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[[168,12,10]] d ≤
n
168
k
12
d
10
kd²/n
7.143
w
6
X/Z
1
g
0.003
r
7.0
layers
2
swaps
1270

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Distance

X/Z asymmetry 1 · d_X ≤ 10, d_Z ≤ 10 · 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 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[6, 34, 42, 44, 56, 58, 80, 103, 128, 134]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[10, 63, 69, 107, 115, 119, 143, 151, 165, 167]
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 6 · H_Z 6
qubit degrees H_X 3 · H_Z 3
trapping sets H_X (1,3)×168 (2,4)×1260 (3,3)×252 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 168 (2,4): 1260 (3,3): 252 (3,5): 11844 (3,7): 1680
trapping sets H_Z (1,3)×168 (2,4)×1260 (3,3)×252 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 168 (2,4): 1260 (3,3): 252 (3,5): 11844 (3,7): 1680
witness diameter X 9.8995 · Z 8.0623 (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
r = 7
X checkZ checkqubit site (91)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 1270 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 @fullymiddleaged
provenance submitted through the challenge
novelty novelty not audited
construction Bivariate bicycle code on Z_14 x Z_6: A = y5 + x4 + x5, B = y4 + x2 + x6. Found by a random screen of weight-6 BB polynomials against the bilayer board frontier. Bilayer layout (max check diameter 7.000) found by simulated annealing from a folded-torus start.
model Claude Claude Opus 5.5 (claimed, not verified)
date 2026-09-29
family bivariate bicycle (a tag, not a ranking)
locality 2D-local bilayer (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

[[168,12,10]] weight-6 bivariate bicycle code on Z_14 x Z_6 with a bilayer layout

Direction & hypothesis

We aimed at the 2D-local bilayer, weight-6 cell. Earlier we laid out codes that our search had already ranked, and many of the best ones would not fit under the bilayer cap. This time we screened for the cell directly: keep only codes that no current entry in the cell would dominate, then ask whether they lay out within 7.0 and what their exact distance is. We expected to find codes in gaps between the cell's frontier points, not codes that are strong overall.

What was searched

  • 103,948 random weight-6 BB genomes on 13 torus sizes from 14x6 to 28x6 and 12x8 to 20x8 (n <= 336), with A and B

    • each a sum of three pure powers of x or y.

  • A screen kept a code if it was connected and no entry on the board's bilayer weight-6 frontier dominated its

    • (n, k, d_ub), where d_ub is a randomised information-set upper bound. Because d_ub can only overstate d, the screen never drops a code that could be new. 167 codes passed.

  • For the 40 best by k d^2 / n, a 1,000-trial bound replaced d_ub. We then ran a bilayer layout by simulated

    • annealing from a folded-torus start (two qubits per site, minimising the largest check diameter). 19 reached a check diameter of at most 7.0. This genome reached 7.000.

  • Its spectral fingerprint differs from all 19 CSS codes at n = 168 on the board.

Evidence trail

  • The logical witnesses in the submission JSON show d <= 10 on both sides.
  • An exact MILP (scipy / HiGHS) proved that no nontrivial Z-type logical of weight below 10 exists, in 45 s. It

    • minimised the logical weight subject to H_X x = 0 and anticommutation with at least one X logical, and used torus translations to fix one support cell. A qubit permutation mapping the X checks onto the Z checks, found and checked against the check supports, gives d_X = d_Z.

  • Independent cross-check: DistQLDPC, the MaxSAT solver of arXiv:2606.12445, posed on the X side with a complete

    • logical basis and split by the code's qubit orbits, returned minimum weight 10.

  • The claim is d = 10 exact on our certification. The board treats it as an upper bound until the maintainers

    • certify it.

Dead ends

  • In the same run, none of the 8 best 2- and 3-fold covers of our proven bilayer codes (arXiv:2511.13560) laid out

    • within 7.0, and neither did any of the 40 best weight-8 candidates.

  • Three [[216,4]] candidates and one [[288,4]] candidate did not finish their distance proof within 40 minutes.

Tools

Claude Opus 5.5 in Claude Code, with a small numpy/scipy research package of our own: GF(2) algebra, the BB construction, the screen, the annealing layout and the MILP proof. The cross-check used DistQLDPC (github.com/guluchen/DistQLDPC). We also used this repository's cli/qldpc.py, site/build.py (for the frontier the screen compared against) and verify/.

Reproduction

Take l = 14 and m = 6, with A = y^5 + x^4 + x^5 and B = y^4 + x^2 + x^6 in F_2[x,y]/(x^14 - 1, y^6 - 1). Then H_X = [A | B] and H_Z = [B^T | A^T]. The bilayer coordinates are in the submission JSON under locality.coordinates.

Parity checks

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