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[[264,6,12]] d =
n
264
k
6
d
12
kd²/n
3.273
w
6
g
0.0113
r
4.1231
layers
2

Share this result

Distance

d_X 12 · witness weight 12 (claimed exact)
witness operator (support, 12 qubits)
[18, 53, 66, 77, 78, 139, 173, 197, 210, 222, 245, 255]
d_Z 12 · witness weight 12 (claimed exact)
witness operator (support, 12 qubits)
[84, 86, 88, 90, 92, 94, 216, 218, 220, 222, 224, 226]
certificate exact, d = 12 · scipy/HiGHS MILP
X: no logical < 12 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 (132)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, (11,12) 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 140 · Z-checks 141
H_X (140 checks, sparse supports)
[2, 12, 25, 132, 146, 157] [3, 13, 26, 133, 147, 158] [4, 14, 27, 134, 148, 159] [5, 15, 28, 135, 149, 160] [6, 16, 29, 136, 150, 161] [7, 17, 30, 137, 151, 162] [8, 18, 31, 138, 152, 163] [9, 19, 32, 139, 153, 164] [10, 20, 33, 140, 154, 165] [11, 21, 34, 141, 155, 166] [14, 24, 37, 144, 158, 169] [15, 25, 38, 145, 159, 170] [16, 26, 39, 146, 160, 171] [17, 27, 40, 147, 161, 172] [18, 28, 41, 148, 162, 173] [19, 29, 42, 149, 163, 174] [20, 30, 43, 150, 164, 175] [21, 31, 44, 151, 165, 176] [22, 32, 45, 152, 166, 177] [23, 33, 46, 153, 167, 178] [26, 36, 49, 156, 170, 181] [27, 37, 50, 157, 171, 182] [28, 38, 51, 158, 172, 183] [29, 39, 52, 159, 173, 184] [30, 40, 53, 160, 174, 185] [31, 41, 54, 161, 175, 186] [32, 42, 55, 162, 176, 187] [33, 43, 56, 163, 177, 188] [34, 44, 57, 164, 178, 189] [35, 45, 58, 165, 179, 190] [38, 48, 61, 168, 182, 193] [39, 49, 62, 169, 183, 194] [40, 50, 63, 170, 184, 195] [41, 51, 64, 171, 185, 196] [42, 52, 65, 172, 186, 197] [43, 53, 66, 173, 187, 198] [44, 54, 67, 174, 188, 199] [45, 55, 68, 175, 189, 200] [46, 56, 69, 176, 190, 201] [47, 57, 70, 177, 191, 202] [50, 60, 73, 180, 194, 205] [51, 61, 74, 181, 195, 206] [52, 62, 75, 182, 196, 207] [53, 63, 76, 183, 197, 208] [54, 64, 77, 184, 198, 209] [55, 65, 78, 185, 199, 210] [56, 66, 79, 186, 200, 211] [57, 67, 80, 187, 201, 212] [58, 68, 81, 188, 202, 213] [59, 69, 82, 189, 203, 214] [62, 72, 85, 192, 206, 217] [63, 73, 86, 193, 207, 218] [64, 74, 87, 194, 208, 219] [65, 75, 88, 195, 209, 220] [66, 76, 89, 196, 210, 221] [67, 77, 90, 197, 211, 222] [68, 78, 91, 198, 212, 223] [69, 79, 92, 199, 213, 224] [70, 80, 93, 200, 214, 225] [71, 81, 94, 201, 215, 226] [74, 84, 97, 204, 218, 229] [75, 85, 98, 205, 219, 230] [76, 86, 99, 206, 220, 231] [77, 87, 100, 207, 221, 232] [78, 88, 101, 208, 222, 233] [79, 89, 102, 209, 223, 234] [80, 90, 103, 210, 224, 235] [81, 91, 104, 211, 225, 236] [82, 92, 105, 212, 226, 237] [83, 93, 106, 213, 227, 238] [86, 96, 109, 216, 230, 241] [87, 97, 110, 217, 231, 242] [88, 98, 111, 218, 232, 243] [89, 99, 112, 219, 233, 244] [90, 100, 113, 220, 234, 245] [91, 101, 114, 221, 235, 246] [92, 102, 115, 222, 236, 247] [93, 103, 116, 223, 237, 248] [94, 104, 117, 224, 238, 249] [95, 105, 118, 225, 239, 250] [98, 108, 121, 228, 242, 253] [99, 109, 122, 229, 243, 254] [100, 110, 123, 230, 244, 255] [101, 111, 124, 231, 245, 256] [102, 112, 125, 232, 246, 257] [103, 113, 126, 233, 247, 258] [104, 114, 127, 234, 248, 259] [105, 115, 128, 235, 249, 260] [106, 116, 129, 236, 250, 261] [107, 117, 130, 237, 251, 262] [0, 132] [1, 133] [2, 134] [3, 135] [4, 136] [5, 137] [6, 138] [7, 139] [8, 140] [9, 141] [10, 142] [11, 143] [122, 252] [123, 253] [124, 254] [125, 255] [126, 256] [127, 257] [128, 258] [129, 259] [130, 260] [131, 261] [13, 132, 134, 145] [14, 133, 135, 146] [15, 134, 136, 147] [16, 135, 137, 148] [17, 136, 138, 149] [18, 137, 139, 150] [19, 138, 140, 151] [20, 139, 141, 152] [21, 140, 142, 153] [22, 141, 143, 154] [112, 242, 252, 256] [113, 243, 253, 257] [114, 244, 254, 258] [115, 245, 255, 259] [116, 246, 256, 260] [117, 247, 257, 261] [118, 248, 258, 262] [119, 249, 259, 263] [110, 120, 240, 254] [111, 121, 241, 255] [112, 122, 242, 256] [113, 123, 243, 257] [114, 124, 244, 258] [115, 125, 245, 259] [116, 126, 246, 260] [117, 127, 247, 261] [118, 128, 248, 262] [119, 129, 249, 263]
H_Z (141 checks, sparse supports)
[1, 12, 26, 133, 146, 156] [2, 13, 27, 134, 147, 157] [3, 14, 28, 135, 148, 158] [4, 15, 29, 136, 149, 159] [5, 16, 30, 137, 150, 160] [6, 17, 31, 138, 151, 161] [7, 18, 32, 139, 152, 162] [8, 19, 33, 140, 153, 163] [9, 20, 34, 141, 154, 164] [10, 21, 35, 142, 155, 165] [13, 24, 38, 145, 158, 168] [14, 25, 39, 146, 159, 169] [15, 26, 40, 147, 160, 170] [16, 27, 41, 148, 161, 171] [17, 28, 42, 149, 162, 172] [18, 29, 43, 150, 163, 173] [19, 30, 44, 151, 164, 174] [20, 31, 45, 152, 165, 175] [21, 32, 46, 153, 166, 176] [22, 33, 47, 154, 167, 177] [25, 36, 50, 157, 170, 180] [26, 37, 51, 158, 171, 181] [27, 38, 52, 159, 172, 182] [28, 39, 53, 160, 173, 183] [29, 40, 54, 161, 174, 184] [30, 41, 55, 162, 175, 185] [31, 42, 56, 163, 176, 186] [32, 43, 57, 164, 177, 187] [33, 44, 58, 165, 178, 188] [34, 45, 59, 166, 179, 189] [37, 48, 62, 169, 182, 192] [38, 49, 63, 170, 183, 193] [39, 50, 64, 171, 184, 194] [40, 51, 65, 172, 185, 195] [41, 52, 66, 173, 186, 196] [42, 53, 67, 174, 187, 197] [43, 54, 68, 175, 188, 198] [44, 55, 69, 176, 189, 199] [45, 56, 70, 177, 190, 200] [46, 57, 71, 178, 191, 201] [49, 60, 74, 181, 194, 204] [50, 61, 75, 182, 195, 205] [51, 62, 76, 183, 196, 206] [52, 63, 77, 184, 197, 207] [53, 64, 78, 185, 198, 208] [54, 65, 79, 186, 199, 209] [55, 66, 80, 187, 200, 210] [56, 67, 81, 188, 201, 211] [57, 68, 82, 189, 202, 212] [58, 69, 83, 190, 203, 213] [61, 72, 86, 193, 206, 216] [62, 73, 87, 194, 207, 217] [63, 74, 88, 195, 208, 218] [64, 75, 89, 196, 209, 219] [65, 76, 90, 197, 210, 220] [66, 77, 91, 198, 211, 221] [67, 78, 92, 199, 212, 222] [68, 79, 93, 200, 213, 223] [69, 80, 94, 201, 214, 224] [70, 81, 95, 202, 215, 225] [73, 84, 98, 205, 218, 228] [74, 85, 99, 206, 219, 229] [75, 86, 100, 207, 220, 230] [76, 87, 101, 208, 221, 231] [77, 88, 102, 209, 222, 232] [78, 89, 103, 210, 223, 233] [79, 90, 104, 211, 224, 234] [80, 91, 105, 212, 225, 235] [81, 92, 106, 213, 226, 236] [82, 93, 107, 214, 227, 237] [85, 96, 110, 217, 230, 240] [86, 97, 111, 218, 231, 241] [87, 98, 112, 219, 232, 242] [88, 99, 113, 220, 233, 243] [89, 100, 114, 221, 234, 244] [90, 101, 115, 222, 235, 245] [91, 102, 116, 223, 236, 246] [92, 103, 117, 224, 237, 247] [93, 104, 118, 225, 238, 248] [94, 105, 119, 226, 239, 249] [97, 108, 122, 229, 242, 252] [98, 109, 123, 230, 243, 253] [99, 110, 124, 231, 244, 254] [100, 111, 125, 232, 245, 255] [101, 112, 126, 233, 246, 256] [102, 113, 127, 234, 247, 257] [103, 114, 128, 235, 248, 258] [104, 115, 129, 236, 249, 259] [105, 116, 130, 237, 250, 260] [106, 117, 131, 238, 251, 261] [24, 144] [36, 156] [48, 168] [60, 180] [72, 192] [84, 204] [96, 216] [108, 228] [120, 240] [23, 167] [35, 179] [47, 191] [59, 203] [71, 215] [83, 227] [95, 239] [107, 251] [119, 263] [49, 144, 156, 169] [61, 156, 168, 181] [73, 168, 180, 193] [85, 180, 192, 205] [97, 192, 204, 217] [109, 204, 216, 229] [121, 216, 228, 241] [0, 25, 132, 145] [12, 37, 144, 157] [24, 49, 156, 169] [36, 61, 168, 181] [48, 73, 180, 193] [60, 85, 192, 205] [72, 97, 204, 217] [84, 109, 216, 229] [96, 121, 228, 241] [34, 155, 167, 178] [46, 167, 179, 190] [58, 179, 191, 202] [70, 191, 203, 214] [82, 203, 215, 226] [94, 215, 227, 238] [106, 227, 239, 250] [118, 239, 251, 262] [11, 22, 143, 166] [23, 34, 155, 178] [35, 46, 167, 190] [47, 58, 179, 202] [59, 70, 191, 214] [71, 82, 203, 226] [83, 94, 215, 238] [95, 106, 227, 250] [107, 118, 239, 262]
raw submission JSON