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

Share this result

Distance

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 ≤)

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

GAP script + matrices + validation artifacts: research/candidates/campaign-20260714/baseline-240-16-20/ (in the campaign branch history); construction string in codes/240-16-20.json.

Parity checks

X-checks 120 · Z-checks 120
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]
raw submission JSON