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[[266,6,22]] d ≤
n
266
k
6
d
22
kd²/n
10.917
w
6

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Distance

d_X 22 · witness weight 22 (claimed upper_bound)
witness operator (support, 22 qubits)
[5, 30, 34, 38, 39, 47, 65, 68, 74, 92, 109, 112, 116, 133, 187, 194, 202, 210, 225, 232, 250, 251]
d_Z 22 · witness weight 22 (claimed upper_bound)
witness operator (support, 22 qubits)
[16, 28, 30, 35, 52, 59, 66, 74, 75, 82, 88, 89, 95, 97, 156, 164, 176, 189, 191, 199, 236, 263]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

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+x-1y-1, g(x,y)=1+y+x5 on the abelian quotient group Z2/L with twist basis a_1=[0, 7], a_2=[19, 2] (n=2|det[a_1,a_2]|=2*133). 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 133 · Z-checks 133
H_X (133 checks, sparse supports)
[0, 1, 112, 133, 135, 148] [1, 3, 27, 134, 137, 154] [2, 4, 118, 135, 138, 155] [3, 6, 34, 136, 140, 161] [0, 4, 7, 137, 141, 162] [5, 8, 123, 138, 142, 163] [6, 10, 41, 139, 144, 168] [1, 7, 11, 140, 145, 169] [2, 8, 12, 141, 146, 170] [9, 13, 127, 142, 147, 171] [10, 15, 48, 143, 149, 175] [3, 11, 16, 144, 150, 176] [4, 12, 17, 145, 151, 177] [5, 13, 18, 146, 152, 178] [14, 19, 130, 147, 153, 179] [15, 21, 55, 148, 155, 182] [6, 16, 22, 149, 156, 183] [7, 17, 23, 150, 157, 184] [8, 18, 24, 151, 158, 185] [9, 19, 25, 152, 159, 186] [20, 26, 132, 153, 160, 187] [21, 28, 62, 154, 162, 189] [10, 22, 29, 155, 163, 190] [11, 23, 30, 156, 164, 191] [12, 24, 31, 157, 165, 192] [13, 25, 32, 158, 166, 193] [14, 26, 33, 159, 167, 194] [27, 34, 105, 133, 160, 195] [28, 35, 69, 161, 169, 196] [15, 29, 36, 162, 170, 197] [16, 30, 37, 163, 171, 198] [17, 31, 38, 164, 172, 199] [18, 32, 39, 165, 173, 200] [19, 33, 40, 166, 174, 201] [20, 34, 41, 134, 167, 202] [35, 42, 76, 168, 176, 203] [21, 36, 43, 169, 177, 204] [22, 37, 44, 170, 178, 205] [23, 38, 45, 171, 179, 206] [24, 39, 46, 172, 180, 207] [25, 40, 47, 173, 181, 208] [26, 41, 48, 136, 174, 209] [42, 49, 83, 175, 183, 210] [28, 43, 50, 176, 184, 211] [29, 44, 51, 177, 185, 212] [30, 45, 52, 178, 186, 213] [31, 46, 53, 179, 187, 214] [32, 47, 54, 180, 188, 215] [33, 48, 55, 139, 181, 216] [49, 56, 90, 182, 190, 217] [35, 50, 57, 183, 191, 218] [36, 51, 58, 184, 192, 219] [37, 52, 59, 185, 193, 220] [38, 53, 60, 186, 194, 221] [39, 54, 61, 187, 195, 222] [40, 55, 62, 143, 188, 223] [56, 63, 97, 189, 197, 224] [42, 57, 64, 190, 198, 225] [43, 58, 65, 191, 199, 226] [44, 59, 66, 192, 200, 227] [45, 60, 67, 193, 201, 228] [46, 61, 68, 194, 202, 229] [47, 62, 69, 148, 195, 230] [63, 70, 104, 196, 204, 231] [49, 64, 71, 197, 205, 232] [50, 65, 72, 198, 206, 233] [51, 66, 73, 199, 207, 234] [52, 67, 74, 200, 208, 235] [53, 68, 75, 201, 209, 236] [54, 69, 76, 154, 202, 237] [70, 77, 111, 203, 211, 238] [56, 71, 78, 204, 212, 239] [57, 72, 79, 205, 213, 240] [58, 73, 80, 206, 214, 241] [59, 74, 81, 207, 215, 242] [60, 75, 82, 208, 216, 243] [61, 76, 83, 161, 209, 244] [77, 84, 117, 153, 210, 218] [63, 78, 85, 211, 219, 245] [64, 79, 86, 212, 220, 246] [65, 80, 87, 213, 221, 247] [66, 81, 88, 214, 222, 248] [67, 82, 89, 215, 223, 249] [68, 83, 90, 168, 216, 250] [84, 91, 122, 159, 217, 225] [70, 85, 92, 160, 218, 226] [71, 86, 93, 219, 227, 251] [72, 87, 94, 220, 228, 252] [73, 88, 95, 221, 229, 253] [74, 89, 96, 222, 230, 254] [75, 90, 97, 175, 223, 255] [91, 98, 126, 166, 224, 232] [77, 92, 99, 167, 225, 233] [78, 93, 100, 133, 226, 234] [79, 94, 101, 227, 235, 256] [80, 95, 102, 228, 236, 257] [81, 96, 103, 229, 237, 258] [82, 97, 104, 182, 230, 259] [98, 105, 129, 173, 231, 239] [84, 99, 106, 174, 232, 240] [85, 100, 107, 134, 233, 241] [86, 101, 108, 135, 234, 242] [87, 102, 109, 235, 243, 260] [88, 103, 110, 236, 244, 261] [89, 104, 111, 189, 237, 262] [20, 105, 131, 180, 238, 245] [91, 106, 112, 181, 239, 246] [92, 107, 113, 136, 240, 247] [93, 108, 114, 137, 241, 248] [94, 109, 115, 138, 242, 249] [95, 110, 116, 243, 250, 263] [96, 111, 117, 196, 244, 264] [27, 98, 112, 188, 245, 251] [99, 113, 118, 139, 246, 252] [100, 114, 119, 140, 247, 253] [101, 115, 120, 141, 248, 254] [102, 116, 121, 142, 249, 255] [103, 117, 122, 203, 250, 265] [0, 106, 118, 143, 251, 256] [107, 119, 123, 144, 252, 257] [108, 120, 124, 145, 253, 258] [109, 121, 125, 146, 254, 259] [110, 122, 126, 147, 210, 255] [2, 113, 123, 149, 256, 260] [114, 124, 127, 150, 257, 261] [115, 125, 128, 151, 258, 262] [116, 126, 129, 152, 217, 259] [5, 119, 127, 156, 260, 263] [120, 128, 130, 157, 261, 264] [121, 129, 131, 158, 224, 262] [9, 124, 130, 164, 263, 265] [125, 131, 132, 165, 231, 264] [14, 128, 132, 172, 238, 265]
H_Z (133 checks, sparse supports)
[0, 27, 93, 133, 137, 251] [1, 34, 100, 133, 134, 140] [0, 2, 101, 135, 141, 256] [3, 41, 107, 134, 136, 144] [1, 4, 108, 135, 137, 145] [2, 5, 109, 138, 146, 260] [6, 48, 113, 136, 139, 149] [3, 7, 114, 137, 140, 150] [4, 8, 115, 138, 141, 151] [5, 9, 116, 142, 152, 263] [10, 55, 118, 139, 143, 155] [6, 11, 119, 140, 144, 156] [7, 12, 120, 141, 145, 157] [8, 13, 121, 142, 146, 158] [9, 14, 122, 147, 159, 265] [0, 15, 62, 143, 148, 162] [10, 16, 123, 144, 149, 163] [11, 17, 124, 145, 150, 164] [12, 18, 125, 146, 151, 165] [13, 19, 126, 147, 152, 166] [14, 20, 77, 153, 167, 238] [1, 21, 69, 148, 154, 169] [2, 15, 22, 149, 155, 170] [16, 23, 127, 150, 156, 171] [17, 24, 128, 151, 157, 172] [18, 25, 129, 152, 158, 173] [19, 26, 84, 153, 159, 174] [20, 27, 85, 134, 160, 245] [3, 28, 76, 154, 161, 176] [4, 21, 29, 155, 162, 177] [5, 22, 30, 156, 163, 178] [23, 31, 130, 157, 164, 179] [24, 32, 131, 158, 165, 180] [25, 33, 91, 159, 166, 181] [26, 34, 92, 136, 160, 167] [6, 35, 83, 161, 168, 183] [7, 28, 36, 162, 169, 184] [8, 29, 37, 163, 170, 185] [9, 30, 38, 164, 171, 186] [31, 39, 132, 165, 172, 187] [32, 40, 98, 166, 173, 188] [33, 41, 99, 139, 167, 174] [10, 42, 90, 168, 175, 190] [11, 35, 43, 169, 176, 191] [12, 36, 44, 170, 177, 192] [13, 37, 45, 171, 178, 193] [14, 38, 46, 172, 179, 194] [39, 47, 105, 173, 180, 195] [40, 48, 106, 143, 174, 181] [15, 49, 97, 175, 182, 197] [16, 42, 50, 176, 183, 198] [17, 43, 51, 177, 184, 199] [18, 44, 52, 178, 185, 200] [19, 45, 53, 179, 186, 201] [20, 46, 54, 180, 187, 202] [47, 55, 112, 148, 181, 188] [21, 56, 104, 182, 189, 204] [22, 49, 57, 183, 190, 205] [23, 50, 58, 184, 191, 206] [24, 51, 59, 185, 192, 207] [25, 52, 60, 186, 193, 208] [26, 53, 61, 187, 194, 209] [27, 54, 62, 154, 188, 195] [28, 63, 111, 189, 196, 211] [29, 56, 64, 190, 197, 212] [30, 57, 65, 191, 198, 213] [31, 58, 66, 192, 199, 214] [32, 59, 67, 193, 200, 215] [33, 60, 68, 194, 201, 216] [34, 61, 69, 161, 195, 202] [35, 70, 117, 196, 203, 218] [36, 63, 71, 197, 204, 219] [37, 64, 72, 198, 205, 220] [38, 65, 73, 199, 206, 221] [39, 66, 74, 200, 207, 222] [40, 67, 75, 201, 208, 223] [41, 68, 76, 168, 202, 209] [42, 77, 122, 203, 210, 225] [43, 70, 78, 204, 211, 226] [44, 71, 79, 205, 212, 227] [45, 72, 80, 206, 213, 228] [46, 73, 81, 207, 214, 229] [47, 74, 82, 208, 215, 230] [48, 75, 83, 175, 209, 216] [49, 84, 126, 210, 217, 232] [50, 77, 85, 211, 218, 233] [51, 78, 86, 212, 219, 234] [52, 79, 87, 213, 220, 235] [53, 80, 88, 214, 221, 236] [54, 81, 89, 215, 222, 237] [55, 82, 90, 182, 216, 223] [56, 91, 129, 217, 224, 239] [57, 84, 92, 218, 225, 240] [58, 85, 93, 219, 226, 241] [59, 86, 94, 220, 227, 242] [60, 87, 95, 221, 228, 243] [61, 88, 96, 222, 229, 244] [62, 89, 97, 189, 223, 230] [63, 98, 131, 224, 231, 245] [64, 91, 99, 225, 232, 246] [65, 92, 100, 226, 233, 247] [66, 93, 101, 227, 234, 248] [67, 94, 102, 228, 235, 249] [68, 95, 103, 229, 236, 250] [69, 96, 104, 196, 230, 237] [70, 105, 132, 160, 231, 238] [71, 98, 106, 232, 239, 251] [72, 99, 107, 233, 240, 252] [73, 100, 108, 234, 241, 253] [74, 101, 109, 235, 242, 254] [75, 102, 110, 236, 243, 255] [76, 103, 111, 203, 237, 244] [78, 105, 112, 133, 239, 245] [79, 106, 113, 240, 246, 256] [80, 107, 114, 241, 247, 257] [81, 108, 115, 242, 248, 258] [82, 109, 116, 243, 249, 259] [83, 110, 117, 210, 244, 250] [86, 112, 118, 135, 246, 251] [87, 113, 119, 247, 252, 260] [88, 114, 120, 248, 253, 261] [89, 115, 121, 249, 254, 262] [90, 116, 122, 217, 250, 255] [94, 118, 123, 138, 252, 256] [95, 119, 124, 253, 257, 263] [96, 120, 125, 254, 258, 264] [97, 121, 126, 224, 255, 259] [102, 123, 127, 142, 257, 260] [103, 124, 128, 258, 261, 265] [104, 125, 129, 231, 259, 262] [110, 127, 130, 147, 261, 263] [111, 128, 131, 238, 262, 264] [117, 130, 132, 153, 264, 265]
raw submission JSON