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[[216,15,11]] d =
n
216
k
15
d
11
kd²/n
8.403
w
8
g
0.01
r
5.3852
layers
2

Share this result

Distance

d_X 11 · witness weight 11 (claimed upper_bound)
witness operator (support, 11 qubits)
[16, 18, 26, 28, 35, 37, 68, 101, 120, 121, 197]
d_Z 11 · witness weight 11 (claimed upper_bound)
witness operator (support, 11 qubits)
[12, 25, 28, 41, 49, 55, 90, 112, 127, 156, 207]
certificate exact, d = 11 · CryptoMiniSat 5.14 SAT
X: no logical < 11 exists (CryptoMiniSat, XOR + sequential-counter cardinality); Z: no logical < 11 exists (CryptoMiniSat, XOR + sequential-counter cardinality)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 5.385
X checkZ checkqubit site (114)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

Construction & provenance

authors @FarLab
provenance submitted through the challenge
novelty novelty not audited
construction Open-boundary planar weight-8 bivariate-bicycle code, f supports {(0,0),(-1,-2),(2,1),(2,-2)}, g supports {(0,0),(-3,-1),(-1,0),(-3,1)}, on an 11x11 lattice (boundary-engine build, [[234,15,11]]), then r=1 lattice grafting (arXiv:2504.08887 Sec. III E) removing 18 qubits at a d>=11 floor, and generating-set weight reduction (12 -> 8, rowspace-preserving). Bilayer grid layout: A/B sublattices stacked, kept-qubit coordinates recovered by deterministic replay of the seeded graft chain.
model Claude Claude Fable 5 (claimed, not verified)
builds on arXiv:2504.08887 (planar BB + grafting technique), arXiv:2410.11942 (open-boundary theory)
date 2026-07-08
notes d_X = 11 is EXACT (scipy/HiGHS MILP certificate, 15/15 subproblems proved, witness verified) computed on the pre-reduction matrices; row reduction preserves the code, so d_X carries over. Claimed d=11 additionally corroborated by deep RIS: flat at 11 across 1M/4M/16M trials/side x 2 seeds (~66M side-trials) with a validated weight-11 X witness -- so d<=11 is proven and no lighter logical was found on either side. Z-side value is the packager's lightest-found (upper bound). Literature novelty of the parameter set not checked.
family bivariate 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 98 · Z-checks 103
H_X (98 checks, sparse supports)
[1, 13, 30, 33, 115, 117, 136, 145] [2, 14, 31, 34, 116, 118, 137, 146] [3, 15, 32, 35, 117, 119, 138, 147] [4, 16, 33, 36, 118, 120, 139, 148] [5, 17, 34, 37, 119, 121, 140, 149] [6, 18, 35, 38, 120, 122, 141, 150] [7, 19, 36, 39, 121, 123, 142, 151] [10, 22, 40, 43, 124, 126, 144, 154] [11, 23, 41, 44, 125, 127, 145, 155] [12, 24, 42, 45, 126, 128, 146, 156] [13, 25, 43, 46, 127, 129, 147, 157] [14, 26, 44, 47, 128, 130, 148, 158] [15, 27, 45, 48, 129, 131, 149, 159] [16, 28, 46, 49, 130, 132, 150, 160] [20, 31, 50, 53, 134, 136, 154, 162] [21, 32, 51, 54, 135, 137, 155, 163] [22, 33, 52, 55, 136, 138, 156, 164] [23, 34, 53, 56, 137, 139, 157, 165] [24, 35, 54, 57, 138, 140, 158, 166] [25, 36, 55, 58, 139, 141, 159, 167] [26, 37, 56, 59, 140, 142, 160, 168] [27, 38, 57, 60, 141, 143, 161, 169] [30, 43, 62, 65, 144, 146, 163, 173] [31, 44, 63, 66, 145, 147, 164, 174] [32, 45, 64, 67, 146, 148, 165, 175] [33, 46, 65, 68, 147, 149, 166, 176] [34, 47, 66, 69, 148, 150, 167, 177] [35, 48, 67, 70, 149, 151, 168, 178] [40, 52, 71, 74, 153, 155, 172, 181] [41, 53, 72, 75, 154, 156, 173, 182] [42, 54, 73, 76, 155, 157, 174, 183] [43, 55, 74, 77, 156, 158, 175, 184] [44, 56, 75, 78, 157, 159, 176, 185] [45, 57, 76, 79, 158, 160, 177, 186] [46, 58, 77, 80, 159, 161, 178, 187] [51, 64, 83, 86, 162, 164, 182, 191] [52, 65, 84, 87, 163, 165, 183, 192] [53, 66, 85, 88, 164, 166, 184, 193] [54, 67, 86, 89, 165, 167, 185, 194] [55, 68, 87, 90, 166, 168, 186, 195] [56, 69, 88, 91, 167, 169, 187, 196] [57, 70, 89, 92, 168, 170, 188, 197] [61, 73, 93, 96, 171, 173, 190, 199] [62, 74, 94, 97, 172, 174, 191, 200] [63, 75, 95, 98, 173, 175, 192, 201] [64, 76, 96, 99, 174, 176, 193, 202] [65, 77, 97, 100, 175, 177, 194, 203] [66, 78, 98, 101, 176, 178, 195, 204] [67, 79, 99, 102, 177, 179, 196, 205] [68, 80, 100, 103, 178, 180, 197, 206] [72, 85, 104, 107, 181, 183, 200, 208] [73, 86, 105, 108, 182, 184, 201, 209] [74, 87, 106, 109, 183, 185, 202, 210] [75, 88, 107, 110, 184, 186, 203, 211] [76, 89, 108, 111, 185, 187, 204, 212] [77, 90, 109, 112, 186, 188, 205, 213] [78, 91, 110, 113, 187, 189, 206, 214] [0, 3, 115] [1, 4, 116] [2, 5, 117] [3, 6, 118] [4, 7, 119] [5, 8, 120] [6, 9, 121] [2, 20, 23, 125, 135] [3, 21, 24, 126, 136] [4, 22, 25, 127, 137] [5, 23, 26, 128, 138] [6, 24, 27, 129, 139] [7, 25, 28, 130, 140] [8, 26, 29, 131, 141] [10, 13, 115, 125] [11, 14, 116, 126] [12, 15, 117, 127] [13, 16, 118, 128] [14, 17, 119, 129] [15, 18, 120, 130] [16, 19, 121, 131] [105, 208, 210] [106, 209, 211] [107, 210, 212] [108, 211, 213] [109, 212, 214] [110, 213, 215] [93, 105, 198, 200] [94, 106, 199, 201] [95, 107, 200, 202] [83, 96, 190, 192, 208] [84, 97, 191, 193, 209] [85, 98, 192, 194, 210] [86, 99, 193, 195, 211] [87, 100, 194, 196, 212] [88, 101, 195, 197, 213] [96, 108, 201, 203] [97, 109, 202, 204] [98, 110, 203, 205] [99, 111, 204, 206] [100, 112, 205, 207]
H_Z (103 checks, sparse supports)
[2, 12, 30, 32, 117, 135, 146] [3, 13, 31, 33, 115, 118, 136, 147] [4, 14, 32, 34, 116, 119, 137, 148] [5, 15, 33, 35, 117, 120, 138, 149] [6, 16, 34, 36, 118, 121, 139, 150] [7, 17, 35, 37, 119, 122, 140, 151] [8, 18, 36, 38, 120, 123, 141] [9, 19, 37, 59, 121] [12, 22, 41, 43, 124, 127, 144, 156] [13, 23, 42, 44, 125, 128, 145, 157] [14, 24, 43, 45, 126, 129, 146, 158] [15, 25, 44, 46, 127, 130, 147, 159] [16, 26, 45, 47, 128, 131, 148, 160] [17, 27, 46, 48, 129, 132, 149, 161] [133] [22, 31, 51, 53, 134, 137, 154, 164] [23, 32, 52, 54, 135, 138, 155, 165] [24, 33, 53, 55, 136, 139, 156, 166] [25, 34, 54, 56, 137, 140, 157, 167] [26, 35, 55, 57, 138, 141, 158, 168] [27, 36, 56, 58, 139, 142, 159, 169] [28, 37, 57, 59, 140, 143, 160, 170] [31, 42, 62, 64, 146, 162, 174] [32, 43, 63, 65, 144, 147, 163, 175] [33, 44, 64, 66, 145, 148, 164, 176] [34, 45, 65, 67, 146, 149, 165, 177] [35, 46, 66, 68, 147, 150, 166, 178] [36, 47, 67, 69, 148, 151, 167, 179] [152] [38, 49, 69, 150, 169] [42, 52, 72, 74, 153, 156, 172, 183] [43, 53, 73, 75, 154, 157, 173, 184] [44, 54, 74, 76, 155, 158, 174, 185] [45, 55, 75, 77, 156, 159, 175, 186] [46, 56, 76, 78, 157, 160, 176, 187] [47, 57, 77, 79, 158, 161, 177, 188] [48, 58, 78, 80, 159, 178, 189] [49, 59, 79, 160, 179] [52, 63, 83, 85, 164, 181, 192] [53, 64, 84, 86, 162, 165, 182, 193] [54, 65, 85, 87, 163, 166, 183, 194] [55, 66, 86, 88, 164, 167, 184, 195] [56, 67, 87, 89, 165, 168, 185, 196] [57, 68, 88, 90, 166, 169, 186, 197] [58, 69, 89, 91, 167, 170, 187] [59, 70, 90, 92, 168, 188] [63, 73, 94, 96, 171, 174, 190, 201] [64, 74, 95, 97, 172, 175, 191, 202] [65, 75, 96, 98, 173, 176, 192, 203] [66, 76, 97, 99, 174, 177, 193, 204] [67, 77, 98, 100, 175, 178, 194, 205] [68, 78, 99, 101, 176, 179, 195, 206] [69, 79, 100, 102, 177, 180, 196, 207] [70, 80, 101, 103, 178, 197] [73, 84, 104, 106, 183, 199, 209] [74, 85, 105, 107, 181, 184, 200, 210] [75, 86, 106, 108, 182, 185, 201, 211] [76, 87, 107, 109, 183, 186, 202, 212] [77, 88, 108, 110, 184, 187, 203, 213] [78, 89, 109, 111, 185, 188, 204, 214] [79, 90, 110, 112, 186, 189, 205, 215] [80, 91, 111, 113, 187, 206] [0, 10, 30, 115, 144] [10, 20, 41, 125, 154] [20, 51, 135, 162] [40, 62, 144, 172] [40, 50, 72, 154, 181] [50, 61, 83, 162, 190] [61, 71, 94, 172, 199] [71, 104, 181] [1, 11, 31, 116, 134, 145] [11, 21, 40, 42, 126, 155] [21, 30, 50, 52, 136, 153, 163] [30, 41, 61, 63, 145, 173] [41, 51, 71, 73, 155, 171, 182] [51, 62, 84, 163, 191] [62, 72, 93, 95, 173, 200] [72, 83, 105, 182, 198, 208] [114] [40, 124, 153] [50, 134] [61, 171] [71, 153] [82] [93, 171, 198] [38, 122, 143] [49, 132] [39, 59, 142] [39, 70, 151, 170] [60, 80, 161, 180] [60, 91, 169, 189] [102, 179] [92, 112, 188, 207] [19, 29, 48, 131, 151] [29, 38, 58, 60, 141, 161] [39, 123] [18, 28, 47, 49, 130, 150] [60, 143] [37, 48, 68, 70, 149, 168, 180] [81] [92, 170] [103, 180] [113, 189]
raw submission JSON