← back to the board
[[336,88,12]] d ≤
n
336
k
88
d
12
kd²/n
37.714
w
8
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · 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 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[7, 46, 59, 100, 120, 160, 172, 249, 252, 272, 295, 333]
d_Z 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[22, 41, 51, 72, 139, 166, 174, 201, 258, 261, 302, 307]
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 3 · H_Z 3
trapping sets H_X (1,3)×336 (2,4)×3528 (3,3)×420 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 336 (2,4): 3528 (3,3): 420 (3,5): 48132 (3,7): 7056
trapping sets H_Z (1,3)×336 (2,4)×3528 (3,3)×420 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 336 (2,4): 3528 (3,3): 420 (3,5): 48132 (3,7): 7056

Construction & provenance

authors @MathysRennela and Okada, Koki and Kasai, Kenta and Maskara, Nishad
provenance literature baseline
construction Pair-partition CPM/GPM CSS code (Okada-Kasai arXiv:2607.14091 catalogue), (J,L,P)=(3,8,42), check weight 8; H_X and H_Z taken verbatim from the published instance at https://kasai.ict.eng.isct.ac.jp/pair_partition_cpm_css_codes_data/distance_records/gpm_pp_frontier_20260811/s3c7_d12/gpm_pp_s3c7_frontier.npz (sha256 4eb37b67d1c19638c9de37b16ed5647534673f7ea73ff2a42354d3f4f5e65484), structure and distance re-verified here
model Mimo-V2.6-Flash (claimed, not verified)
reference arXiv:2607.14091
date 2026-09-29
notes Literature reproduction of a published instance from the Okada-Kasai pair-partition catalogue (https://kasai.ict.eng.isct.ac.jp/pair_partition_cpm_css_codes_20260714.html), not a new construction search. Equivalence checked at submission time: the dedup gate finds no exact or graph-equivalent existing entry, and verify/validate_candidate.py reports nothing dominating this code in its cell.
family pair-partition CPM (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

[[336,88,12]] pair-partition CPM/GPM CSS code, (J,L,P) = (3,8,42)

Direction & hypothesis

Target cell: CSS, weight-8 x unrestricted — check weight 8 and no layout, so the entry ranks on the weight-8 board. The authoritative catalogue that accompanies arXiv:2607.14091 (Okada and Kasai) publishes 44 pair-partition codes with exact distances, and this repository carried only part of them. The hypothesis was deliberately narrow: screen every catalogue row against this repository's admissibility cap and against the board's own Pareto test, and expect survivors to land as new Pareto points rather than displacers, because those rows were published for this family's parameters and were never searched against this board.

What was searched

No construction search. The catalogue rows were screened, not optimised:

  • the catalogue page names 61 distinct [[n,k,d]] triples, and 36 of them sit

    • at n <= 700, inside the blocklength cap;

  • 23 of those 36 were already in codes/ before this batch;
  • 13 were absent. Ten of them cleared verify/validate_candidate.py as

    • non-dominated in their own cell and make up this batch. The other three are [[110,8,12]], [[190,8,18]] and [[200,54,10]], accounted for under dead ends.

The check matrices are the authors' published instance, used byte for byte. Screening reproduced here: CSS commutation H_X H_Z^T = 0 over GF(2); rank(H_X) = rank(H_Z) = 124, hence k = 336 - 248 = 88; every row of both matrices has weight 8. Only after those agreed was a distance search run.

Published description of this instance: GPM-PP frontier addition over the noncyclic S3 x C7 regular action, seed 2026081203, trial 118.

Evidence trail

The submit gate for this instance: 3,000 Python RIS trials per side, then a 2,000,000-trial verify/gf2_fast accelerator pass, both at seed 0; the verifier then re-ran a refutation search against the written entry. Lightest logical found on each side was d = 12, matching the catalogue's claimed distance, with both witnesses written into codes/336-88-12.json.

python verify/qldpc_verify.py codes/336-88-12.json exits 0. The catalogue reports these distances as exact; this repository records only the witness-backed upper bound, because verify/certify.py is out of its envelope at k = 88. CI re-runs a deeper refutation search on the PR and the weekly sweep keeps re-testing, so a wrong d would surface as a refutation rather than a quiet pass.

Dead ends

  • The eight catalogue rows between n = 700 and n = 1000 and the seventeen

    • past n = 1000 were left out of this batch. Past 1000 they fail the blocklength cap outright; between 700 and 1000 they would additionally need w <= 8 and d <= 40, and their check weights were not screened here. That is a stated gap, not a claim of inadmissibility — it is the obvious next batch.

  • Exact certification was not attempted. verify/certify.py is measured to

    • hold only at d <= 13 and k <= 12, and this code runs at k = 88, so the claim here is an upper bound by design rather than a shortfall of the search.

  • No layout was supplied, so the code ranks as unrestricted. A toric-style

    • layout with a smaller neighbourhood would open a different cell, but the published instance is a circulant construction and no coordinates come with it; inventing some would be a separate submission with its own evidence.

  • Three catalogue rows are admissible, absent, and still not submitted here.

    • [[110,8,12]] and [[190,8,18]] are reported dominated by verify/validate_candidate.py (by [[104,8,12]] and [[170,8,20]] respectively), so they would ship as a dominated entry beside a stronger one. [[200,54,10]] appears only as a GPM-PP comparison point in the catalogue's prose and ships no instance file, so there is nothing to submit even if it cleared the frontier.

Tools

Model: Mimo-V2.6-Flash, driving an opencode agent session opened by @MathysRennela, who opened this PR. Repo tooling: ./qldpc submit for the witness search and packaging, verify/qldpc_verify.py for the local gate, verify/check_prose.py and verify/prepush_prose_check.sh for the prose gate. The accelerator is verify/gf2_fast, built from verify/gf2_fast.cpp.

Reproduction

Rebuild (H_X, H_Z) with no other input:

1. Download the published instance from https://kasai.ict.eng.isct.ac.jp/pair_partition_cpm_css_codes_data/distance_records/gpm_pp_frontier_20260811/s3c7_d12/gpm_pp_s3c7_frontier.npz (sha256 4eb37b67d1c19638c9de37b16ed5647534673f7ea73ff2a42354d3f4f5e65484). 2. The file stores dense binary H_X and H_Z of shape 126 x 336 under the keys Hx and Hz; load them directly, no expansion step is needed. 3. Confirm H_X H_Z^T = 0, rank 124 on each side (the 126 rows carry two redundancies), k = 88, and max row weight 8. 4. Only after those agree should a distance search be run; the witnesses in codes/336-88-12.json are the ones this entry stands on.

Parity checks

X-checks 126 (max weight 8) · Z-checks 126 (max weight 8)
H_X (126 checks, sparse supports)
[29, 73, 101, 154, 203, 212, 268, 317] [28, 74, 100, 153, 202, 210, 267, 316] [27, 72, 99, 155, 201, 211, 269, 315] [26, 76, 98, 151, 200, 215, 265, 314] [25, 77, 97, 150, 199, 213, 264, 313] [24, 75, 96, 152, 198, 214, 266, 312] [35, 79, 107, 160, 209, 218, 274, 323] [34, 80, 106, 159, 208, 216, 273, 322] [33, 78, 105, 161, 207, 217, 275, 321] [32, 82, 104, 157, 206, 221, 271, 320] [31, 83, 103, 156, 205, 219, 270, 319] [30, 81, 102, 158, 204, 220, 272, 318] [41, 43, 113, 166, 173, 224, 280, 329] [40, 44, 112, 165, 172, 222, 279, 328] [39, 42, 111, 167, 171, 223, 281, 327] [38, 46, 110, 163, 170, 227, 277, 326] [37, 47, 109, 162, 169, 225, 276, 325] [36, 45, 108, 164, 168, 226, 278, 324] [5, 49, 119, 130, 179, 230, 286, 335] [4, 50, 118, 129, 178, 228, 285, 334] [3, 48, 117, 131, 177, 229, 287, 333] [2, 52, 116, 127, 176, 233, 283, 332] [1, 53, 115, 126, 175, 231, 282, 331] [0, 51, 114, 128, 174, 232, 284, 330] [11, 55, 125, 136, 185, 236, 292, 299] [10, 56, 124, 135, 184, 234, 291, 298] [9, 54, 123, 137, 183, 235, 293, 297] [8, 58, 122, 133, 182, 239, 289, 296] [7, 59, 121, 132, 181, 237, 288, 295] [6, 57, 120, 134, 180, 238, 290, 294] [17, 61, 89, 142, 191, 242, 256, 305] [16, 62, 88, 141, 190, 240, 255, 304] [15, 60, 87, 143, 189, 241, 257, 303] [14, 64, 86, 139, 188, 245, 253, 302] [13, 65, 85, 138, 187, 243, 252, 301] [12, 63, 84, 140, 186, 244, 254, 300] [23, 67, 95, 148, 197, 248, 262, 311] [22, 68, 94, 147, 196, 246, 261, 310] [21, 66, 93, 149, 195, 247, 263, 309] [20, 70, 92, 145, 194, 251, 259, 308] [19, 71, 91, 144, 193, 249, 258, 307] [18, 69, 90, 146, 192, 250, 260, 306] [17, 46, 121, 144, 182, 216, 279, 323] [16, 45, 122, 145, 180, 217, 281, 322] [15, 47, 120, 146, 181, 218, 280, 321] [14, 43, 124, 147, 185, 219, 276, 320] [13, 42, 125, 148, 183, 220, 278, 319] [12, 44, 123, 149, 184, 221, 277, 318] [23, 52, 85, 150, 188, 222, 285, 329] [22, 51, 86, 151, 186, 223, 287, 328] [21, 53, 84, 152, 187, 224, 286, 327] [20, 49, 88, 153, 191, 225, 282, 326] [19, 48, 89, 154, 189, 226, 284, 325] [18, 50, 87, 155, 190, 227, 283, 324] [29, 58, 91, 156, 194, 228, 291, 335] [28, 57, 92, 157, 192, 229, 293, 334] [27, 59, 90, 158, 193, 230, 292, 333] [26, 55, 94, 159, 197, 231, 288, 332] [25, 54, 95, 160, 195, 232, 290, 331] [24, 56, 93, 161, 196, 233, 289, 330] [35, 64, 97, 162, 200, 234, 255, 299] [34, 63, 98, 163, 198, 235, 257, 298] [33, 65, 96, 164, 199, 236, 256, 297] [32, 61, 100, 165, 203, 237, 252, 296] [31, 60, 101, 166, 201, 238, 254, 295] [30, 62, 99, 167, 202, 239, 253, 294] [41, 70, 103, 126, 206, 240, 261, 305] [40, 69, 104, 127, 204, 241, 263, 304] [39, 71, 102, 128, 205, 242, 262, 303] [38, 67, 106, 129, 209, 243, 258, 302] [37, 66, 107, 130, 207, 244, 260, 301] [36, 68, 105, 131, 208, 245, 259, 300] [5, 76, 109, 132, 170, 246, 267, 311] [4, 75, 110, 133, 168, 247, 269, 310] [3, 77, 108, 134, 169, 248, 268, 309] [2, 73, 112, 135, 173, 249, 264, 308] [1, 72, 113, 136, 171, 250, 266, 307] [0, 74, 111, 137, 172, 251, 265, 306] [11, 82, 115, 138, 176, 210, 273, 317] [10, 81, 116, 139, 174, 211, 275, 316] [9, 83, 114, 140, 175, 212, 274, 315] [8, 79, 118, 141, 179, 213, 270, 314] [7, 78, 119, 142, 177, 214, 272, 313] [6, 80, 117, 143, 178, 215, 271, 312] [8, 69, 99, 136, 187, 222, 273, 320] [6, 71, 101, 135, 188, 223, 275, 318] [7, 70, 100, 137, 186, 224, 274, 319] [11, 66, 96, 133, 190, 225, 270, 323] [9, 68, 98, 132, 191, 226, 272, 321] [10, 67, 97, 134, 189, 227, 271, 322] [14, 75, 105, 142, 193, 228, 279, 326] [12, 77, 107, 141, 194, 229, 281, 324] [13, 76, 106, 143, 192, 230, 280, 325] [17, 72, 102, 139, 196, 231, 276, 329] [15, 74, 104, 138, 197, 232, 278, 327] [16, 73, 103, 140, 195, 233, 277, 328] [20, 81, 111, 148, 199, 234, 285, 332] [18, 83, 113, 147, 200, 235, 287, 330] [19, 82, 112, 149, 198, 236, 286, 331] [23, 78, 108, 145, 202, 237, 282, 335] [21, 80, 110, 144, 203, 238, 284, 333] [22, 79, 109, 146, 201, 239, 283, 334] [26, 45, 117, 154, 205, 240, 291, 296] [24, 47, 119, 153, 206, 241, 293, 294] [25, 46, 118, 155, 204, 242, 292, 295] [29, 42, 114, 151, 208, 243, 288, 299] [27, 44, 116, 150, 209, 244, 290, 297] [28, 43, 115, 152, 207, 245, 289, 298] [32, 51, 123, 160, 169, 246, 255, 302] [30, 53, 125, 159, 170, 247, 257, 300] [31, 52, 124, 161, 168, 248, 256, 301] [35, 48, 120, 157, 172, 249, 252, 305] [33, 50, 122, 156, 173, 250, 254, 303] [34, 49, 121, 158, 171, 251, 253, 304] [38, 57, 87, 166, 175, 210, 261, 308] [36, 59, 89, 165, 176, 211, 263, 306] [37, 58, 88, 167, 174, 212, 262, 307] [41, 54, 84, 163, 178, 213, 258, 311] [39, 56, 86, 162, 179, 214, 260, 309] [40, 55, 85, 164, 177, 215, 259, 310] [2, 63, 93, 130, 181, 216, 267, 314] [0, 65, 95, 129, 182, 217, 269, 312] [1, 64, 94, 131, 180, 218, 268, 313] [5, 60, 90, 127, 184, 219, 264, 317] [3, 62, 92, 126, 185, 220, 266, 315] [4, 61, 91, 128, 183, 221, 265, 316]
H_Z (126 checks, sparse supports)
[11, 66, 125, 138, 168, 247, 278, 301] [10, 67, 124, 139, 169, 248, 276, 302] [9, 68, 123, 140, 170, 246, 277, 300] [8, 69, 122, 141, 171, 250, 281, 304] [7, 70, 121, 142, 172, 251, 279, 305] [6, 71, 120, 143, 173, 249, 280, 303] [17, 72, 89, 144, 174, 211, 284, 307] [16, 73, 88, 145, 175, 212, 282, 308] [15, 74, 87, 146, 176, 210, 283, 306] [14, 75, 86, 147, 177, 214, 287, 310] [13, 76, 85, 148, 178, 215, 285, 311] [12, 77, 84, 149, 179, 213, 286, 309] [23, 78, 95, 150, 180, 217, 290, 313] [22, 79, 94, 151, 181, 218, 288, 314] [21, 80, 93, 152, 182, 216, 289, 312] [20, 81, 92, 153, 183, 220, 293, 316] [19, 82, 91, 154, 184, 221, 291, 317] [18, 83, 90, 155, 185, 219, 292, 315] [29, 42, 101, 156, 186, 223, 254, 319] [28, 43, 100, 157, 187, 224, 252, 320] [27, 44, 99, 158, 188, 222, 253, 318] [26, 45, 98, 159, 189, 226, 257, 322] [25, 46, 97, 160, 190, 227, 255, 323] [24, 47, 96, 161, 191, 225, 256, 321] [35, 48, 107, 162, 192, 229, 260, 325] [34, 49, 106, 163, 193, 230, 258, 326] [33, 50, 105, 164, 194, 228, 259, 324] [32, 51, 104, 165, 195, 232, 263, 328] [31, 52, 103, 166, 196, 233, 261, 329] [30, 53, 102, 167, 197, 231, 262, 327] [41, 54, 113, 126, 198, 235, 266, 331] [40, 55, 112, 127, 199, 236, 264, 332] [39, 56, 111, 128, 200, 234, 265, 330] [38, 57, 110, 129, 201, 238, 269, 334] [37, 58, 109, 130, 202, 239, 267, 335] [36, 59, 108, 131, 203, 237, 268, 333] [5, 60, 119, 132, 204, 241, 272, 295] [4, 61, 118, 133, 205, 242, 270, 296] [3, 62, 117, 134, 206, 240, 271, 294] [2, 63, 116, 135, 207, 244, 275, 298] [1, 64, 115, 136, 208, 245, 273, 299] [0, 65, 114, 137, 209, 243, 274, 297] [18, 69, 87, 147, 206, 241, 261, 320] [19, 71, 89, 149, 204, 242, 263, 318] [20, 70, 88, 148, 205, 240, 262, 319] [21, 66, 84, 144, 209, 244, 258, 323] [22, 68, 86, 146, 207, 245, 260, 321] [23, 67, 85, 145, 208, 243, 259, 322] [24, 75, 93, 153, 170, 247, 267, 326] [25, 77, 95, 155, 168, 248, 269, 324] [26, 76, 94, 154, 169, 246, 268, 325] [27, 72, 90, 150, 173, 250, 264, 329] [28, 74, 92, 152, 171, 251, 266, 327] [29, 73, 91, 151, 172, 249, 265, 328] [30, 81, 99, 159, 176, 211, 273, 332] [31, 83, 101, 161, 174, 212, 275, 330] [32, 82, 100, 160, 175, 210, 274, 331] [33, 78, 96, 156, 179, 214, 270, 335] [34, 80, 98, 158, 177, 215, 272, 333] [35, 79, 97, 157, 178, 213, 271, 334] [36, 45, 105, 165, 182, 217, 279, 296] [37, 47, 107, 167, 180, 218, 281, 294] [38, 46, 106, 166, 181, 216, 280, 295] [39, 42, 102, 162, 185, 220, 276, 299] [40, 44, 104, 164, 183, 221, 278, 297] [41, 43, 103, 163, 184, 219, 277, 298] [0, 51, 111, 129, 188, 223, 285, 302] [1, 53, 113, 131, 186, 224, 287, 300] [2, 52, 112, 130, 187, 222, 286, 301] [3, 48, 108, 126, 191, 226, 282, 305] [4, 50, 110, 128, 189, 227, 284, 303] [5, 49, 109, 127, 190, 225, 283, 304] [6, 57, 117, 135, 194, 229, 291, 308] [7, 59, 119, 137, 192, 230, 293, 306] [8, 58, 118, 136, 193, 228, 292, 307] [9, 54, 114, 132, 197, 232, 288, 311] [10, 56, 116, 134, 195, 233, 290, 309] [11, 55, 115, 133, 196, 231, 289, 310] [12, 63, 123, 141, 200, 235, 255, 314] [13, 65, 125, 143, 198, 236, 257, 312] [14, 64, 124, 142, 199, 234, 256, 313] [15, 60, 120, 138, 203, 238, 252, 317] [16, 62, 122, 140, 201, 239, 254, 315] [17, 61, 121, 139, 202, 237, 253, 316] [34, 63, 121, 159, 208, 216, 288, 300] [33, 65, 122, 161, 207, 217, 289, 301] [35, 64, 120, 160, 209, 218, 290, 302] [31, 60, 124, 156, 205, 219, 291, 303] [30, 62, 125, 158, 204, 220, 292, 304] [32, 61, 123, 157, 206, 221, 293, 305] [40, 69, 85, 165, 172, 222, 252, 306] [39, 71, 86, 167, 171, 223, 253, 307] [41, 70, 84, 166, 173, 224, 254, 308] [37, 66, 88, 162, 169, 225, 255, 309] [36, 68, 89, 164, 168, 226, 256, 310] [38, 67, 87, 163, 170, 227, 257, 311] [4, 75, 91, 129, 178, 228, 258, 312] [3, 77, 92, 131, 177, 229, 259, 313] [5, 76, 90, 130, 179, 230, 260, 314] [1, 72, 94, 126, 175, 231, 261, 315] [0, 74, 95, 128, 174, 232, 262, 316] [2, 73, 93, 127, 176, 233, 263, 317] [10, 81, 97, 135, 184, 234, 264, 318] [9, 83, 98, 137, 183, 235, 265, 319] [11, 82, 96, 136, 185, 236, 266, 320] [7, 78, 100, 132, 181, 237, 267, 321] [6, 80, 101, 134, 180, 238, 268, 322] [8, 79, 99, 133, 182, 239, 269, 323] [16, 45, 103, 141, 190, 240, 270, 324] [15, 47, 104, 143, 189, 241, 271, 325] [17, 46, 102, 142, 191, 242, 272, 326] [13, 42, 106, 138, 187, 243, 273, 327] [12, 44, 107, 140, 186, 244, 274, 328] [14, 43, 105, 139, 188, 245, 275, 329] [22, 51, 109, 147, 196, 246, 276, 330] [21, 53, 110, 149, 195, 247, 277, 331] [23, 52, 108, 148, 197, 248, 278, 332] [19, 48, 112, 144, 193, 249, 279, 333] [18, 50, 113, 146, 192, 250, 280, 334] [20, 49, 111, 145, 194, 251, 281, 335] [28, 57, 115, 153, 202, 210, 282, 294] [27, 59, 116, 155, 201, 211, 283, 295] [29, 58, 114, 154, 203, 212, 284, 296] [25, 54, 118, 150, 199, 213, 285, 297] [24, 56, 119, 152, 198, 214, 286, 298] [26, 55, 117, 151, 200, 215, 287, 299]
Code ID 336-88-12 · download JSON · raw on GitHub