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[[288,16,18]] d ≤
n
288
k
16
d
18
kd²/n
18.0
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 18 · witness weight 18 (claimed upper_bound)
witness operator (support, 18 qubits)
[2, 14, 32, 38, 41, 47, 50, 62, 80, 86, 89, 95, 98, 110, 128, 134, 137, 143]
d_Z 18 · witness weight 18 (claimed upper_bound)
witness operator (support, 18 qubits)
[151, 160, 163, 166, 169, 187, 199, 208, 211, 214, 217, 235, 247, 256, 259, 262, 265, 283]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 7 · H_Z 7
qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)
trapping sets H_X (1,3)×144 (2,4)×432 (3,3)×144 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 144 (1,4): 144 (2,4): 432 (2,5): 1728 (2,6): 864 (3,3): 144 (3,4): 288 (3,5): 1296 (3,6): 14976 (3,7): 22608 (3,8): 6048 (3,9): 2592 (3,10): 576
trapping sets H_Z (1,3)×144 (2,4)×432 (3,3)×144 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 144 (1,4): 144 (2,4): 432 (2,5): 1728 (2,6): 864 (3,3): 144 (3,4): 288 (3,5): 1296 (3,6): 14976 (3,7): 22608 (3,8): 6048 (3,9): 2592 (3,10): 576

Construction & provenance

authors Qian, Dongheng and Li, Tianyi
provenance literature baseline
construction Lifted product (2BGA) over Z_12 x Z_12, reconstructed from arXiv:2608.08996 Table 1 (coset-orbit balanced product over Z_12 x Z_48 with normal stabilizer K=<y12>, reduced to G/K). A = y2 + y7 + x, B = y3 + x + x2 + x5 y9. n=2*12*12=288.
model classical construction (no AI model)
reference arXiv:2608.08996
date 2026
notes Literature baseline. Reconstructed from the paper's published construction data (Table 1, w=7): the balanced product over Z_12 x Z_48 with normal stabilizer K=<y^12> reduces to a lifted product over G/K = Z_12 x Z_12, with A = y^2 + y^7 + x and B = y^3 + x + x^2 + x^5 y^9 (y-exponents reduced mod 12). Paper reports d = 18 as MILP-exact; here recorded as a witness-backed upper bound (not server-certified).
family lifted product (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[288,16,18]] — lifted product over Z_12 × Z_12, from arXiv:2608.08996

Direction & hypothesis

The paper *Multi-agent discovery of practical quantum LDPC codes* (arXiv:2608.08996, Table 1) reports a [[288,16,18]] code at overall weight w=7 with kd²/n = 18.00, which it claims is the strongest known code in that weight class. The board currently holds [[288,16,16]] (2BGA, kd²/n = 14.2) at the same n,k — so this paper's code strictly dominates it on distance at identical block length and logical count. Reconstructing and verifying it is a direct frontier advance for the unrestricted / weight-8 cell.

The paper's construction is a coset-orbit balanced product over G = Z_12 × Z_48 with normal stabilizer K = ⟨y^12⟩ (order 4). Because K is normal, the balanced product reduces to an ordinary lifted product (2BGA) over G/K ≅ Z_12 × Z_12, with n = 2·|G/K| = 2·144 = 288. The protograph entries A, B ∈ F₂[G] project onto F₂[G/K] by reducing the y-exponent mod 12:

A = y² + y⁷ + x            ->  a = {(1,0),(0,2),(0,7)}
B = y³ + x + x² + x⁵y⁹     ->  b = {(1,0),(2,0),(0,3),(5,9)}

What was searched

No search was needed — this is a direct reconstruction of a published code. The 2BGA on Z_12 × Z_12 with supports a, b above was built with the kit's bb.build_bb, and its parameters recomputed from the parity-check matrices (the same facts the verifier checks).

Evidence trail

Reconstruction checks (all recomputed from H_X, H_Z):

  • n = 288, k = 16, CSS commutation H_X H_Zᵀ = 0: pass
  • max check weight: X = 7, Z = 7 → overall weight w = 7 (matches paper)
  • distance (RIS upper bound, gf2_fast): d ≤ 18, stable at both

    • 20,000 and 100,000 trials, witness weight 18 on the X side.

The paper reports d = 18 as MILP-exact; here it is recorded as a witness-backed upper bound (confidence: upper_bound), which is the honest tier for a submission without server certification. The witness is a genuine nontrivial logical of weight 18, verified by the gate.

Dead ends

None — this is a single reconstruction. The only subtlety is the reduction: the paper states the code as a balanced product over Z_12 × Z_48 with a normal K, which is equivalent to the 2BGA over Z_12 × Z_12 used here. The y-exponents in A, B (7, 9) exceed 12 and must be reduced mod 12 to land in G/K; doing so reproduces the claimed [[288,16,18]] exactly.

Tools

  • Model: none (deterministic reconstruction from the paper's published

    • construction data).

  • Repo tooling: research/kit/bb.py (2BGA builder), research/kit/css.py

    • (rank / CSS / k), research/kit/surrogate_fast.py + gf2_fast (distance upper bound), cli/qldpc.py submit (verifier + packaging).

  • Compute: trivial; distance search a few seconds at 100k trials.

Reproduction

from research.kit.bb import build_bb
from research.kit.css import compute_k, verify_css

HX, HZ = build_bb(12, 12,
                  A_terms=[(1,0),(0,2),(0,7)],   # y² + y⁷ + x
                  B_terms=[(1,0),(2,0),(0,3),(5,9)])  # y³ + x + x² + x⁵y⁹
assert verify_css(HX, HZ)
assert compute_k(HX, HZ) == 16

n = 2·12·12 = 288. Then `./qldpc submit recon_288_16_18.npz --authors @handle --family lifted-product --construction "..."` reproduces the submission.

Parity checks

X-checks 144 (max weight 7) · Z-checks 144 (max weight 7)
H_X (144 checks, sparse supports)
[2, 7, 12, 147, 156, 168, 213] [3, 8, 13, 148, 157, 169, 214] [4, 9, 14, 149, 158, 170, 215] [5, 10, 15, 150, 159, 171, 204] [6, 11, 16, 151, 160, 172, 205] [0, 7, 17, 152, 161, 173, 206] [1, 8, 18, 153, 162, 174, 207] [2, 9, 19, 154, 163, 175, 208] [3, 10, 20, 155, 164, 176, 209] [4, 11, 21, 144, 165, 177, 210] [0, 5, 22, 145, 166, 178, 211] [1, 6, 23, 146, 167, 179, 212] [14, 19, 24, 159, 168, 180, 225] [15, 20, 25, 160, 169, 181, 226] [16, 21, 26, 161, 170, 182, 227] [17, 22, 27, 162, 171, 183, 216] [18, 23, 28, 163, 172, 184, 217] [12, 19, 29, 164, 173, 185, 218] [13, 20, 30, 165, 174, 186, 219] [14, 21, 31, 166, 175, 187, 220] [15, 22, 32, 167, 176, 188, 221] [16, 23, 33, 156, 177, 189, 222] [12, 17, 34, 157, 178, 190, 223] [13, 18, 35, 158, 179, 191, 224] [26, 31, 36, 171, 180, 192, 237] [27, 32, 37, 172, 181, 193, 238] [28, 33, 38, 173, 182, 194, 239] [29, 34, 39, 174, 183, 195, 228] [30, 35, 40, 175, 184, 196, 229] [24, 31, 41, 176, 185, 197, 230] [25, 32, 42, 177, 186, 198, 231] [26, 33, 43, 178, 187, 199, 232] [27, 34, 44, 179, 188, 200, 233] [28, 35, 45, 168, 189, 201, 234] [24, 29, 46, 169, 190, 202, 235] [25, 30, 47, 170, 191, 203, 236] [38, 43, 48, 183, 192, 204, 249] [39, 44, 49, 184, 193, 205, 250] [40, 45, 50, 185, 194, 206, 251] [41, 46, 51, 186, 195, 207, 240] [42, 47, 52, 187, 196, 208, 241] [36, 43, 53, 188, 197, 209, 242] [37, 44, 54, 189, 198, 210, 243] [38, 45, 55, 190, 199, 211, 244] [39, 46, 56, 191, 200, 212, 245] [40, 47, 57, 180, 201, 213, 246] [36, 41, 58, 181, 202, 214, 247] [37, 42, 59, 182, 203, 215, 248] [50, 55, 60, 195, 204, 216, 261] [51, 56, 61, 196, 205, 217, 262] [52, 57, 62, 197, 206, 218, 263] [53, 58, 63, 198, 207, 219, 252] [54, 59, 64, 199, 208, 220, 253] [48, 55, 65, 200, 209, 221, 254] [49, 56, 66, 201, 210, 222, 255] [50, 57, 67, 202, 211, 223, 256] [51, 58, 68, 203, 212, 224, 257] [52, 59, 69, 192, 213, 225, 258] [48, 53, 70, 193, 214, 226, 259] [49, 54, 71, 194, 215, 227, 260] [62, 67, 72, 207, 216, 228, 273] [63, 68, 73, 208, 217, 229, 274] [64, 69, 74, 209, 218, 230, 275] [65, 70, 75, 210, 219, 231, 264] [66, 71, 76, 211, 220, 232, 265] [60, 67, 77, 212, 221, 233, 266] [61, 68, 78, 213, 222, 234, 267] [62, 69, 79, 214, 223, 235, 268] [63, 70, 80, 215, 224, 236, 269] [64, 71, 81, 204, 225, 237, 270] [60, 65, 82, 205, 226, 238, 271] [61, 66, 83, 206, 227, 239, 272] [74, 79, 84, 219, 228, 240, 285] [75, 80, 85, 220, 229, 241, 286] [76, 81, 86, 221, 230, 242, 287] [77, 82, 87, 222, 231, 243, 276] [78, 83, 88, 223, 232, 244, 277] [72, 79, 89, 224, 233, 245, 278] [73, 80, 90, 225, 234, 246, 279] [74, 81, 91, 226, 235, 247, 280] [75, 82, 92, 227, 236, 248, 281] [76, 83, 93, 216, 237, 249, 282] [72, 77, 94, 217, 238, 250, 283] [73, 78, 95, 218, 239, 251, 284] [86, 91, 96, 153, 231, 240, 252] [87, 92, 97, 154, 232, 241, 253] [88, 93, 98, 155, 233, 242, 254] [89, 94, 99, 144, 234, 243, 255] [90, 95, 100, 145, 235, 244, 256] [84, 91, 101, 146, 236, 245, 257] [85, 92, 102, 147, 237, 246, 258] [86, 93, 103, 148, 238, 247, 259] [87, 94, 104, 149, 239, 248, 260] [88, 95, 105, 150, 228, 249, 261] [84, 89, 106, 151, 229, 250, 262] [85, 90, 107, 152, 230, 251, 263] [98, 103, 108, 165, 243, 252, 264] [99, 104, 109, 166, 244, 253, 265] [100, 105, 110, 167, 245, 254, 266] [101, 106, 111, 156, 246, 255, 267] [102, 107, 112, 157, 247, 256, 268] [96, 103, 113, 158, 248, 257, 269] [97, 104, 114, 159, 249, 258, 270] [98, 105, 115, 160, 250, 259, 271] [99, 106, 116, 161, 251, 260, 272] [100, 107, 117, 162, 240, 261, 273] [96, 101, 118, 163, 241, 262, 274] [97, 102, 119, 164, 242, 263, 275] [110, 115, 120, 177, 255, 264, 276] [111, 116, 121, 178, 256, 265, 277] [112, 117, 122, 179, 257, 266, 278] [113, 118, 123, 168, 258, 267, 279] [114, 119, 124, 169, 259, 268, 280] [108, 115, 125, 170, 260, 269, 281] [109, 116, 126, 171, 261, 270, 282] [110, 117, 127, 172, 262, 271, 283] [111, 118, 128, 173, 263, 272, 284] [112, 119, 129, 174, 252, 273, 285] [108, 113, 130, 175, 253, 274, 286] [109, 114, 131, 176, 254, 275, 287] [122, 127, 132, 144, 189, 267, 276] [123, 128, 133, 145, 190, 268, 277] [124, 129, 134, 146, 191, 269, 278] [125, 130, 135, 147, 180, 270, 279] [126, 131, 136, 148, 181, 271, 280] [120, 127, 137, 149, 182, 272, 281] [121, 128, 138, 150, 183, 273, 282] [122, 129, 139, 151, 184, 274, 283] [123, 130, 140, 152, 185, 275, 284] [124, 131, 141, 153, 186, 264, 285] [120, 125, 142, 154, 187, 265, 286] [121, 126, 143, 155, 188, 266, 287] [0, 134, 139, 144, 156, 201, 279] [1, 135, 140, 145, 157, 202, 280] [2, 136, 141, 146, 158, 203, 281] [3, 137, 142, 147, 159, 192, 282] [4, 138, 143, 148, 160, 193, 283] [5, 132, 139, 149, 161, 194, 284] [6, 133, 140, 150, 162, 195, 285] [7, 134, 141, 151, 163, 196, 286] [8, 135, 142, 152, 164, 197, 287] [9, 136, 143, 153, 165, 198, 276] [10, 132, 137, 154, 166, 199, 277] [11, 133, 138, 155, 167, 200, 278]
H_Z (144 checks, sparse supports)
[9, 87, 120, 132, 149, 154, 276] [10, 88, 121, 133, 150, 155, 277] [11, 89, 122, 134, 144, 151, 278] [0, 90, 123, 135, 145, 152, 279] [1, 91, 124, 136, 146, 153, 280] [2, 92, 125, 137, 147, 154, 281] [3, 93, 126, 138, 148, 155, 282] [4, 94, 127, 139, 144, 149, 283] [5, 95, 128, 140, 145, 150, 284] [6, 84, 129, 141, 146, 151, 285] [7, 85, 130, 142, 147, 152, 286] [8, 86, 131, 143, 148, 153, 287] [0, 21, 99, 132, 144, 161, 166] [1, 22, 100, 133, 145, 162, 167] [2, 23, 101, 134, 146, 156, 163] [3, 12, 102, 135, 147, 157, 164] [4, 13, 103, 136, 148, 158, 165] [5, 14, 104, 137, 149, 159, 166] [6, 15, 105, 138, 150, 160, 167] [7, 16, 106, 139, 151, 156, 161] [8, 17, 107, 140, 152, 157, 162] [9, 18, 96, 141, 153, 158, 163] [10, 19, 97, 142, 154, 159, 164] [11, 20, 98, 143, 155, 160, 165] [0, 12, 33, 111, 156, 173, 178] [1, 13, 34, 112, 157, 174, 179] [2, 14, 35, 113, 158, 168, 175] [3, 15, 24, 114, 159, 169, 176] [4, 16, 25, 115, 160, 170, 177] [5, 17, 26, 116, 161, 171, 178] [6, 18, 27, 117, 162, 172, 179] [7, 19, 28, 118, 163, 168, 173] [8, 20, 29, 119, 164, 169, 174] [9, 21, 30, 108, 165, 170, 175] [10, 22, 31, 109, 166, 171, 176] [11, 23, 32, 110, 167, 172, 177] [12, 24, 45, 123, 168, 185, 190] [13, 25, 46, 124, 169, 186, 191] [14, 26, 47, 125, 170, 180, 187] [15, 27, 36, 126, 171, 181, 188] [16, 28, 37, 127, 172, 182, 189] [17, 29, 38, 128, 173, 183, 190] [18, 30, 39, 129, 174, 184, 191] [19, 31, 40, 130, 175, 180, 185] [20, 32, 41, 131, 176, 181, 186] [21, 33, 42, 120, 177, 182, 187] [22, 34, 43, 121, 178, 183, 188] [23, 35, 44, 122, 179, 184, 189] [24, 36, 57, 135, 180, 197, 202] [25, 37, 58, 136, 181, 198, 203] [26, 38, 59, 137, 182, 192, 199] [27, 39, 48, 138, 183, 193, 200] [28, 40, 49, 139, 184, 194, 201] [29, 41, 50, 140, 185, 195, 202] [30, 42, 51, 141, 186, 196, 203] [31, 43, 52, 142, 187, 192, 197] [32, 44, 53, 143, 188, 193, 198] [33, 45, 54, 132, 189, 194, 199] [34, 46, 55, 133, 190, 195, 200] [35, 47, 56, 134, 191, 196, 201] [3, 36, 48, 69, 192, 209, 214] [4, 37, 49, 70, 193, 210, 215] [5, 38, 50, 71, 194, 204, 211] [6, 39, 51, 60, 195, 205, 212] [7, 40, 52, 61, 196, 206, 213] [8, 41, 53, 62, 197, 207, 214] [9, 42, 54, 63, 198, 208, 215] [10, 43, 55, 64, 199, 204, 209] [11, 44, 56, 65, 200, 205, 210] [0, 45, 57, 66, 201, 206, 211] [1, 46, 58, 67, 202, 207, 212] [2, 47, 59, 68, 203, 208, 213] [15, 48, 60, 81, 204, 221, 226] [16, 49, 61, 82, 205, 222, 227] [17, 50, 62, 83, 206, 216, 223] [18, 51, 63, 72, 207, 217, 224] [19, 52, 64, 73, 208, 218, 225] [20, 53, 65, 74, 209, 219, 226] [21, 54, 66, 75, 210, 220, 227] [22, 55, 67, 76, 211, 216, 221] [23, 56, 68, 77, 212, 217, 222] [12, 57, 69, 78, 213, 218, 223] [13, 58, 70, 79, 214, 219, 224] [14, 59, 71, 80, 215, 220, 225] [27, 60, 72, 93, 216, 233, 238] [28, 61, 73, 94, 217, 234, 239] [29, 62, 74, 95, 218, 228, 235] [30, 63, 75, 84, 219, 229, 236] [31, 64, 76, 85, 220, 230, 237] [32, 65, 77, 86, 221, 231, 238] [33, 66, 78, 87, 222, 232, 239] [34, 67, 79, 88, 223, 228, 233] [35, 68, 80, 89, 224, 229, 234] [24, 69, 81, 90, 225, 230, 235] [25, 70, 82, 91, 226, 231, 236] [26, 71, 83, 92, 227, 232, 237] [39, 72, 84, 105, 228, 245, 250] [40, 73, 85, 106, 229, 246, 251] [41, 74, 86, 107, 230, 240, 247] [42, 75, 87, 96, 231, 241, 248] [43, 76, 88, 97, 232, 242, 249] [44, 77, 89, 98, 233, 243, 250] [45, 78, 90, 99, 234, 244, 251] [46, 79, 91, 100, 235, 240, 245] [47, 80, 92, 101, 236, 241, 246] [36, 81, 93, 102, 237, 242, 247] [37, 82, 94, 103, 238, 243, 248] [38, 83, 95, 104, 239, 244, 249] [51, 84, 96, 117, 240, 257, 262] [52, 85, 97, 118, 241, 258, 263] [53, 86, 98, 119, 242, 252, 259] [54, 87, 99, 108, 243, 253, 260] [55, 88, 100, 109, 244, 254, 261] [56, 89, 101, 110, 245, 255, 262] [57, 90, 102, 111, 246, 256, 263] [58, 91, 103, 112, 247, 252, 257] [59, 92, 104, 113, 248, 253, 258] [48, 93, 105, 114, 249, 254, 259] [49, 94, 106, 115, 250, 255, 260] [50, 95, 107, 116, 251, 256, 261] [63, 96, 108, 129, 252, 269, 274] [64, 97, 109, 130, 253, 270, 275] [65, 98, 110, 131, 254, 264, 271] [66, 99, 111, 120, 255, 265, 272] [67, 100, 112, 121, 256, 266, 273] [68, 101, 113, 122, 257, 267, 274] [69, 102, 114, 123, 258, 268, 275] [70, 103, 115, 124, 259, 264, 269] [71, 104, 116, 125, 260, 265, 270] [60, 105, 117, 126, 261, 266, 271] [61, 106, 118, 127, 262, 267, 272] [62, 107, 119, 128, 263, 268, 273] [75, 108, 120, 141, 264, 281, 286] [76, 109, 121, 142, 265, 282, 287] [77, 110, 122, 143, 266, 276, 283] [78, 111, 123, 132, 267, 277, 284] [79, 112, 124, 133, 268, 278, 285] [80, 113, 125, 134, 269, 279, 286] [81, 114, 126, 135, 270, 280, 287] [82, 115, 127, 136, 271, 276, 281] [83, 116, 128, 137, 272, 277, 282] [72, 117, 129, 138, 273, 278, 283] [73, 118, 130, 139, 274, 279, 284] [74, 119, 131, 140, 275, 280, 285]
Code ID 288-16-18 · download JSON · raw on GitHub