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[[292,18,8]] d =
n
292
k
18
d
8
kd²/n
3.945
w
6

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Distance

d_X 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[9, 14, 30, 55, 65, 90, 155, 201]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[57, 118, 171, 193, 203, 229, 254, 264]
certificate exact, d = 8 · CryptoMiniSat 5.14 SAT
X: no logical < 8 exists (CryptoMiniSat, XOR + sequential-counter cardinality); Z: no logical < 8 exists (CryptoMiniSat, XOR + sequential-counter cardinality)

Construction & provenance

authors Zijian Liang and Ke Liu and Hao Song and Yu-An Chen
provenance literature baseline
construction Twisted-torus bivariate-bicycle code from Generalized toric codes on twisted tori for quantum error correction (PRX Quantum 6, 020357, 2025). Stabilizers f(x,y)=1+x+y2, g(x,y)=1+y+x-4y on the abelian quotient group Z2/L with twist basis a_1=[0, 73], a_2=[2, 32] (n=2|det[a_1,a_2]|=2*146). Reconstructed from the published polynomials and twist; CSS form H_X=[f|g], H_Z=[gbar|fbar].
model classical construction (no AI model)
reference arXiv:2503.03827
date 2025
notes Reconstructed baseline from Liang, Liu, Song, Chen, Generalized toric codes on twisted tori for quantum error correction (PRX Quantum 6, 020357, 2025), arXiv:2503.03827. The [[n,k,d]] parameter set and stabilizer polynomials are published in that paper; this entry reproduces them. Distance is an upper bound per the paper (probabilistic for d>20); the witness is the surrogate logical of that weight. Seeded to fill a board coverage gap at this block size, not a discovery.
family bivariate bicycle (a tag, not a ranking)
locality unrestricted (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 146 · Z-checks 146
H_X (146 checks, sparse supports)
[0, 1, 5, 146, 148, 260] [1, 3, 8, 147, 150, 269] [2, 4, 9, 148, 151, 270] [3, 6, 12, 149, 153, 277] [4, 7, 13, 150, 154, 278] [5, 8, 14, 151, 155, 279] [6, 10, 17, 152, 157, 283] [7, 11, 18, 153, 158, 284] [8, 12, 19, 154, 159, 285] [9, 13, 20, 155, 160, 236] [10, 15, 23, 148, 156, 162] [11, 16, 24, 157, 163, 288] [12, 17, 25, 158, 164, 289] [13, 18, 26, 159, 165, 245] [14, 19, 27, 160, 166, 246] [15, 21, 30, 150, 161, 168] [16, 22, 31, 151, 162, 169] [17, 23, 32, 163, 170, 291] [18, 24, 33, 164, 171, 254] [19, 25, 34, 165, 172, 255] [20, 26, 35, 166, 173, 256] [21, 28, 38, 153, 167, 175] [22, 29, 39, 154, 168, 176] [23, 30, 40, 155, 169, 177] [24, 31, 41, 170, 178, 263] [25, 32, 42, 171, 179, 264] [26, 33, 43, 172, 180, 265] [27, 34, 44, 173, 181, 266] [28, 36, 47, 157, 174, 183] [29, 37, 48, 158, 175, 184] [30, 38, 49, 159, 176, 185] [31, 39, 50, 160, 177, 186] [32, 40, 51, 178, 187, 272] [33, 41, 52, 179, 188, 273] [34, 42, 53, 180, 189, 274] [10, 35, 43, 181, 190, 275] [36, 45, 56, 162, 182, 192] [37, 46, 57, 163, 183, 193] [38, 47, 58, 164, 184, 194] [39, 48, 59, 165, 185, 195] [40, 49, 60, 166, 186, 196] [41, 50, 61, 187, 197, 280] [42, 51, 62, 188, 198, 281] [15, 43, 52, 189, 199, 282] [16, 44, 53, 146, 156, 190] [45, 54, 65, 168, 191, 201] [46, 55, 66, 169, 192, 202] [47, 56, 67, 170, 193, 203] [48, 57, 68, 171, 194, 204] [49, 58, 69, 172, 195, 205] [50, 59, 70, 173, 196, 206] [51, 60, 71, 197, 207, 286] [21, 52, 61, 198, 208, 287] [22, 53, 62, 147, 161, 199] [54, 63, 74, 175, 200, 210] [55, 64, 75, 176, 201, 211] [56, 65, 76, 177, 202, 212] [57, 66, 77, 178, 203, 213] [58, 67, 78, 179, 204, 214] [59, 68, 79, 180, 205, 215] [60, 69, 80, 181, 206, 216] [28, 61, 70, 207, 217, 290] [29, 62, 71, 149, 167, 208] [63, 72, 83, 183, 209, 219] [64, 73, 84, 184, 210, 220] [65, 74, 85, 185, 211, 221] [66, 75, 86, 186, 212, 222] [67, 76, 87, 187, 213, 223] [68, 77, 88, 188, 214, 224] [69, 78, 89, 189, 215, 225] [36, 70, 79, 190, 216, 226] [37, 71, 80, 152, 174, 217] [72, 81, 92, 192, 218, 228] [73, 82, 93, 193, 219, 229] [74, 83, 94, 194, 220, 230] [75, 84, 95, 195, 221, 231] [76, 85, 96, 196, 222, 232] [77, 86, 97, 197, 223, 233] [78, 87, 98, 198, 224, 234] [45, 79, 88, 199, 225, 235] [46, 80, 89, 156, 182, 226] [81, 90, 101, 201, 227, 237] [82, 91, 102, 202, 228, 238] [83, 92, 103, 203, 229, 239] [84, 93, 104, 204, 230, 240] [85, 94, 105, 205, 231, 241] [86, 95, 106, 206, 232, 242] [87, 96, 107, 207, 233, 243] [54, 88, 97, 208, 234, 244] [55, 89, 98, 161, 191, 235] [90, 99, 110, 210, 236, 246] [91, 100, 111, 211, 237, 247] [92, 101, 112, 212, 238, 248] [93, 102, 113, 213, 239, 249] [94, 103, 114, 214, 240, 250] [95, 104, 115, 215, 241, 251] [96, 105, 116, 216, 242, 252] [63, 97, 106, 217, 243, 253] [64, 98, 107, 167, 200, 244] [99, 108, 119, 219, 245, 255] [100, 109, 120, 220, 246, 256] [101, 110, 121, 221, 247, 257] [102, 111, 122, 222, 248, 258] [103, 112, 123, 223, 249, 259] [104, 113, 124, 224, 250, 260] [105, 114, 125, 225, 251, 261] [72, 106, 115, 226, 252, 262] [73, 107, 116, 174, 209, 253] [108, 117, 127, 228, 254, 264] [109, 118, 128, 229, 255, 265] [110, 119, 129, 230, 256, 266] [111, 120, 130, 231, 257, 267] [112, 121, 131, 232, 258, 268] [113, 122, 132, 233, 259, 269] [114, 123, 133, 234, 260, 270] [81, 115, 124, 235, 261, 271] [82, 116, 125, 182, 218, 262] [14, 117, 134, 237, 263, 272] [118, 126, 135, 238, 264, 273] [119, 127, 136, 239, 265, 274] [0, 120, 128, 240, 266, 275] [121, 129, 137, 241, 267, 276] [122, 130, 138, 242, 268, 277] [123, 131, 139, 243, 269, 278] [90, 124, 132, 244, 270, 279] [91, 125, 133, 191, 227, 271] [20, 126, 140, 247, 272, 280] [127, 134, 141, 248, 273, 281] [1, 128, 135, 249, 274, 282] [2, 129, 136, 146, 250, 275] [0, 130, 142, 251, 276, 283] [131, 137, 143, 252, 277, 284] [99, 132, 138, 253, 278, 285] [100, 133, 139, 200, 236, 279] [27, 134, 144, 257, 280, 286] [3, 135, 140, 258, 281, 287] [4, 136, 141, 147, 259, 282] [2, 137, 145, 261, 283, 288] [108, 138, 142, 262, 284, 289] [109, 139, 143, 209, 245, 285] [6, 35, 140, 267, 286, 290] [7, 141, 144, 149, 268, 287] [5, 117, 142, 271, 288, 291] [118, 143, 145, 218, 254, 289] [11, 44, 144, 152, 276, 290] [9, 126, 145, 227, 263, 291]
H_Z (146 checks, sparse supports)
[0, 44, 129, 146, 266, 276] [1, 53, 136, 146, 147, 274] [0, 2, 10, 148, 275, 283] [3, 62, 141, 147, 149, 281] [1, 4, 15, 148, 150, 282] [2, 5, 16, 146, 151, 288] [6, 71, 144, 149, 152, 286] [3, 7, 21, 150, 153, 287] [4, 8, 22, 147, 151, 154] [5, 9, 23, 148, 155, 291] [10, 44, 80, 152, 156, 181] [6, 11, 28, 153, 157, 290] [7, 12, 29, 149, 154, 158] [8, 13, 30, 150, 155, 159] [9, 14, 31, 151, 160, 263] [15, 53, 89, 156, 161, 189] [10, 16, 36, 157, 162, 190] [11, 17, 37, 152, 158, 163] [12, 18, 38, 153, 159, 164] [13, 19, 39, 154, 160, 165] [14, 20, 40, 155, 166, 272] [21, 62, 98, 161, 167, 198] [15, 22, 45, 162, 168, 199] [16, 23, 46, 156, 163, 169] [17, 24, 47, 157, 164, 170] [18, 25, 48, 158, 165, 171] [19, 26, 49, 159, 166, 172] [20, 27, 50, 160, 173, 280] [28, 71, 107, 167, 174, 207] [21, 29, 54, 168, 175, 208] [22, 30, 55, 161, 169, 176] [23, 31, 56, 162, 170, 177] [24, 32, 57, 163, 171, 178] [25, 33, 58, 164, 172, 179] [26, 34, 59, 165, 173, 180] [27, 35, 60, 166, 181, 286] [36, 80, 116, 174, 182, 216] [28, 37, 63, 175, 183, 217] [29, 38, 64, 167, 176, 184] [30, 39, 65, 168, 177, 185] [31, 40, 66, 169, 178, 186] [32, 41, 67, 170, 179, 187] [33, 42, 68, 171, 180, 188] [34, 43, 69, 172, 181, 189] [35, 44, 70, 173, 190, 290] [45, 89, 125, 182, 191, 225] [36, 46, 72, 183, 192, 226] [37, 47, 73, 174, 184, 193] [38, 48, 74, 175, 185, 194] [39, 49, 75, 176, 186, 195] [40, 50, 76, 177, 187, 196] [41, 51, 77, 178, 188, 197] [42, 52, 78, 179, 189, 198] [43, 53, 79, 180, 190, 199] [54, 98, 133, 191, 200, 234] [45, 55, 81, 192, 201, 235] [46, 56, 82, 182, 193, 202] [47, 57, 83, 183, 194, 203] [48, 58, 84, 184, 195, 204] [49, 59, 85, 185, 196, 205] [50, 60, 86, 186, 197, 206] [51, 61, 87, 187, 198, 207] [52, 62, 88, 188, 199, 208] [63, 107, 139, 200, 209, 243] [54, 64, 90, 201, 210, 244] [55, 65, 91, 191, 202, 211] [56, 66, 92, 192, 203, 212] [57, 67, 93, 193, 204, 213] [58, 68, 94, 194, 205, 214] [59, 69, 95, 195, 206, 215] [60, 70, 96, 196, 207, 216] [61, 71, 97, 197, 208, 217] [72, 116, 143, 209, 218, 252] [63, 73, 99, 210, 219, 253] [64, 74, 100, 200, 211, 220] [65, 75, 101, 201, 212, 221] [66, 76, 102, 202, 213, 222] [67, 77, 103, 203, 214, 223] [68, 78, 104, 204, 215, 224] [69, 79, 105, 205, 216, 225] [70, 80, 106, 206, 217, 226] [81, 125, 145, 218, 227, 261] [72, 82, 108, 219, 228, 262] [73, 83, 109, 209, 220, 229] [74, 84, 110, 210, 221, 230] [75, 85, 111, 211, 222, 231] [76, 86, 112, 212, 223, 232] [77, 87, 113, 213, 224, 233] [78, 88, 114, 214, 225, 234] [79, 89, 115, 215, 226, 235] [9, 90, 133, 227, 236, 270] [81, 91, 117, 228, 237, 271] [82, 92, 118, 218, 229, 238] [83, 93, 119, 219, 230, 239] [84, 94, 120, 220, 231, 240] [85, 95, 121, 221, 232, 241] [86, 96, 122, 222, 233, 242] [87, 97, 123, 223, 234, 243] [88, 98, 124, 224, 235, 244] [13, 99, 139, 236, 245, 278] [14, 90, 100, 237, 246, 279] [91, 101, 126, 227, 238, 247] [92, 102, 127, 228, 239, 248] [93, 103, 128, 229, 240, 249] [94, 104, 129, 230, 241, 250] [95, 105, 130, 231, 242, 251] [96, 106, 131, 232, 243, 252] [97, 107, 132, 233, 244, 253] [18, 108, 143, 245, 254, 284] [19, 99, 109, 246, 255, 285] [20, 100, 110, 236, 247, 256] [101, 111, 134, 237, 248, 257] [102, 112, 135, 238, 249, 258] [103, 113, 136, 239, 250, 259] [0, 104, 114, 240, 251, 260] [105, 115, 137, 241, 252, 261] [106, 116, 138, 242, 253, 262] [24, 117, 145, 254, 263, 288] [25, 108, 118, 255, 264, 289] [26, 109, 119, 245, 256, 265] [27, 110, 120, 246, 257, 266] [111, 121, 140, 247, 258, 267] [112, 122, 141, 248, 259, 268] [1, 113, 123, 249, 260, 269] [2, 114, 124, 250, 261, 270] [115, 125, 142, 251, 262, 271] [32, 117, 126, 264, 272, 291] [33, 118, 127, 254, 265, 273] [34, 119, 128, 255, 266, 274] [35, 120, 129, 256, 267, 275] [121, 130, 144, 257, 268, 276] [3, 122, 131, 258, 269, 277] [4, 123, 132, 259, 270, 278] [5, 124, 133, 260, 271, 279] [41, 126, 134, 263, 273, 280] [42, 127, 135, 264, 274, 281] [43, 128, 136, 265, 275, 282] [6, 130, 137, 267, 277, 283] [7, 131, 138, 268, 278, 284] [8, 132, 139, 269, 279, 285] [51, 134, 140, 272, 281, 286] [52, 135, 141, 273, 282, 287] [11, 137, 142, 276, 284, 288] [12, 138, 143, 277, 285, 289] [61, 140, 144, 280, 287, 290] [17, 142, 145, 283, 289, 291]
raw submission JSON