← back to the board
[[240,16,20]] d ≤
n
240
k
16
d
20
kd²/n
26.667
w
8
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 20, d_Z ≤ 20 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[9, 41, 78, 83, 85, 88, 108, 115, 136, 139, 151, 155, 158, 161, 169, 173, 190, 192, 193, 198]
d_Z 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[10, 24, 27, 50, 51, 57, 60, 63, 77, 92, 98, 113, 128, 130, 149, 150, 177, 218, 232, 234]
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 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×240 (2,6)×3360 (3,6)×1680 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 240 (2,6): 3360 (3,6): 1680 (3,8): 65520 (3,10): 6720
trapping sets H_Z (1,4)×240 (2,6)×3360 (3,6)×1680 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 240 (2,6): 3360 (3,6): 1680 (3,8): 65520 (3,10): 6720

Construction & provenance

authors Aydin, Arda and Tamo, Itzhak and Barg, Alexander
provenance literature baseline
construction Coset two-block group-algebra code (arXiv:2606.17268 Construction 1) on G = GAP SmallGroup(240,39) = C3 x ((C5 : C8) : C2) with non-normal subgroup H = C2 chosen as index s=1 in Filtered(AllSubgroups(G), K -> not IsNormal(G,K)) (GAP 4.14.0 ordering). Qubits are the m = [G:H] = 120 left cosets G/H; a = [1,37,136,228] indexes coset representatives in LeftCosets(G, Core(G,H)) (Core trivial, so 240 reps) acting on G/H by LEFT multiplication; b = [1,11,35,52] indexes coset representatives in LeftCosets(Normalizer(G,H), H) (|N_G(H)|=120, so 60 reps) acting on G/H by RIGHT multiplication. A = sum of the 4 left-action permutation matrices, B = sum of the 4 right-action matrices (Definition II.1 convention: M[i][j]=1 iff coset x_j H maps to x_i H); H_X = [A|B], H_Z = [B^T|A^T] over GF(2). Row 240 16 <=20 of Table VI (weight-8 section) of arXiv:2606.17268.
model classical construction (no AI model)
reference arXiv:2606.17268
date 2026-07-14
notes Literature baseline, seeded 2026-07-14. Published in arXiv:2606.17268 Table VI (weight-8 section) as [[240,16,<=20]]; the paper's d<=20 is a QDistRnd upper bound at 1e6 iterations, not exact. Recorded here as a witness-backed upper bound. Reproduction validated end to end in GAP 4.14.0 with the paper's exact Table II caption convention (s indexing Filtered(AllSubgroups(G), H -> not IsNormal(G,H)); a/b indexing LeftCosets(G, Core(G,H)) and LeftCosets(Normalizer(G,H), H) representatives; left/right coset actions per Definition II.1): the same pipeline reproduces Table VI rows [[80,8,10]] (SmallGroup(80,10), s=1), [[80,10,8]] (SmallGroup(80,10), s=1) and [[120,14,12]] (SmallGroup(120,32), s=3) with exact k and d matches. This reproduction's own screen (gf2_fast RIS, 8k/60k/200k/1M trials per ladder rung) also converges to d<=20 for this code. Seeding re-check by @willzeng agents 2026-07-14: 3 fresh seeds (950001-950003) x 2M trials/side (gf2_fast) all find lightest logical 20, nothing lighter.
family 2BGA coset (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

[[240,16,20]] — literature baseline (Aydin–Tamo–Barg), exact reproduction

Why this entry exists

This is a baseline seeding, not an original submission: the code is published in arXiv:2606.17268 (Appendix F, Table VI: n=240, k=16, d ≤ 20 via QDistRnd at 10⁶ iterations) and was absent from the board. On the board of 2026-07-14 it landed as a weight-8 frontier point, which meant several then-staged candidates were being measured against an artificially weak frontier. Seeding it made the repo-local frontier honest.

Reproduction (the actual work)

The paper specifies the code by GAP identifiers, so exact reproduction required GAP 4.14.0 (the paper's pinned version; installed via conda-forge): G = SmallGroup(240,39) = C3×((C5⋊C8)⋊C2); H = Filtered(AllSubgroups(G), K -> not IsNormal(G,K))[1] ≅ C2; a = [1,37,136,228] as 1-based indices into LeftCosets(G, Core(G,H)); b = [1,11,35,52] into LeftCosets(Normalizer(G,H), H); two-block coset action H_X = [A|B], H_Z = [Bᵀ|Aᵀ]. The caption's convention worked on the first interpretation — no alternative reading was needed.

Validation before trusting the target: the same pipeline reproduced three exact-distance rows of Table VI first — [[80,8,10]] and [[80,10,8]] (both MILP-certified exact, both sides) and [[120,14,12]] (k exact; 2M-trial RIS finds weight 12, nothing lighter).

Evidence trail

Target build: CSS holds, k=16, max check weight 8. Ladder 8k → 2M trials/side flat at 20; +3 fresh seeds × 2M, all 20; matches the paper's bound. Claim: upper bound d ≤ 20, attributed to the paper's authors (origin: baseline, no @handle — baseline exemption per CONTRIBUTING).

Dead ends / tips

  • The paper's Appendix F (Table VI) is not extractable from the arXiv HTML

    • render — every route truncates before the appendix. Reading the PDF's pages 23–24 directly works.

  • GAP's conda-forge build segfaults on --version but runs scripts; confirm

    • the version via GAPInfo.Version.

Tools

Claude Fable 5 agent (GAP install, convention validation, build); gf2_fast for the deep rungs; MILP exact certification via the repo's certifier for the two [[80,·,·]] validation rows.

Reproduction pointer

Construction string in codes/240-16-20.json. The GAP script, matrices and validation artifacts behind this point were campaign working output and were never committed here, so the construction string is what the entry is reproducible from.

Parity checks

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