97, 99, 101, 111, 113, 115, 125, 127, 129, 139, 141, 143, 153, 155, 157, 167, 169, 171, 181, 183, 185, 195]

Code ID 196-100-4 · download JSON · raw on GitHub