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[[240,6,11]] d =
n
240
k
6
d
11
kd²/n
3.025
w
6
g
0.0105
r
4.1231
layers
2

Share this result

Distance

d_X 11 · witness weight 11 (claimed exact)
witness operator (support, 11 qubits)
[12, 37, 61, 72, 84, 109, 121, 169, 180, 192, 193]
d_Z 12 · witness weight 12 (claimed exact)
witness operator (support, 12 qubits)
[0, 2, 4, 6, 8, 10, 120, 122, 124, 126, 128, 130]
certificate exact, d = 11 · scipy/HiGHS MILP
X: no logical < 11 exists; Z: no logical < 12 exists

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 4.123
X checkZ checkqubit site (120)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 and @vprusso
provenance submitted through the challenge
novelty novelty not audited
construction Generalized weight-6 form f = 1 + xy + x-1 y2, g = 1 + x-1 y + x-2 y-1; open boundaries via annulus-type boundary engine, (10,12) lattice. New instance of the Liang-Eberhardt-Chen open-boundary family (arXiv:2504.08887), outside the paper's enumerated form; the paper's k=6 family stops at [[221,6,10]].
model Claude Claude Opus 4.8 (claimed, not verified)
builds on arXiv:2504.08887
date 2026-06-19
notes Distance proven exact by cutoff MILP (scipy/HiGHS); no lighter logical exists on either side.
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

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

Parity checks

X-checks 130 · Z-checks 125
H_X (130 checks, sparse supports)
[2, 12, 25, 120, 134, 145] [3, 13, 26, 121, 135, 146] [4, 14, 27, 122, 136, 147] [5, 15, 28, 123, 137, 148] [6, 16, 29, 124, 138, 149] [7, 17, 30, 125, 139, 150] [8, 18, 31, 126, 140, 151] [9, 19, 32, 127, 141, 152] [10, 20, 33, 128, 142, 153] [11, 21, 34, 129, 143, 154] [14, 24, 37, 132, 146, 157] [15, 25, 38, 133, 147, 158] [16, 26, 39, 134, 148, 159] [17, 27, 40, 135, 149, 160] [18, 28, 41, 136, 150, 161] [19, 29, 42, 137, 151, 162] [20, 30, 43, 138, 152, 163] [21, 31, 44, 139, 153, 164] [22, 32, 45, 140, 154, 165] [23, 33, 46, 141, 155, 166] [26, 36, 49, 144, 158, 169] [27, 37, 50, 145, 159, 170] [28, 38, 51, 146, 160, 171] [29, 39, 52, 147, 161, 172] [30, 40, 53, 148, 162, 173] [31, 41, 54, 149, 163, 174] [32, 42, 55, 150, 164, 175] [33, 43, 56, 151, 165, 176] [34, 44, 57, 152, 166, 177] [35, 45, 58, 153, 167, 178] [38, 48, 61, 156, 170, 181] [39, 49, 62, 157, 171, 182] [40, 50, 63, 158, 172, 183] [41, 51, 64, 159, 173, 184] [42, 52, 65, 160, 174, 185] [43, 53, 66, 161, 175, 186] [44, 54, 67, 162, 176, 187] [45, 55, 68, 163, 177, 188] [46, 56, 69, 164, 178, 189] [47, 57, 70, 165, 179, 190] [50, 60, 73, 168, 182, 193] [51, 61, 74, 169, 183, 194] [52, 62, 75, 170, 184, 195] [53, 63, 76, 171, 185, 196] [54, 64, 77, 172, 186, 197] [55, 65, 78, 173, 187, 198] [56, 66, 79, 174, 188, 199] [57, 67, 80, 175, 189, 200] [58, 68, 81, 176, 190, 201] [59, 69, 82, 177, 191, 202] [62, 72, 85, 180, 194, 205] [63, 73, 86, 181, 195, 206] [64, 74, 87, 182, 196, 207] [65, 75, 88, 183, 197, 208] [66, 76, 89, 184, 198, 209] [67, 77, 90, 185, 199, 210] [68, 78, 91, 186, 200, 211] [69, 79, 92, 187, 201, 212] [70, 80, 93, 188, 202, 213] [71, 81, 94, 189, 203, 214] [74, 84, 97, 192, 206, 217] [75, 85, 98, 193, 207, 218] [76, 86, 99, 194, 208, 219] [77, 87, 100, 195, 209, 220] [78, 88, 101, 196, 210, 221] [79, 89, 102, 197, 211, 222] [80, 90, 103, 198, 212, 223] [81, 91, 104, 199, 213, 224] [82, 92, 105, 200, 214, 225] [83, 93, 106, 201, 215, 226] [86, 96, 109, 204, 218, 229] [87, 97, 110, 205, 219, 230] [88, 98, 111, 206, 220, 231] [89, 99, 112, 207, 221, 232] [90, 100, 113, 208, 222, 233] [91, 101, 114, 209, 223, 234] [92, 102, 115, 210, 224, 235] [93, 103, 116, 211, 225, 236] [94, 104, 117, 212, 226, 237] [95, 105, 118, 213, 227, 238] [0, 120] [1, 121] [2, 122] [3, 123] [4, 124] [5, 125] [6, 126] [7, 127] [8, 128] [9, 129] [10, 130] [11, 131] [110, 228] [111, 229] [112, 230] [113, 231] [114, 232] [115, 233] [116, 234] [117, 235] [118, 236] [119, 237] [13, 120, 122, 133] [14, 121, 123, 134] [15, 122, 124, 135] [16, 123, 125, 136] [17, 124, 126, 137] [18, 125, 127, 138] [19, 126, 128, 139] [20, 127, 129, 140] [21, 128, 130, 141] [22, 129, 131, 142] [100, 218, 228, 232] [101, 219, 229, 233] [102, 220, 230, 234] [103, 221, 231, 235] [104, 222, 232, 236] [105, 223, 233, 237] [106, 224, 234, 238] [107, 225, 235, 239] [98, 108, 216, 230] [99, 109, 217, 231] [100, 110, 218, 232] [101, 111, 219, 233] [102, 112, 220, 234] [103, 113, 221, 235] [104, 114, 222, 236] [105, 115, 223, 237] [106, 116, 224, 238] [107, 117, 225, 239]
H_Z (125 checks, sparse supports)
[1, 12, 26, 121, 134, 144] [2, 13, 27, 122, 135, 145] [3, 14, 28, 123, 136, 146] [4, 15, 29, 124, 137, 147] [5, 16, 30, 125, 138, 148] [6, 17, 31, 126, 139, 149] [7, 18, 32, 127, 140, 150] [8, 19, 33, 128, 141, 151] [9, 20, 34, 129, 142, 152] [10, 21, 35, 130, 143, 153] [13, 24, 38, 133, 146, 156] [14, 25, 39, 134, 147, 157] [15, 26, 40, 135, 148, 158] [16, 27, 41, 136, 149, 159] [17, 28, 42, 137, 150, 160] [18, 29, 43, 138, 151, 161] [19, 30, 44, 139, 152, 162] [20, 31, 45, 140, 153, 163] [21, 32, 46, 141, 154, 164] [22, 33, 47, 142, 155, 165] [25, 36, 50, 145, 158, 168] [26, 37, 51, 146, 159, 169] [27, 38, 52, 147, 160, 170] [28, 39, 53, 148, 161, 171] [29, 40, 54, 149, 162, 172] [30, 41, 55, 150, 163, 173] [31, 42, 56, 151, 164, 174] [32, 43, 57, 152, 165, 175] [33, 44, 58, 153, 166, 176] [34, 45, 59, 154, 167, 177] [37, 48, 62, 157, 170, 180] [38, 49, 63, 158, 171, 181] [39, 50, 64, 159, 172, 182] [40, 51, 65, 160, 173, 183] [41, 52, 66, 161, 174, 184] [42, 53, 67, 162, 175, 185] [43, 54, 68, 163, 176, 186] [44, 55, 69, 164, 177, 187] [45, 56, 70, 165, 178, 188] [46, 57, 71, 166, 179, 189] [49, 60, 74, 169, 182, 192] [50, 61, 75, 170, 183, 193] [51, 62, 76, 171, 184, 194] [52, 63, 77, 172, 185, 195] [53, 64, 78, 173, 186, 196] [54, 65, 79, 174, 187, 197] [55, 66, 80, 175, 188, 198] [56, 67, 81, 176, 189, 199] [57, 68, 82, 177, 190, 200] [58, 69, 83, 178, 191, 201] [61, 72, 86, 181, 194, 204] [62, 73, 87, 182, 195, 205] [63, 74, 88, 183, 196, 206] [64, 75, 89, 184, 197, 207] [65, 76, 90, 185, 198, 208] [66, 77, 91, 186, 199, 209] [67, 78, 92, 187, 200, 210] [68, 79, 93, 188, 201, 211] [69, 80, 94, 189, 202, 212] [70, 81, 95, 190, 203, 213] [73, 84, 98, 193, 206, 216] [74, 85, 99, 194, 207, 217] [75, 86, 100, 195, 208, 218] [76, 87, 101, 196, 209, 219] [77, 88, 102, 197, 210, 220] [78, 89, 103, 198, 211, 221] [79, 90, 104, 199, 212, 222] [80, 91, 105, 200, 213, 223] [81, 92, 106, 201, 214, 224] [82, 93, 107, 202, 215, 225] [85, 96, 110, 205, 218, 228] [86, 97, 111, 206, 219, 229] [87, 98, 112, 207, 220, 230] [88, 99, 113, 208, 221, 231] [89, 100, 114, 209, 222, 232] [90, 101, 115, 210, 223, 233] [91, 102, 116, 211, 224, 234] [92, 103, 117, 212, 225, 235] [93, 104, 118, 213, 226, 236] [94, 105, 119, 214, 227, 237] [24, 132] [36, 144] [48, 156] [60, 168] [72, 180] [84, 192] [96, 204] [108, 216] [23, 155] [35, 167] [47, 179] [59, 191] [71, 203] [83, 215] [95, 227] [107, 239] [49, 132, 144, 157] [61, 144, 156, 169] [73, 156, 168, 181] [85, 168, 180, 193] [97, 180, 192, 205] [109, 192, 204, 217] [0, 25, 120, 133] [12, 37, 132, 145] [24, 49, 144, 157] [36, 61, 156, 169] [48, 73, 168, 181] [60, 85, 180, 193] [72, 97, 192, 205] [84, 109, 204, 217] [34, 143, 155, 166] [46, 155, 167, 178] [58, 167, 179, 190] [70, 179, 191, 202] [82, 191, 203, 214] [94, 203, 215, 226] [106, 215, 227, 238] [11, 22, 131, 154] [23, 34, 143, 166] [35, 46, 155, 178] [47, 58, 167, 190] [59, 70, 179, 202] [71, 82, 191, 214] [83, 94, 203, 226] [95, 106, 215, 238]
raw submission JSON