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[[246,4,22]] d ≤
n
246
k
4
d
22
kd²/n
7.87
w
6

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Distance

d_X 22 · witness weight 22 (claimed upper_bound)
witness operator (support, 22 qubits)
[12, 16, 18, 50, 63, 77, 80, 82, 84, 93, 97, 125, 133, 138, 140, 142, 160, 176, 197, 201, 217, 234]
d_Z 22 · witness weight 22 (claimed upper_bound)
witness operator (support, 22 qubits)
[2, 5, 8, 12, 17, 33, 54, 59, 64, 71, 89, 90, 99, 104, 107, 147, 178, 180, 182, 203, 205, 244]
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+x3y, g(x,y)=1+y+x2y-2 on the abelian quotient group Z2/L with twist basis a_1=[0, 123], a_2=[1, 22] (n=2|det[a_1,a_2]|=2*123). 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 123 · Z-checks 123
H_X (123 checks, sparse supports)
[0, 1, 11, 123, 125, 232] [1, 3, 16, 124, 127, 238] [2, 4, 17, 125, 128, 239] [3, 6, 22, 126, 130, 242] [4, 7, 23, 127, 131, 243] [5, 8, 24, 126, 128, 132] [6, 10, 29, 129, 134, 199] [7, 11, 30, 130, 135, 245] [8, 12, 31, 129, 131, 136] [9, 13, 32, 130, 132, 137] [10, 15, 37, 133, 139, 209] [11, 16, 38, 134, 140, 210] [12, 17, 39, 133, 135, 141] [13, 18, 40, 134, 136, 142] [14, 19, 41, 135, 137, 143] [15, 21, 46, 138, 145, 219] [16, 22, 47, 139, 146, 220] [17, 23, 48, 138, 140, 147] [18, 24, 49, 139, 141, 148] [19, 25, 50, 140, 142, 149] [20, 26, 51, 141, 143, 150] [21, 28, 56, 144, 152, 228] [22, 29, 57, 145, 153, 229] [23, 30, 58, 144, 146, 154] [24, 31, 59, 145, 147, 155] [25, 32, 60, 146, 148, 156] [26, 33, 61, 147, 149, 157] [27, 34, 62, 148, 150, 158] [28, 36, 66, 151, 160, 235] [29, 37, 67, 152, 161, 236] [30, 38, 68, 151, 153, 162] [31, 39, 69, 152, 154, 163] [32, 40, 70, 153, 155, 164] [33, 41, 71, 154, 156, 165] [34, 42, 72, 155, 157, 166] [35, 43, 73, 156, 158, 167] [20, 36, 45, 159, 169, 240] [37, 46, 77, 160, 170, 241] [38, 47, 78, 159, 161, 171] [39, 48, 79, 160, 162, 172] [40, 49, 80, 161, 163, 173] [41, 50, 81, 162, 164, 174] [42, 51, 82, 163, 165, 175] [43, 52, 83, 164, 166, 176] [44, 53, 84, 165, 167, 177] [26, 45, 55, 128, 168, 179] [27, 46, 56, 169, 180, 244] [47, 57, 88, 168, 170, 181] [48, 58, 89, 169, 171, 182] [49, 59, 90, 170, 172, 183] [50, 60, 91, 171, 173, 184] [51, 61, 92, 172, 174, 185] [52, 62, 93, 173, 175, 186] [53, 63, 94, 174, 176, 187] [54, 64, 95, 175, 177, 188] [14, 33, 55, 131, 178, 189] [34, 56, 66, 132, 179, 190] [35, 57, 67, 178, 180, 191] [58, 68, 98, 179, 181, 192] [59, 69, 99, 180, 182, 193] [60, 70, 100, 181, 183, 194] [61, 71, 101, 182, 184, 195] [62, 72, 102, 183, 185, 196] [63, 73, 103, 184, 186, 197] [64, 74, 104, 185, 187, 198] [65, 75, 105, 186, 188, 199] [20, 42, 66, 136, 189, 200] [43, 67, 77, 137, 190, 201] [44, 68, 78, 189, 191, 202] [69, 79, 107, 190, 192, 203] [70, 80, 108, 191, 193, 204] [71, 81, 109, 192, 194, 205] [72, 82, 110, 193, 195, 206] [0, 73, 83, 194, 196, 207] [74, 84, 111, 195, 197, 208] [75, 85, 112, 196, 198, 209] [76, 86, 113, 197, 199, 210] [27, 52, 77, 142, 200, 211] [53, 78, 88, 143, 201, 212] [54, 79, 89, 200, 202, 213] [80, 90, 114, 201, 203, 214] [81, 91, 115, 202, 204, 215] [82, 92, 116, 203, 205, 216] [1, 83, 93, 204, 206, 217] [2, 84, 94, 205, 207, 218] [85, 95, 117, 206, 208, 219] [86, 96, 118, 207, 209, 220] [36, 87, 97, 138, 208, 210] [35, 63, 88, 149, 211, 221] [64, 89, 98, 150, 212, 222] [65, 90, 99, 211, 213, 223] [91, 100, 119, 212, 214, 224] [92, 101, 120, 213, 215, 225] [3, 93, 102, 214, 216, 226] [4, 94, 103, 215, 217, 227] [5, 95, 104, 216, 218, 228] [96, 105, 121, 217, 219, 229] [45, 97, 106, 144, 218, 220] [44, 74, 98, 157, 221, 230] [75, 99, 107, 158, 222, 231] [76, 100, 108, 221, 223, 232] [101, 109, 122, 222, 224, 233] [6, 102, 110, 123, 223, 225] [0, 7, 103, 224, 226, 234] [8, 104, 111, 225, 227, 235] [9, 105, 112, 226, 228, 236] [55, 106, 113, 151, 227, 229] [54, 85, 107, 166, 230, 237] [86, 108, 114, 167, 231, 238] [87, 109, 115, 230, 232, 239] [10, 110, 116, 124, 231, 233] [2, 12, 111, 233, 234, 240] [13, 112, 117, 123, 235, 241] [14, 113, 118, 159, 234, 236] [65, 96, 114, 176, 237, 242] [97, 115, 119, 177, 238, 243] [15, 116, 120, 126, 237, 239] [5, 18, 117, 124, 240, 244] [19, 118, 121, 125, 168, 241] [76, 106, 119, 187, 242, 245] [21, 120, 122, 129, 188, 243] [9, 25, 121, 127, 178, 244] [28, 87, 122, 133, 198, 245]
H_Z (123 checks, sparse supports)
[0, 102, 112, 123, 196, 226] [1, 110, 117, 123, 124, 206] [0, 2, 118, 125, 207, 234] [3, 5, 116, 124, 126, 216] [1, 4, 121, 125, 127, 217] [2, 5, 45, 128, 218, 240] [6, 8, 120, 126, 129, 225] [3, 7, 9, 127, 130, 226] [4, 8, 55, 128, 131, 227] [5, 9, 56, 132, 228, 244] [10, 12, 122, 129, 133, 233] [6, 11, 13, 123, 130, 134] [7, 12, 14, 131, 135, 234] [8, 13, 66, 132, 136, 235] [9, 14, 67, 137, 178, 236] [15, 17, 87, 133, 138, 239] [10, 16, 18, 124, 134, 139] [11, 17, 19, 125, 135, 140] [12, 18, 20, 136, 141, 240] [13, 19, 77, 137, 142, 241] [14, 20, 78, 143, 159, 189] [21, 23, 97, 138, 144, 243] [15, 22, 24, 126, 139, 145] [16, 23, 25, 127, 140, 146] [17, 24, 26, 128, 141, 147] [18, 25, 27, 142, 148, 244] [19, 26, 88, 143, 149, 168] [20, 27, 89, 150, 169, 200] [28, 30, 106, 144, 151, 245] [21, 29, 31, 129, 145, 152] [22, 30, 32, 130, 146, 153] [23, 31, 33, 131, 147, 154] [24, 32, 34, 132, 148, 155] [25, 33, 35, 149, 156, 178] [26, 34, 98, 150, 157, 179] [27, 35, 99, 158, 180, 211] [36, 38, 113, 151, 159, 210] [28, 37, 39, 133, 152, 160] [29, 38, 40, 134, 153, 161] [30, 39, 41, 135, 154, 162] [31, 40, 42, 136, 155, 163] [32, 41, 43, 137, 156, 164] [33, 42, 44, 157, 165, 189] [34, 43, 107, 158, 166, 190] [35, 44, 108, 167, 191, 221] [45, 47, 118, 159, 168, 220] [36, 46, 48, 138, 160, 169] [37, 47, 49, 139, 161, 170] [38, 48, 50, 140, 162, 171] [39, 49, 51, 141, 163, 172] [40, 50, 52, 142, 164, 173] [41, 51, 53, 143, 165, 174] [42, 52, 54, 166, 175, 200] [43, 53, 114, 167, 176, 201] [44, 54, 115, 177, 202, 230] [55, 57, 121, 168, 178, 229] [45, 56, 58, 144, 169, 179] [46, 57, 59, 145, 170, 180] [47, 58, 60, 146, 171, 181] [48, 59, 61, 147, 172, 182] [49, 60, 62, 148, 173, 183] [50, 61, 63, 149, 174, 184] [51, 62, 64, 150, 175, 185] [52, 63, 65, 176, 186, 211] [53, 64, 119, 177, 187, 212] [54, 65, 120, 188, 213, 237] [55, 66, 68, 151, 179, 189] [56, 67, 69, 152, 180, 190] [57, 68, 70, 153, 181, 191] [58, 69, 71, 154, 182, 192] [59, 70, 72, 155, 183, 193] [60, 71, 73, 156, 184, 194] [61, 72, 74, 157, 185, 195] [62, 73, 75, 158, 186, 196] [63, 74, 76, 187, 197, 221] [64, 75, 122, 188, 198, 222] [6, 65, 76, 199, 223, 242] [66, 77, 79, 160, 190, 200] [67, 78, 80, 161, 191, 201] [68, 79, 81, 162, 192, 202] [69, 80, 82, 163, 193, 203] [70, 81, 83, 164, 194, 204] [71, 82, 84, 165, 195, 205] [72, 83, 85, 166, 196, 206] [73, 84, 86, 167, 197, 207] [74, 85, 87, 198, 208, 230] [10, 75, 86, 199, 209, 231] [11, 76, 87, 210, 232, 245] [77, 88, 90, 170, 201, 211] [78, 89, 91, 171, 202, 212] [79, 90, 92, 172, 203, 213] [80, 91, 93, 173, 204, 214] [81, 92, 94, 174, 205, 215] [82, 93, 95, 175, 206, 216] [83, 94, 96, 176, 207, 217] [84, 95, 97, 177, 208, 218] [15, 85, 96, 209, 219, 237] [16, 86, 97, 210, 220, 238] [88, 98, 100, 181, 212, 221] [89, 99, 101, 182, 213, 222] [90, 100, 102, 183, 214, 223] [91, 101, 103, 184, 215, 224] [92, 102, 104, 185, 216, 225] [93, 103, 105, 186, 217, 226] [94, 104, 106, 187, 218, 227] [21, 95, 105, 188, 219, 228] [22, 96, 106, 220, 229, 242] [98, 107, 109, 192, 222, 230] [99, 108, 110, 193, 223, 231] [0, 100, 109, 194, 224, 232] [101, 110, 111, 195, 225, 233] [103, 111, 113, 197, 227, 234] [28, 104, 112, 198, 228, 235] [29, 105, 113, 199, 229, 236] [107, 114, 116, 203, 231, 237] [1, 108, 115, 204, 232, 238] [2, 109, 116, 205, 233, 239] [36, 111, 117, 208, 235, 240] [37, 112, 118, 209, 236, 241] [3, 114, 119, 214, 238, 242] [4, 115, 120, 215, 239, 243] [46, 117, 121, 219, 241, 244] [7, 119, 122, 224, 243, 245]
raw submission JSON