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[[224,6,20]] d ≤
n
224
k
6
d
20
kd²/n
10.714
w
6

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Distance

d_X 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[4, 7, 23, 33, 39, 55, 64, 74, 89, 96, 120, 138, 139, 161, 174, 180, 197, 201, 207, 213]
d_Z 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[14, 17, 21, 84, 85, 88, 89, 90, 93, 109, 134, 141, 148, 155, 166, 180, 194, 197, 202, 206]
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-3y2, g(x,y)=1+y+x-3y-1 on the abelian quotient group Z2/L with twist basis a_1=[0, 28], a_2=[4, -6] (n=2|det[a_1,a_2]|=2*112). 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 112 · Z-checks 112
H_X (112 checks, sparse supports)
[0, 1, 87, 112, 114, 196] [1, 3, 93, 113, 116, 202] [2, 4, 94, 114, 117, 203] [3, 6, 99, 115, 119, 208] [4, 7, 100, 116, 120, 209] [5, 8, 101, 117, 121, 210] [5, 6, 10, 118, 123, 214] [7, 11, 105, 119, 124, 215] [8, 12, 106, 120, 125, 216] [9, 13, 107, 121, 126, 199] [8, 10, 15, 122, 128, 219] [9, 11, 16, 112, 123, 129] [12, 17, 109, 124, 130, 220] [13, 18, 110, 125, 131, 205] [1, 14, 19, 126, 132, 206] [12, 15, 21, 127, 134, 222] [13, 16, 22, 113, 128, 135] [14, 17, 23, 114, 129, 136] [18, 24, 111, 130, 137, 211] [3, 19, 25, 131, 138, 212] [4, 20, 26, 122, 132, 213] [17, 21, 27, 133, 140, 223] [18, 22, 28, 115, 134, 141] [19, 23, 29, 116, 135, 142] [20, 24, 30, 117, 136, 143] [6, 25, 31, 137, 144, 217] [7, 26, 32, 127, 138, 218] [23, 27, 33, 132, 139, 146] [24, 28, 34, 118, 140, 147] [25, 29, 35, 119, 141, 148] [26, 30, 36, 120, 142, 149] [10, 31, 37, 121, 143, 150] [11, 32, 38, 133, 144, 221] [29, 33, 39, 138, 145, 152] [30, 34, 40, 122, 146, 153] [31, 35, 41, 123, 147, 154] [32, 36, 42, 124, 148, 155] [15, 37, 43, 125, 149, 156] [16, 38, 44, 126, 139, 150] [35, 39, 45, 144, 151, 158] [36, 40, 46, 127, 152, 159] [37, 41, 47, 128, 153, 160] [38, 42, 48, 129, 154, 161] [21, 43, 49, 130, 155, 162] [22, 44, 50, 131, 145, 156] [41, 45, 51, 150, 157, 164] [42, 46, 52, 133, 158, 165] [43, 47, 53, 134, 159, 166] [44, 48, 54, 135, 160, 167] [27, 49, 55, 136, 161, 168] [28, 50, 56, 137, 151, 162] [47, 51, 57, 156, 163, 170] [48, 52, 58, 139, 164, 171] [49, 53, 59, 140, 165, 172] [50, 54, 60, 141, 166, 173] [33, 55, 61, 142, 167, 174] [34, 56, 62, 143, 157, 168] [53, 57, 63, 162, 169, 176] [54, 58, 64, 145, 170, 177] [55, 59, 65, 146, 171, 178] [56, 60, 66, 147, 172, 179] [39, 61, 67, 148, 173, 180] [40, 62, 68, 149, 163, 174] [59, 63, 69, 168, 175, 182] [60, 64, 70, 151, 176, 183] [61, 65, 71, 152, 177, 184] [62, 66, 72, 153, 178, 185] [45, 67, 73, 154, 179, 186] [46, 68, 74, 155, 169, 180] [65, 69, 75, 174, 181, 188] [66, 70, 76, 157, 182, 189] [67, 71, 77, 158, 183, 190] [68, 72, 78, 159, 184, 191] [51, 73, 79, 160, 185, 192] [52, 74, 80, 161, 175, 186] [71, 75, 81, 180, 187, 194] [72, 76, 82, 163, 188, 195] [73, 77, 83, 164, 189, 196] [74, 78, 84, 165, 190, 197] [57, 79, 85, 166, 191, 198] [58, 80, 86, 167, 181, 192] [77, 81, 87, 186, 193, 200] [78, 82, 88, 169, 194, 201] [79, 83, 89, 170, 195, 202] [80, 84, 90, 171, 196, 203] [63, 85, 91, 172, 197, 204] [64, 86, 92, 173, 187, 198] [83, 87, 93, 192, 199, 206] [84, 88, 94, 175, 200, 207] [85, 89, 95, 176, 201, 208] [86, 90, 96, 177, 202, 209] [69, 91, 97, 178, 203, 210] [70, 92, 98, 179, 193, 204] [89, 93, 99, 198, 205, 212] [90, 94, 100, 181, 206, 213] [91, 95, 101, 182, 207, 214] [92, 96, 102, 183, 208, 215] [75, 97, 103, 184, 209, 216] [76, 98, 104, 185, 199, 210] [5, 95, 99, 204, 211, 217] [96, 100, 105, 187, 212, 218] [97, 101, 106, 188, 213, 219] [98, 102, 107, 112, 189, 214] [0, 81, 103, 190, 215, 220] [82, 104, 108, 191, 205, 216] [9, 102, 105, 193, 217, 221] [103, 106, 109, 194, 218, 222] [104, 107, 110, 113, 195, 219] [2, 88, 108, 197, 211, 220] [0, 14, 109, 200, 221, 223] [108, 110, 111, 115, 201, 222] [2, 20, 111, 118, 207, 223]
H_Z (112 checks, sparse supports)
[0, 11, 102, 112, 215, 221] [1, 16, 107, 112, 113, 126] [0, 2, 17, 114, 220, 223] [3, 22, 110, 113, 115, 131] [1, 4, 23, 114, 116, 132] [2, 5, 24, 117, 118, 211] [6, 28, 111, 115, 118, 137] [3, 7, 29, 116, 119, 138] [4, 8, 30, 117, 120, 122] [5, 9, 31, 121, 123, 217] [10, 20, 34, 118, 122, 143] [6, 11, 35, 119, 123, 144] [7, 12, 36, 120, 124, 127] [8, 13, 37, 121, 125, 128] [9, 14, 38, 126, 129, 221] [15, 26, 40, 122, 127, 149] [10, 16, 41, 123, 128, 150] [11, 17, 42, 124, 129, 133] [12, 18, 43, 125, 130, 134] [13, 19, 44, 126, 131, 135] [14, 20, 27, 132, 136, 223] [21, 32, 46, 127, 133, 155] [15, 22, 47, 128, 134, 156] [16, 23, 48, 129, 135, 139] [17, 24, 49, 130, 136, 140] [18, 25, 50, 131, 137, 141] [19, 26, 33, 132, 138, 142] [27, 38, 52, 133, 139, 161] [21, 28, 53, 134, 140, 162] [22, 29, 54, 135, 141, 145] [23, 30, 55, 136, 142, 146] [24, 31, 56, 137, 143, 147] [25, 32, 39, 138, 144, 148] [33, 44, 58, 139, 145, 167] [27, 34, 59, 140, 146, 168] [28, 35, 60, 141, 147, 151] [29, 36, 61, 142, 148, 152] [30, 37, 62, 143, 149, 153] [31, 38, 45, 144, 150, 154] [39, 50, 64, 145, 151, 173] [33, 40, 65, 146, 152, 174] [34, 41, 66, 147, 153, 157] [35, 42, 67, 148, 154, 158] [36, 43, 68, 149, 155, 159] [37, 44, 51, 150, 156, 160] [45, 56, 70, 151, 157, 179] [39, 46, 71, 152, 158, 180] [40, 47, 72, 153, 159, 163] [41, 48, 73, 154, 160, 164] [42, 49, 74, 155, 161, 165] [43, 50, 57, 156, 162, 166] [51, 62, 76, 157, 163, 185] [45, 52, 77, 158, 164, 186] [46, 53, 78, 159, 165, 169] [47, 54, 79, 160, 166, 170] [48, 55, 80, 161, 167, 171] [49, 56, 63, 162, 168, 172] [57, 68, 82, 163, 169, 191] [51, 58, 83, 164, 170, 192] [52, 59, 84, 165, 171, 175] [53, 60, 85, 166, 172, 176] [54, 61, 86, 167, 173, 177] [55, 62, 69, 168, 174, 178] [63, 74, 88, 169, 175, 197] [57, 64, 89, 170, 176, 198] [58, 65, 90, 171, 177, 181] [59, 66, 91, 172, 178, 182] [60, 67, 92, 173, 179, 183] [61, 68, 75, 174, 180, 184] [69, 80, 94, 175, 181, 203] [63, 70, 95, 176, 182, 204] [64, 71, 96, 177, 183, 187] [65, 72, 97, 178, 184, 188] [66, 73, 98, 179, 185, 189] [67, 74, 81, 180, 186, 190] [75, 86, 100, 181, 187, 209] [69, 76, 101, 182, 188, 210] [70, 77, 102, 183, 189, 193] [71, 78, 103, 184, 190, 194] [72, 79, 104, 185, 191, 195] [73, 80, 87, 186, 192, 196] [81, 92, 105, 187, 193, 215] [75, 82, 106, 188, 194, 216] [76, 83, 107, 189, 195, 199] [0, 77, 84, 190, 196, 200] [78, 85, 108, 191, 197, 201] [79, 86, 93, 192, 198, 202] [9, 87, 98, 112, 193, 199] [81, 88, 109, 194, 200, 220] [82, 89, 110, 195, 201, 205] [1, 83, 90, 196, 202, 206] [2, 84, 91, 197, 203, 207] [85, 92, 99, 198, 204, 208] [13, 93, 104, 113, 199, 205] [14, 87, 94, 114, 200, 206] [88, 95, 111, 201, 207, 211] [3, 89, 96, 202, 208, 212] [4, 90, 97, 203, 209, 213] [5, 91, 98, 204, 210, 214] [18, 99, 108, 115, 205, 211] [19, 93, 100, 116, 206, 212] [20, 94, 101, 117, 207, 213] [6, 95, 102, 208, 214, 217] [7, 96, 103, 209, 215, 218] [8, 97, 104, 210, 216, 219] [25, 99, 105, 119, 212, 217] [26, 100, 106, 120, 213, 218] [10, 101, 107, 121, 214, 219] [12, 103, 108, 216, 220, 222] [32, 105, 109, 124, 218, 221] [15, 106, 110, 125, 219, 222] [21, 109, 111, 130, 222, 223]
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