← back to the board
[[252,12,16]] d ≤
n
252
k
12
d
16
kd²/n
12.19
w
6

Share this result

Distance

d_X 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[6, 11, 17, 29, 31, 40, 132, 136, 185, 191, 196, 220, 229, 243, 250, 251]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[4, 6, 17, 23, 62, 73, 83, 84, 94, 105, 164, 174, 176, 196, 208, 220]
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-3y-1, g(x,y)=1+y+x2y-2 on the abelian quotient group Z2/L with twist basis a_1=[0, 18], a_2=[7, 7] (n=2|det[a_1,a_2]|=2*126). 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 126 · Z-checks 126
H_X (126 checks, sparse supports)
[0, 1, 61, 126, 128, 229] [1, 3, 72, 127, 130, 237] [2, 4, 73, 128, 131, 238] [3, 6, 83, 129, 133, 243] [4, 7, 84, 130, 134, 244] [5, 8, 85, 129, 131, 135] [6, 10, 94, 132, 137, 247] [7, 11, 95, 133, 138, 248] [8, 12, 96, 132, 134, 139] [9, 13, 97, 133, 135, 140] [10, 15, 104, 136, 142, 249] [0, 11, 16, 137, 143, 250] [12, 17, 105, 136, 138, 144] [13, 18, 106, 137, 139, 145] [14, 19, 107, 138, 140, 146] [15, 21, 112, 141, 148, 180] [1, 16, 22, 142, 149, 251] [2, 17, 23, 141, 143, 150] [18, 24, 113, 142, 144, 151] [19, 25, 114, 143, 145, 152] [20, 26, 66, 144, 146, 153] [21, 28, 118, 147, 155, 190] [3, 22, 29, 148, 156, 191] [4, 23, 30, 147, 149, 157] [5, 24, 31, 148, 150, 158] [25, 32, 119, 149, 151, 159] [26, 33, 77, 150, 152, 160] [27, 34, 78, 151, 153, 161] [28, 36, 122, 154, 163, 201] [6, 29, 37, 155, 164, 202] [7, 30, 38, 154, 156, 165] [8, 31, 39, 155, 157, 166] [9, 32, 40, 156, 158, 167] [33, 41, 88, 157, 159, 168] [34, 42, 89, 158, 160, 169] [35, 43, 90, 159, 161, 170] [36, 45, 124, 162, 172, 212] [10, 37, 46, 163, 173, 213] [11, 38, 47, 162, 164, 174] [12, 39, 48, 163, 165, 175] [13, 40, 49, 164, 166, 176] [14, 41, 50, 165, 167, 177] [42, 51, 99, 166, 168, 178] [43, 52, 100, 167, 169, 179] [44, 53, 101, 168, 170, 180] [45, 55, 125, 171, 182, 223] [15, 46, 56, 172, 183, 224] [16, 47, 57, 171, 173, 184] [17, 48, 58, 172, 174, 185] [18, 49, 59, 173, 175, 186] [19, 50, 60, 174, 176, 187] [20, 51, 61, 175, 177, 188] [52, 62, 108, 176, 178, 189] [53, 63, 109, 177, 179, 190] [54, 64, 110, 178, 180, 191] [55, 65, 66, 181, 193, 232] [21, 56, 67, 182, 194, 233] [22, 57, 68, 181, 183, 195] [23, 58, 69, 182, 184, 196] [24, 59, 70, 183, 185, 197] [25, 60, 71, 184, 186, 198] [26, 61, 72, 185, 187, 199] [27, 62, 73, 186, 188, 200] [63, 74, 115, 187, 189, 201] [64, 75, 116, 188, 190, 202] [65, 76, 117, 154, 189, 191] [66, 76, 77, 192, 204, 239] [28, 67, 78, 193, 205, 240] [29, 68, 79, 192, 194, 206] [30, 69, 80, 193, 195, 207] [31, 70, 81, 194, 196, 208] [32, 71, 82, 195, 197, 209] [33, 72, 83, 196, 198, 210] [34, 73, 84, 197, 199, 211] [35, 74, 85, 198, 200, 212] [75, 86, 120, 199, 201, 213] [76, 87, 121, 162, 200, 202] [77, 87, 88, 131, 203, 215] [36, 78, 89, 204, 216, 245] [37, 79, 90, 203, 205, 217] [38, 80, 91, 204, 206, 218] [39, 81, 92, 205, 207, 219] [40, 82, 93, 206, 208, 220] [41, 83, 94, 207, 209, 221] [42, 84, 95, 208, 210, 222] [43, 85, 96, 209, 211, 223] [44, 86, 97, 210, 212, 224] [87, 98, 123, 171, 211, 213] [14, 88, 98, 134, 214, 225] [45, 89, 99, 135, 215, 226] [46, 90, 100, 214, 216, 227] [47, 91, 101, 215, 217, 228] [48, 92, 102, 216, 218, 229] [49, 93, 103, 217, 219, 230] [50, 94, 104, 126, 218, 220] [0, 51, 95, 219, 221, 231] [52, 96, 105, 220, 222, 232] [53, 97, 106, 221, 223, 233] [54, 98, 107, 181, 222, 224] [20, 55, 99, 139, 225, 234] [56, 100, 108, 140, 226, 235] [57, 101, 109, 225, 227, 236] [58, 102, 110, 226, 228, 237] [59, 103, 111, 227, 229, 238] [60, 104, 112, 127, 228, 230] [2, 62, 105, 230, 231, 239] [63, 106, 113, 126, 232, 240] [64, 107, 114, 192, 231, 233] [27, 67, 108, 145, 234, 241] [68, 109, 115, 146, 235, 242] [69, 110, 116, 234, 236, 243] [70, 111, 117, 235, 237, 244] [71, 112, 118, 129, 236, 238] [5, 74, 113, 127, 239, 245] [75, 114, 119, 128, 203, 240] [35, 79, 115, 152, 241, 246] [80, 116, 120, 153, 242, 247] [81, 117, 121, 241, 243, 248] [82, 118, 122, 132, 242, 244] [9, 86, 119, 130, 214, 245] [44, 91, 120, 160, 246, 249] [92, 121, 123, 161, 247, 250] [93, 122, 124, 136, 246, 248] [54, 102, 123, 169, 249, 251] [103, 124, 125, 141, 170, 250] [65, 111, 125, 147, 179, 251]
H_Z (126 checks, sparse supports)
[0, 94, 106, 126, 137, 221] [1, 104, 113, 126, 127, 142] [0, 2, 114, 128, 143, 231] [3, 5, 112, 127, 129, 148] [1, 4, 119, 128, 130, 149] [2, 5, 77, 131, 150, 239] [6, 8, 118, 129, 132, 155] [3, 7, 9, 130, 133, 156] [4, 8, 88, 131, 134, 157] [5, 9, 89, 135, 158, 245] [10, 12, 122, 132, 136, 163] [6, 11, 13, 133, 137, 164] [7, 12, 14, 134, 138, 165] [8, 13, 99, 135, 139, 166] [9, 14, 100, 140, 167, 214] [15, 17, 124, 136, 141, 172] [10, 16, 18, 137, 142, 173] [11, 17, 19, 138, 143, 174] [12, 18, 20, 139, 144, 175] [13, 19, 108, 140, 145, 176] [14, 20, 109, 146, 177, 225] [21, 23, 125, 141, 147, 182] [15, 22, 24, 142, 148, 183] [16, 23, 25, 143, 149, 184] [17, 24, 26, 144, 150, 185] [18, 25, 27, 145, 151, 186] [19, 26, 115, 146, 152, 187] [20, 27, 116, 153, 188, 234] [28, 30, 65, 147, 154, 193] [21, 29, 31, 148, 155, 194] [22, 30, 32, 149, 156, 195] [23, 31, 33, 150, 157, 196] [24, 32, 34, 151, 158, 197] [25, 33, 35, 152, 159, 198] [26, 34, 120, 153, 160, 199] [27, 35, 121, 161, 200, 241] [36, 38, 76, 154, 162, 204] [28, 37, 39, 155, 163, 205] [29, 38, 40, 156, 164, 206] [30, 39, 41, 157, 165, 207] [31, 40, 42, 158, 166, 208] [32, 41, 43, 159, 167, 209] [33, 42, 44, 160, 168, 210] [34, 43, 123, 161, 169, 211] [35, 44, 124, 170, 212, 246] [45, 47, 87, 162, 171, 215] [36, 46, 48, 163, 172, 216] [37, 47, 49, 164, 173, 217] [38, 48, 50, 165, 174, 218] [39, 49, 51, 166, 175, 219] [40, 50, 52, 167, 176, 220] [41, 51, 53, 168, 177, 221] [42, 52, 54, 169, 178, 222] [43, 53, 125, 170, 179, 223] [15, 44, 54, 180, 224, 249] [55, 57, 98, 171, 181, 225] [45, 56, 58, 172, 182, 226] [46, 57, 59, 173, 183, 227] [47, 58, 60, 174, 184, 228] [48, 59, 61, 175, 185, 229] [49, 60, 62, 176, 186, 230] [50, 61, 63, 126, 177, 187] [51, 62, 64, 178, 188, 231] [52, 63, 65, 179, 189, 232] [21, 53, 64, 180, 190, 233] [22, 54, 65, 181, 191, 251] [66, 68, 107, 146, 181, 192] [55, 67, 69, 182, 193, 234] [56, 68, 70, 183, 194, 235] [57, 69, 71, 184, 195, 236] [58, 70, 72, 185, 196, 237] [59, 71, 73, 186, 197, 238] [60, 72, 74, 127, 187, 198] [61, 73, 75, 128, 188, 199] [62, 74, 76, 189, 200, 239] [28, 63, 75, 190, 201, 240] [29, 64, 76, 191, 192, 202] [77, 79, 114, 152, 192, 203] [66, 78, 80, 153, 193, 204] [67, 79, 81, 194, 205, 241] [68, 80, 82, 195, 206, 242] [69, 81, 83, 196, 207, 243] [70, 82, 84, 197, 208, 244] [71, 83, 85, 129, 198, 209] [72, 84, 86, 130, 199, 210] [73, 85, 87, 131, 200, 211] [36, 74, 86, 201, 212, 245] [37, 75, 87, 202, 203, 213] [88, 90, 119, 159, 203, 214] [77, 89, 91, 160, 204, 215] [78, 90, 92, 161, 205, 216] [79, 91, 93, 206, 217, 246] [80, 92, 94, 207, 218, 247] [81, 93, 95, 208, 219, 248] [82, 94, 96, 132, 209, 220] [83, 95, 97, 133, 210, 221] [84, 96, 98, 134, 211, 222] [45, 85, 97, 135, 212, 223] [46, 86, 98, 213, 214, 224] [88, 99, 101, 168, 215, 225] [89, 100, 102, 169, 216, 226] [90, 101, 103, 170, 217, 227] [91, 102, 104, 218, 228, 249] [0, 92, 103, 219, 229, 250] [93, 104, 105, 136, 220, 230] [95, 105, 107, 138, 222, 231] [55, 96, 106, 139, 223, 232] [56, 97, 107, 140, 224, 233] [99, 108, 110, 178, 226, 234] [100, 109, 111, 179, 227, 235] [101, 110, 112, 180, 228, 236] [1, 102, 111, 229, 237, 251] [2, 103, 112, 141, 230, 238] [66, 105, 113, 144, 232, 239] [67, 106, 114, 145, 233, 240] [108, 115, 117, 189, 235, 241] [109, 116, 118, 190, 236, 242] [3, 110, 117, 191, 237, 243] [4, 111, 118, 147, 238, 244] [78, 113, 119, 151, 240, 245] [115, 120, 122, 201, 242, 246] [6, 116, 121, 202, 243, 247] [7, 117, 122, 154, 244, 248] [10, 120, 123, 213, 247, 249] [11, 121, 124, 162, 248, 250] [16, 123, 125, 171, 250, 251]
raw submission JSON