[[216,4,18]] · QEC Challenge

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[[216,4,18]] d ≤

216

kd²/n

6.0

X/Z

0.0025

7.0

layers

swaps

1416

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Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · 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 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[17, 32, 34, 54, 73, 75, 97, 99, 101, 113, 117, 129, 139, 147, 175, 179, 191, 205]

d_Z 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[0, 16, 28, 32, 42, 62, 72, 74, 106, 111, 113, 128, 150, 152, 154, 174, 178, 193]

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)×216 (2,4)×1620 (3,3)×216 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 216 (2,4): 1620 (3,3): 216 (3,5): 15552 (3,7): 2160

trapping sets H_Z (1,3)×216 (2,4)×1620 (3,3)×216 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 216 (2,4): 1620 (3,3): 216 (3,5): 15552 (3,7): 2160

witness diameter X 10.8167 · Z 10.198 (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

X checkZ checkqubit site (116)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 1416 nearest-neighbor SWAPs per round in total, at most 11 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_18 x Z_6: A = y² + x⁸ + x¹³, B = y + x⁷ + x¹¹. 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 Opus 5.5 (claimed, not verified)

date 2026-09-30

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

[[216,4,18]] weight-6 bivariate bicycle code on Z_18 x Z_6 with a bilayer layout

Direction & hypothesis

We aimed at the 2D-local bilayer, weight-6 cell by screening for it 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. Every code on the board with n <= 216, k >= 4 and d >= 18 has no 2D-local layout, so a bilayer code there would be a new point.

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. 167 codes passed.

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

a folded-torus start (two qubits per site, minimising the largest check diameter). 19 reached at most 7.0, 15 of them [[216,4,<=18]] genomes. This genome reached 7.000.

Its spectral fingerprint differs from all 6 CSS codes at n = 216 on the board.

Evidence trail

The logical witnesses in the submission JSON show d <= 18 on both sides.
An exact MILP (scipy / HiGHS) proved that no nontrivial Z-type logical of weight below 18 exists. 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. It took 44,043 s (12.2 h) on one core. 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 18.

The claim is d = 18 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.

A first MILP run stopped at its 4 h cap with 13 <= d <= 18; the proof needed a 14 h cap.

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 = 18 and m = 6, with A = y^2 + x^8 + x^13 and B = y + x^7 + x^11 in F_2[x,y]/(x^18 - 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 108 (max weight 6) · Z-checks 108 (max weight 6)

H_X (108 checks, sparse supports)

[2, 48, 78, 109, 150, 174] [3, 49, 79, 110, 151, 175] [4, 50, 80, 111, 152, 176] [5, 51, 81, 112, 153, 177] [0, 52, 82, 113, 154, 178] [1, 53, 83, 108, 155, 179] [8, 54, 84, 115, 156, 180] [9, 55, 85, 116, 157, 181] [10, 56, 86, 117, 158, 182] [11, 57, 87, 118, 159, 183] [6, 58, 88, 119, 160, 184] [7, 59, 89, 114, 161, 185] [14, 60, 90, 121, 162, 186] [15, 61, 91, 122, 163, 187] [16, 62, 92, 123, 164, 188] [17, 63, 93, 124, 165, 189] [12, 64, 94, 125, 166, 190] [13, 65, 95, 120, 167, 191] [20, 66, 96, 127, 168, 192] [21, 67, 97, 128, 169, 193] [22, 68, 98, 129, 170, 194] [23, 69, 99, 130, 171, 195] [18, 70, 100, 131, 172, 196] [19, 71, 101, 126, 173, 197] [26, 72, 102, 133, 174, 198] [27, 73, 103, 134, 175, 199] [28, 74, 104, 135, 176, 200] [29, 75, 105, 136, 177, 201] [24, 76, 106, 137, 178, 202] [25, 77, 107, 132, 179, 203] [0, 32, 78, 139, 180, 204] [1, 33, 79, 140, 181, 205] [2, 34, 80, 141, 182, 206] [3, 35, 81, 142, 183, 207] [4, 30, 82, 143, 184, 208] [5, 31, 83, 138, 185, 209] [6, 38, 84, 145, 186, 210] [7, 39, 85, 146, 187, 211] [8, 40, 86, 147, 188, 212] [9, 41, 87, 148, 189, 213] [10, 36, 88, 149, 190, 214] [11, 37, 89, 144, 191, 215] [12, 44, 90, 108, 151, 192] [13, 45, 91, 109, 152, 193] [14, 46, 92, 110, 153, 194] [15, 47, 93, 111, 154, 195] [16, 42, 94, 112, 155, 196] [17, 43, 95, 113, 150, 197] [18, 50, 96, 114, 157, 198] [19, 51, 97, 115, 158, 199] [20, 52, 98, 116, 159, 200] [21, 53, 99, 117, 160, 201] [22, 48, 100, 118, 161, 202] [23, 49, 101, 119, 156, 203] [24, 56, 102, 120, 163, 204] [25, 57, 103, 121, 164, 205] [26, 58, 104, 122, 165, 206] [27, 59, 105, 123, 166, 207] [28, 54, 106, 124, 167, 208] [29, 55, 107, 125, 162, 209] [0, 30, 62, 126, 169, 210] [1, 31, 63, 127, 170, 211] [2, 32, 64, 128, 171, 212] [3, 33, 65, 129, 172, 213] [4, 34, 60, 130, 173, 214] [5, 35, 61, 131, 168, 215] [6, 36, 68, 108, 132, 175] [7, 37, 69, 109, 133, 176] [8, 38, 70, 110, 134, 177] [9, 39, 71, 111, 135, 178] [10, 40, 66, 112, 136, 179] [11, 41, 67, 113, 137, 174] [12, 42, 74, 114, 138, 181] [13, 43, 75, 115, 139, 182] [14, 44, 76, 116, 140, 183] [15, 45, 77, 117, 141, 184] [16, 46, 72, 118, 142, 185] [17, 47, 73, 119, 143, 180] [18, 48, 80, 120, 144, 187] [19, 49, 81, 121, 145, 188] [20, 50, 82, 122, 146, 189] [21, 51, 83, 123, 147, 190] [22, 52, 78, 124, 148, 191] [23, 53, 79, 125, 149, 186] [24, 54, 86, 126, 150, 193] [25, 55, 87, 127, 151, 194] [26, 56, 88, 128, 152, 195] [27, 57, 89, 129, 153, 196] [28, 58, 84, 130, 154, 197] [29, 59, 85, 131, 155, 192] [30, 60, 92, 132, 156, 199] [31, 61, 93, 133, 157, 200] [32, 62, 94, 134, 158, 201] [33, 63, 95, 135, 159, 202] [34, 64, 90, 136, 160, 203] [35, 65, 91, 137, 161, 198] [36, 66, 98, 138, 162, 205] [37, 67, 99, 139, 163, 206] [38, 68, 100, 140, 164, 207] [39, 69, 101, 141, 165, 208] [40, 70, 96, 142, 166, 209] [41, 71, 97, 143, 167, 204] [42, 72, 104, 144, 168, 211] [43, 73, 105, 145, 169, 212] [44, 74, 106, 146, 170, 213] [45, 75, 107, 147, 171, 214] [46, 76, 102, 148, 172, 215] [47, 77, 103, 149, 173, 210]

H_Z (108 checks, sparse supports)

[5, 42, 66, 112, 138, 168] [0, 43, 67, 113, 139, 169] [1, 44, 68, 108, 140, 170] [2, 45, 69, 109, 141, 171] [3, 46, 70, 110, 142, 172] [4, 47, 71, 111, 143, 173] [11, 48, 72, 118, 144, 174] [6, 49, 73, 119, 145, 175] [7, 50, 74, 114, 146, 176] [8, 51, 75, 115, 147, 177] [9, 52, 76, 116, 148, 178] [10, 53, 77, 117, 149, 179] [17, 54, 78, 124, 150, 180] [12, 55, 79, 125, 151, 181] [13, 56, 80, 120, 152, 182] [14, 57, 81, 121, 153, 183] [15, 58, 82, 122, 154, 184] [16, 59, 83, 123, 155, 185] [23, 60, 84, 130, 156, 186] [18, 61, 85, 131, 157, 187] [19, 62, 86, 126, 158, 188] [20, 63, 87, 127, 159, 189] [21, 64, 88, 128, 160, 190] [22, 65, 89, 129, 161, 191] [29, 66, 90, 136, 162, 192] [24, 67, 91, 137, 163, 193] [25, 68, 92, 132, 164, 194] [26, 69, 93, 133, 165, 195] [27, 70, 94, 134, 166, 196] [28, 71, 95, 135, 167, 197] [35, 72, 96, 142, 168, 198] [30, 73, 97, 143, 169, 199] [31, 74, 98, 138, 170, 200] [32, 75, 99, 139, 171, 201] [33, 76, 100, 140, 172, 202] [34, 77, 101, 141, 173, 203] [41, 78, 102, 148, 174, 204] [36, 79, 103, 149, 175, 205] [37, 80, 104, 144, 176, 206] [38, 81, 105, 145, 177, 207] [39, 82, 106, 146, 178, 208] [40, 83, 107, 147, 179, 209] [0, 47, 84, 154, 180, 210] [1, 42, 85, 155, 181, 211] [2, 43, 86, 150, 182, 212] [3, 44, 87, 151, 183, 213] [4, 45, 88, 152, 184, 214] [5, 46, 89, 153, 185, 215] [6, 53, 90, 108, 160, 186] [7, 48, 91, 109, 161, 187] [8, 49, 92, 110, 156, 188] [9, 50, 93, 111, 157, 189] [10, 51, 94, 112, 158, 190] [11, 52, 95, 113, 159, 191] [12, 59, 96, 114, 166, 192] [13, 54, 97, 115, 167, 193] [14, 55, 98, 116, 162, 194] [15, 56, 99, 117, 163, 195] [16, 57, 100, 118, 164, 196] [17, 58, 101, 119, 165, 197] [18, 65, 102, 120, 172, 198] [19, 60, 103, 121, 173, 199] [20, 61, 104, 122, 168, 200] [21, 62, 105, 123, 169, 201] [22, 63, 106, 124, 170, 202] [23, 64, 107, 125, 171, 203] [0, 24, 71, 126, 178, 204] [1, 25, 66, 127, 179, 205] [2, 26, 67, 128, 174, 206] [3, 27, 68, 129, 175, 207] [4, 28, 69, 130, 176, 208] [5, 29, 70, 131, 177, 209] [6, 30, 77, 132, 184, 210] [7, 31, 72, 133, 185, 211] [8, 32, 73, 134, 180, 212] [9, 33, 74, 135, 181, 213] [10, 34, 75, 136, 182, 214] [11, 35, 76, 137, 183, 215] [12, 36, 83, 108, 138, 190] [13, 37, 78, 109, 139, 191] [14, 38, 79, 110, 140, 186] [15, 39, 80, 111, 141, 187] [16, 40, 81, 112, 142, 188] [17, 41, 82, 113, 143, 189] [18, 42, 89, 114, 144, 196] [19, 43, 84, 115, 145, 197] [20, 44, 85, 116, 146, 192] [21, 45, 86, 117, 147, 193] [22, 46, 87, 118, 148, 194] [23, 47, 88, 119, 149, 195] [24, 48, 95, 120, 150, 202] [25, 49, 90, 121, 151, 203] [26, 50, 91, 122, 152, 198] [27, 51, 92, 123, 153, 199] [28, 52, 93, 124, 154, 200] [29, 53, 94, 125, 155, 201] [30, 54, 101, 126, 156, 208] [31, 55, 96, 127, 157, 209] [32, 56, 97, 128, 158, 204] [33, 57, 98, 129, 159, 205] [34, 58, 99, 130, 160, 206] [35, 59, 100, 131, 161, 207] [36, 60, 107, 132, 162, 214] [37, 61, 102, 133, 163, 215] [38, 62, 103, 134, 164, 210] [39, 63, 104, 135, 165, 211] [40, 64, 105, 136, 166, 212] [41, 65, 106, 137, 167, 213]

Code ID 216-4-18 · download JSON · raw on GitHub