[[299,5,13]] · QEC Challenge

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[[299,5,13]] d =

299

kd²/n

2.826

0.0283

3.1623

layers

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Distance

d_X 13 · witness weight 13 (claimed exact)

witness operator (support, 13 qubits)

[34, 46, 47, 58, 83, 109, 121, 164, 178, 227, 240, 252, 290]

d_Z 13 · witness weight 13 (claimed exact)

witness operator (support, 13 qubits)

[89, 97, 104, 231, 232, 239, 242, 250, 251, 252, 253, 254, 261]

certificate exact, d = 13 · scipy/HiGHS MILP

X: no logical < 13 exists; Z: no logical < 13 exists

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 (156)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, (12,13) open-boundary lattice. New instance of the Liang-Eberhardt-Chen open-boundary family (arXiv:2504.08887); the paper has no k=5 family.

model 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. 2D-local bilayer layout (reconstructed 2026-07-02): A/B sublattices of 12x13 and 11x13 rows on one grid, physical row = (+i+1) mod 12 on A and (+i+0) mod 11 on B; interaction radius 3.162.

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 144 · Z-checks 157

H_X (144 checks, sparse supports)

[0, 14, 27, 157, 158, 169] [1, 15, 28, 158, 159, 170] [2, 16, 29, 159, 160, 171] [3, 17, 30, 160, 161, 172] [4, 18, 31, 161, 162, 173] [5, 19, 32, 162, 163, 174] [6, 20, 33, 163, 164, 175] [7, 21, 34, 164, 165, 176] [8, 22, 35, 165, 166, 177] [9, 23, 36, 166, 167, 178] [10, 24, 37, 167, 168, 179] [13, 27, 40, 170, 171, 182] [14, 28, 41, 171, 172, 183] [15, 29, 42, 172, 173, 184] [16, 30, 43, 173, 174, 185] [17, 31, 44, 174, 175, 186] [18, 32, 45, 175, 176, 187] [19, 33, 46, 176, 177, 188] [20, 34, 47, 177, 178, 189] [21, 35, 48, 178, 179, 190] [22, 36, 49, 179, 180, 191] [23, 37, 50, 180, 181, 192] [26, 40, 53, 183, 184, 195] [27, 41, 54, 184, 185, 196] [28, 42, 55, 185, 186, 197] [29, 43, 56, 186, 187, 198] [30, 44, 57, 187, 188, 199] [31, 45, 58, 188, 189, 200] [32, 46, 59, 189, 190, 201] [33, 47, 60, 190, 191, 202] [34, 48, 61, 191, 192, 203] [35, 49, 62, 192, 193, 204] [36, 50, 63, 193, 194, 205] [39, 53, 66, 196, 197, 208] [40, 54, 67, 197, 198, 209] [41, 55, 68, 198, 199, 210] [42, 56, 69, 199, 200, 211] [43, 57, 70, 200, 201, 212] [44, 58, 71, 201, 202, 213] [45, 59, 72, 202, 203, 214] [46, 60, 73, 203, 204, 215] [47, 61, 74, 204, 205, 216] [48, 62, 75, 205, 206, 217] [49, 63, 76, 206, 207, 218] [52, 66, 79, 209, 210, 221] [53, 67, 80, 210, 211, 222] [54, 68, 81, 211, 212, 223] [55, 69, 82, 212, 213, 224] [56, 70, 83, 213, 214, 225] [57, 71, 84, 214, 215, 226] [58, 72, 85, 215, 216, 227] [59, 73, 86, 216, 217, 228] [60, 74, 87, 217, 218, 229] [61, 75, 88, 218, 219, 230] [62, 76, 89, 219, 220, 231] [65, 79, 92, 222, 223, 234] [66, 80, 93, 223, 224, 235] [67, 81, 94, 224, 225, 236] [68, 82, 95, 225, 226, 237] [69, 83, 96, 226, 227, 238] [70, 84, 97, 227, 228, 239] [71, 85, 98, 228, 229, 240] [72, 86, 99, 229, 230, 241] [73, 87, 100, 230, 231, 242] [74, 88, 101, 231, 232, 243] [75, 89, 102, 232, 233, 244] [78, 92, 105, 235, 236, 247] [79, 93, 106, 236, 237, 248] [80, 94, 107, 237, 238, 249] [81, 95, 108, 238, 239, 250] [82, 96, 109, 239, 240, 251] [83, 97, 110, 240, 241, 252] [84, 98, 111, 241, 242, 253] [85, 99, 112, 242, 243, 254] [86, 100, 113, 243, 244, 255] [87, 101, 114, 244, 245, 256] [88, 102, 115, 245, 246, 257] [91, 105, 118, 248, 249, 260] [92, 106, 119, 249, 250, 261] [93, 107, 120, 250, 251, 262] [94, 108, 121, 251, 252, 263] [95, 109, 122, 252, 253, 264] [96, 110, 123, 253, 254, 265] [97, 111, 124, 254, 255, 266] [98, 112, 125, 255, 256, 267] [99, 113, 126, 256, 257, 268] [100, 114, 127, 257, 258, 269] [101, 115, 128, 258, 259, 270] [104, 118, 131, 261, 262, 273] [105, 119, 132, 262, 263, 274] [106, 120, 133, 263, 264, 275] [107, 121, 134, 264, 265, 276] [108, 122, 135, 265, 266, 277] [109, 123, 136, 266, 267, 278] [110, 124, 137, 267, 268, 279] [111, 125, 138, 268, 269, 280] [112, 126, 139, 269, 270, 281] [113, 127, 140, 270, 271, 282] [114, 128, 141, 271, 272, 283] [1, 143] [2, 144] [3, 145] [4, 146] [5, 147] [6, 148] [7, 149] [8, 150] [9, 151] [10, 152] [11, 153] [12, 154] [130, 287, 288] [131, 288, 289] [132, 289, 290] [133, 290, 291] [134, 291, 292] [135, 292, 293] [136, 293, 294] [137, 294, 295] [138, 295, 296] [139, 296, 297] [140, 297, 298] [14, 143, 144, 145, 156] [15, 144, 145, 146, 157] [16, 145, 146, 147, 158] [17, 146, 147, 148, 159] [18, 147, 148, 149, 160] [19, 148, 149, 150, 161] [20, 149, 150, 151, 162] [21, 150, 151, 152, 163] [22, 151, 152, 153, 164] [23, 152, 153, 154, 165] [24, 153, 154, 155, 166] [117, 131, 274, 275, 286] [118, 132, 275, 276, 287] [119, 133, 276, 277, 288] [120, 134, 277, 278, 289] [121, 135, 278, 279, 290] [122, 136, 279, 280, 291] [123, 137, 280, 281, 292] [124, 138, 281, 282, 293] [125, 139, 282, 283, 294] [126, 140, 283, 284, 295] [127, 141, 284, 285, 296]

H_Z (157 checks, sparse supports)

[2, 13, 14, 144, 157, 171] [3, 14, 15, 145, 158, 172] [4, 15, 16, 146, 159, 173] [5, 16, 17, 147, 160, 174] [6, 17, 18, 148, 161, 175] [7, 18, 19, 149, 162, 176] [8, 19, 20, 150, 163, 177] [9, 20, 21, 151, 164, 178] [10, 21, 22, 152, 165, 179] [11, 22, 23, 153, 166, 180] [12, 23, 24, 154, 167, 181] [15, 26, 27, 157, 170, 184] [16, 27, 28, 158, 171, 185] [17, 28, 29, 159, 172, 186] [18, 29, 30, 160, 173, 187] [19, 30, 31, 161, 174, 188] [20, 31, 32, 162, 175, 189] [21, 32, 33, 163, 176, 190] [22, 33, 34, 164, 177, 191] [23, 34, 35, 165, 178, 192] [24, 35, 36, 166, 179, 193] [25, 36, 37, 167, 180, 194] [28, 39, 40, 170, 183, 197] [29, 40, 41, 171, 184, 198] [30, 41, 42, 172, 185, 199] [31, 42, 43, 173, 186, 200] [32, 43, 44, 174, 187, 201] [33, 44, 45, 175, 188, 202] [34, 45, 46, 176, 189, 203] [35, 46, 47, 177, 190, 204] [36, 47, 48, 178, 191, 205] [37, 48, 49, 179, 192, 206] [38, 49, 50, 180, 193, 207] [41, 52, 53, 183, 196, 210] [42, 53, 54, 184, 197, 211] [43, 54, 55, 185, 198, 212] [44, 55, 56, 186, 199, 213] [45, 56, 57, 187, 200, 214] [46, 57, 58, 188, 201, 215] [47, 58, 59, 189, 202, 216] [48, 59, 60, 190, 203, 217] [49, 60, 61, 191, 204, 218] [50, 61, 62, 192, 205, 219] [51, 62, 63, 193, 206, 220] [54, 65, 66, 196, 209, 223] [55, 66, 67, 197, 210, 224] [56, 67, 68, 198, 211, 225] [57, 68, 69, 199, 212, 226] [58, 69, 70, 200, 213, 227] [59, 70, 71, 201, 214, 228] [60, 71, 72, 202, 215, 229] [61, 72, 73, 203, 216, 230] [62, 73, 74, 204, 217, 231] [63, 74, 75, 205, 218, 232] [64, 75, 76, 206, 219, 233] [67, 78, 79, 209, 222, 236] [68, 79, 80, 210, 223, 237] [69, 80, 81, 211, 224, 238] [70, 81, 82, 212, 225, 239] [71, 82, 83, 213, 226, 240] [72, 83, 84, 214, 227, 241] [73, 84, 85, 215, 228, 242] [74, 85, 86, 216, 229, 243] [75, 86, 87, 217, 230, 244] [76, 87, 88, 218, 231, 245] [77, 88, 89, 219, 232, 246] [80, 91, 92, 222, 235, 249] [81, 92, 93, 223, 236, 250] [82, 93, 94, 224, 237, 251] [83, 94, 95, 225, 238, 252] [84, 95, 96, 226, 239, 253] [85, 96, 97, 227, 240, 254] [86, 97, 98, 228, 241, 255] [87, 98, 99, 229, 242, 256] [88, 99, 100, 230, 243, 257] [89, 100, 101, 231, 244, 258] [90, 101, 102, 232, 245, 259] [93, 104, 105, 235, 248, 262] [94, 105, 106, 236, 249, 263] [95, 106, 107, 237, 250, 264] [96, 107, 108, 238, 251, 265] [97, 108, 109, 239, 252, 266] [98, 109, 110, 240, 253, 267] [99, 110, 111, 241, 254, 268] [100, 111, 112, 242, 255, 269] [101, 112, 113, 243, 256, 270] [102, 113, 114, 244, 257, 271] [103, 114, 115, 245, 258, 272] [106, 117, 118, 248, 261, 275] [107, 118, 119, 249, 262, 276] [108, 119, 120, 250, 263, 277] [109, 120, 121, 251, 264, 278] [110, 121, 122, 252, 265, 279] [111, 122, 123, 253, 266, 280] [112, 123, 124, 254, 267, 281] [113, 124, 125, 255, 268, 282] [114, 125, 126, 256, 269, 283] [115, 126, 127, 257, 270, 284] [116, 127, 128, 258, 271, 285] [119, 130, 131, 261, 274, 288] [120, 131, 132, 262, 275, 289] [121, 132, 133, 263, 276, 290] [122, 133, 134, 264, 277, 291] [123, 134, 135, 265, 278, 292] [124, 135, 136, 266, 279, 293] [125, 136, 137, 267, 280, 294] [126, 137, 138, 268, 281, 295] [127, 138, 139, 269, 282, 296] [128, 139, 140, 270, 283, 297] [129, 140, 141, 271, 284, 298] [25] [38] [51] [64] [77] [90] [103] [116] [129] [142] [0, 169] [13, 182] [26, 195] [39, 208] [52, 221] [65, 234] [78, 247] [91, 260] [104, 273] [117, 286] [24, 155, 168] [37, 168, 181] [50, 181, 194] [63, 194, 207] [76, 207, 220] [89, 220, 233] [102, 233, 246] [115, 246, 259] [128, 259, 272] [141, 272, 285] [1, 143, 156, 170, 182] [14, 156, 169, 183, 195] [27, 169, 182, 196, 208] [40, 182, 195, 209, 221] [53, 195, 208, 222, 234] [66, 208, 221, 235, 247] [79, 221, 234, 248, 260] [92, 234, 247, 261, 273] [105, 247, 260, 274, 286] [0, 13, 27, 39, 196] [13, 26, 40, 52, 209] [26, 39, 53, 65, 222] [39, 52, 66, 78, 235] [52, 65, 79, 91, 248] [65, 78, 92, 104, 261] [78, 91, 105, 117, 274] [91, 104, 118, 130, 287]

raw submission JSON