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[[210,26,14]] d ≤
n
210
k
26
d
14
kd²/n
24.267
w
8
X/Z
1
g
0.0101
r
7.0
layers
2
swaps
1443

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Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[2, 13, 40, 60, 61, 63, 80, 86, 103, 107, 108, 135, 166, 192]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[1, 21, 58, 73, 93, 94, 105, 138, 140, 141, 142, 156, 176, 178]
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 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×210 (2,6)×2940 (3,6)×1680 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 210 (2,6): 2940 (3,6): 1680 (3,8): 56700 (3,10): 5880
trapping sets H_Z (1,4)×210 (2,6)×2940 (3,6)×1680 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 210 (2,6): 2940 (3,6): 1680 (3,8): 56700 (3,10): 5880
witness diameter X 10.5357 · Z 5.5678 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
layout contributed by @MicheldeMessieres · Simulated annealing of qubit sites on a unit triangular lattice (2 per site) with a hinge cost on checks above the cap; same approach as the board's research/local2d/hinge_anneal.py and fold_layout.py, separate implementation. Search driven by Claude Opus 5.5 (Claude Code). · 2026-10-09
r = 7
X checkZ checkqubit site (110)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 1443 nearest-neighbor SWAPs per round in total, at most 12 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @seunomonije
provenance submitted through the challenge
novelty novelty not audited
construction Generalized bicycle code on Z_105: H_X=[A|B], H_Z=[B^T|A^T], A,B circulants of a = 1+x+x57+x75, b = 1+x2+x17+x84 over GF(2)[x]/(x105-1). Divisor trick: a and b are weight-4 codewords of the classical cyclic code <g>, g = gcd(a,b,x105-1) = x13+x11+x10+x7+x4+x3+x+1 (deg 13), so k = 2*deg g = 26 by the Panteleev-Kalachev closed form; check weight 8 (4+4). Weight-4 codewords enumerated exhaustively per irreducible factor of x105-1 by GF(2^m) root tests + meet-in-the-middle; pair selected by distance screening among 65 sibling pairs.
model Claude Claude Fable 5 (claimed, not verified)
date 2026-07-17
notes Distance evidence: rising RIS ladder 20k/80k/320k/1.28M/2M trials flat at 14; refutation barrage 3x8M trials fresh seeds all 14; both-side weight-14 witnesses from 8M-trial search (Z witness via GB transpose symmetry, verified against ker/rowspace). Literature novelty unverified; nearest known: univariate-bicycle [[234,26,14]] w8 (arXiv:2605.14173), which this does not duplicate (different n, generic GB pair).
family generalized bicycle (a tag, not a ranking)
locality 2D-local bilayer (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

no research note was staged with this submission — notes are requested for new submissions (notes/README.md)

Parity checks

X-checks 105 (max weight 8) · Z-checks 105 (max weight 8)
H_X (105 checks, sparse supports)
[0, 1, 57, 75, 105, 107, 122, 189] [1, 2, 58, 76, 106, 108, 123, 190] [2, 3, 59, 77, 107, 109, 124, 191] [3, 4, 60, 78, 108, 110, 125, 192] [4, 5, 61, 79, 109, 111, 126, 193] [5, 6, 62, 80, 110, 112, 127, 194] [6, 7, 63, 81, 111, 113, 128, 195] [7, 8, 64, 82, 112, 114, 129, 196] [8, 9, 65, 83, 113, 115, 130, 197] [9, 10, 66, 84, 114, 116, 131, 198] [10, 11, 67, 85, 115, 117, 132, 199] [11, 12, 68, 86, 116, 118, 133, 200] [12, 13, 69, 87, 117, 119, 134, 201] [13, 14, 70, 88, 118, 120, 135, 202] [14, 15, 71, 89, 119, 121, 136, 203] [15, 16, 72, 90, 120, 122, 137, 204] [16, 17, 73, 91, 121, 123, 138, 205] [17, 18, 74, 92, 122, 124, 139, 206] [18, 19, 75, 93, 123, 125, 140, 207] [19, 20, 76, 94, 124, 126, 141, 208] [20, 21, 77, 95, 125, 127, 142, 209] [21, 22, 78, 96, 105, 126, 128, 143] [22, 23, 79, 97, 106, 127, 129, 144] [23, 24, 80, 98, 107, 128, 130, 145] [24, 25, 81, 99, 108, 129, 131, 146] [25, 26, 82, 100, 109, 130, 132, 147] [26, 27, 83, 101, 110, 131, 133, 148] [27, 28, 84, 102, 111, 132, 134, 149] [28, 29, 85, 103, 112, 133, 135, 150] [29, 30, 86, 104, 113, 134, 136, 151] [0, 30, 31, 87, 114, 135, 137, 152] [1, 31, 32, 88, 115, 136, 138, 153] [2, 32, 33, 89, 116, 137, 139, 154] [3, 33, 34, 90, 117, 138, 140, 155] [4, 34, 35, 91, 118, 139, 141, 156] [5, 35, 36, 92, 119, 140, 142, 157] [6, 36, 37, 93, 120, 141, 143, 158] [7, 37, 38, 94, 121, 142, 144, 159] [8, 38, 39, 95, 122, 143, 145, 160] [9, 39, 40, 96, 123, 144, 146, 161] [10, 40, 41, 97, 124, 145, 147, 162] [11, 41, 42, 98, 125, 146, 148, 163] [12, 42, 43, 99, 126, 147, 149, 164] [13, 43, 44, 100, 127, 148, 150, 165] [14, 44, 45, 101, 128, 149, 151, 166] [15, 45, 46, 102, 129, 150, 152, 167] [16, 46, 47, 103, 130, 151, 153, 168] [17, 47, 48, 104, 131, 152, 154, 169] [0, 18, 48, 49, 132, 153, 155, 170] [1, 19, 49, 50, 133, 154, 156, 171] [2, 20, 50, 51, 134, 155, 157, 172] [3, 21, 51, 52, 135, 156, 158, 173] [4, 22, 52, 53, 136, 157, 159, 174] [5, 23, 53, 54, 137, 158, 160, 175] [6, 24, 54, 55, 138, 159, 161, 176] [7, 25, 55, 56, 139, 160, 162, 177] [8, 26, 56, 57, 140, 161, 163, 178] [9, 27, 57, 58, 141, 162, 164, 179] [10, 28, 58, 59, 142, 163, 165, 180] [11, 29, 59, 60, 143, 164, 166, 181] [12, 30, 60, 61, 144, 165, 167, 182] [13, 31, 61, 62, 145, 166, 168, 183] [14, 32, 62, 63, 146, 167, 169, 184] [15, 33, 63, 64, 147, 168, 170, 185] [16, 34, 64, 65, 148, 169, 171, 186] [17, 35, 65, 66, 149, 170, 172, 187] [18, 36, 66, 67, 150, 171, 173, 188] [19, 37, 67, 68, 151, 172, 174, 189] [20, 38, 68, 69, 152, 173, 175, 190] [21, 39, 69, 70, 153, 174, 176, 191] [22, 40, 70, 71, 154, 175, 177, 192] [23, 41, 71, 72, 155, 176, 178, 193] [24, 42, 72, 73, 156, 177, 179, 194] [25, 43, 73, 74, 157, 178, 180, 195] [26, 44, 74, 75, 158, 179, 181, 196] [27, 45, 75, 76, 159, 180, 182, 197] [28, 46, 76, 77, 160, 181, 183, 198] [29, 47, 77, 78, 161, 182, 184, 199] [30, 48, 78, 79, 162, 183, 185, 200] [31, 49, 79, 80, 163, 184, 186, 201] [32, 50, 80, 81, 164, 185, 187, 202] [33, 51, 81, 82, 165, 186, 188, 203] [34, 52, 82, 83, 166, 187, 189, 204] [35, 53, 83, 84, 167, 188, 190, 205] [36, 54, 84, 85, 168, 189, 191, 206] [37, 55, 85, 86, 169, 190, 192, 207] [38, 56, 86, 87, 170, 191, 193, 208] [39, 57, 87, 88, 171, 192, 194, 209] [40, 58, 88, 89, 105, 172, 193, 195] [41, 59, 89, 90, 106, 173, 194, 196] [42, 60, 90, 91, 107, 174, 195, 197] [43, 61, 91, 92, 108, 175, 196, 198] [44, 62, 92, 93, 109, 176, 197, 199] [45, 63, 93, 94, 110, 177, 198, 200] [46, 64, 94, 95, 111, 178, 199, 201] [47, 65, 95, 96, 112, 179, 200, 202] [48, 66, 96, 97, 113, 180, 201, 203] [49, 67, 97, 98, 114, 181, 202, 204] [50, 68, 98, 99, 115, 182, 203, 205] [51, 69, 99, 100, 116, 183, 204, 206] [52, 70, 100, 101, 117, 184, 205, 207] [53, 71, 101, 102, 118, 185, 206, 208] [54, 72, 102, 103, 119, 186, 207, 209] [55, 73, 103, 104, 105, 120, 187, 208] [0, 56, 74, 104, 106, 121, 188, 209]
H_Z (105 checks, sparse supports)
[0, 21, 88, 103, 105, 135, 153, 209] [1, 22, 89, 104, 105, 106, 136, 154] [0, 2, 23, 90, 106, 107, 137, 155] [1, 3, 24, 91, 107, 108, 138, 156] [2, 4, 25, 92, 108, 109, 139, 157] [3, 5, 26, 93, 109, 110, 140, 158] [4, 6, 27, 94, 110, 111, 141, 159] [5, 7, 28, 95, 111, 112, 142, 160] [6, 8, 29, 96, 112, 113, 143, 161] [7, 9, 30, 97, 113, 114, 144, 162] [8, 10, 31, 98, 114, 115, 145, 163] [9, 11, 32, 99, 115, 116, 146, 164] [10, 12, 33, 100, 116, 117, 147, 165] [11, 13, 34, 101, 117, 118, 148, 166] [12, 14, 35, 102, 118, 119, 149, 167] [13, 15, 36, 103, 119, 120, 150, 168] [14, 16, 37, 104, 120, 121, 151, 169] [0, 15, 17, 38, 121, 122, 152, 170] [1, 16, 18, 39, 122, 123, 153, 171] [2, 17, 19, 40, 123, 124, 154, 172] [3, 18, 20, 41, 124, 125, 155, 173] [4, 19, 21, 42, 125, 126, 156, 174] [5, 20, 22, 43, 126, 127, 157, 175] [6, 21, 23, 44, 127, 128, 158, 176] [7, 22, 24, 45, 128, 129, 159, 177] [8, 23, 25, 46, 129, 130, 160, 178] [9, 24, 26, 47, 130, 131, 161, 179] [10, 25, 27, 48, 131, 132, 162, 180] [11, 26, 28, 49, 132, 133, 163, 181] [12, 27, 29, 50, 133, 134, 164, 182] [13, 28, 30, 51, 134, 135, 165, 183] [14, 29, 31, 52, 135, 136, 166, 184] [15, 30, 32, 53, 136, 137, 167, 185] [16, 31, 33, 54, 137, 138, 168, 186] [17, 32, 34, 55, 138, 139, 169, 187] [18, 33, 35, 56, 139, 140, 170, 188] [19, 34, 36, 57, 140, 141, 171, 189] [20, 35, 37, 58, 141, 142, 172, 190] [21, 36, 38, 59, 142, 143, 173, 191] [22, 37, 39, 60, 143, 144, 174, 192] [23, 38, 40, 61, 144, 145, 175, 193] [24, 39, 41, 62, 145, 146, 176, 194] [25, 40, 42, 63, 146, 147, 177, 195] [26, 41, 43, 64, 147, 148, 178, 196] [27, 42, 44, 65, 148, 149, 179, 197] [28, 43, 45, 66, 149, 150, 180, 198] [29, 44, 46, 67, 150, 151, 181, 199] [30, 45, 47, 68, 151, 152, 182, 200] [31, 46, 48, 69, 152, 153, 183, 201] [32, 47, 49, 70, 153, 154, 184, 202] [33, 48, 50, 71, 154, 155, 185, 203] [34, 49, 51, 72, 155, 156, 186, 204] [35, 50, 52, 73, 156, 157, 187, 205] [36, 51, 53, 74, 157, 158, 188, 206] [37, 52, 54, 75, 158, 159, 189, 207] [38, 53, 55, 76, 159, 160, 190, 208] [39, 54, 56, 77, 160, 161, 191, 209] [40, 55, 57, 78, 105, 161, 162, 192] [41, 56, 58, 79, 106, 162, 163, 193] [42, 57, 59, 80, 107, 163, 164, 194] [43, 58, 60, 81, 108, 164, 165, 195] [44, 59, 61, 82, 109, 165, 166, 196] [45, 60, 62, 83, 110, 166, 167, 197] [46, 61, 63, 84, 111, 167, 168, 198] [47, 62, 64, 85, 112, 168, 169, 199] [48, 63, 65, 86, 113, 169, 170, 200] [49, 64, 66, 87, 114, 170, 171, 201] [50, 65, 67, 88, 115, 171, 172, 202] [51, 66, 68, 89, 116, 172, 173, 203] [52, 67, 69, 90, 117, 173, 174, 204] [53, 68, 70, 91, 118, 174, 175, 205] [54, 69, 71, 92, 119, 175, 176, 206] [55, 70, 72, 93, 120, 176, 177, 207] [56, 71, 73, 94, 121, 177, 178, 208] [57, 72, 74, 95, 122, 178, 179, 209] [58, 73, 75, 96, 105, 123, 179, 180] [59, 74, 76, 97, 106, 124, 180, 181] [60, 75, 77, 98, 107, 125, 181, 182] [61, 76, 78, 99, 108, 126, 182, 183] [62, 77, 79, 100, 109, 127, 183, 184] [63, 78, 80, 101, 110, 128, 184, 185] [64, 79, 81, 102, 111, 129, 185, 186] [65, 80, 82, 103, 112, 130, 186, 187] [66, 81, 83, 104, 113, 131, 187, 188] [0, 67, 82, 84, 114, 132, 188, 189] [1, 68, 83, 85, 115, 133, 189, 190] [2, 69, 84, 86, 116, 134, 190, 191] [3, 70, 85, 87, 117, 135, 191, 192] [4, 71, 86, 88, 118, 136, 192, 193] [5, 72, 87, 89, 119, 137, 193, 194] [6, 73, 88, 90, 120, 138, 194, 195] [7, 74, 89, 91, 121, 139, 195, 196] [8, 75, 90, 92, 122, 140, 196, 197] [9, 76, 91, 93, 123, 141, 197, 198] [10, 77, 92, 94, 124, 142, 198, 199] [11, 78, 93, 95, 125, 143, 199, 200] [12, 79, 94, 96, 126, 144, 200, 201] [13, 80, 95, 97, 127, 145, 201, 202] [14, 81, 96, 98, 128, 146, 202, 203] [15, 82, 97, 99, 129, 147, 203, 204] [16, 83, 98, 100, 130, 148, 204, 205] [17, 84, 99, 101, 131, 149, 205, 206] [18, 85, 100, 102, 132, 150, 206, 207] [19, 86, 101, 103, 133, 151, 207, 208] [20, 87, 102, 104, 134, 152, 208, 209]
Code ID 210-26-14 · download JSON · raw on GitHub