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[[256,68,12]] d ≤
n
256
k
68
d
12
kd²/n
38.25
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)
[12, 14, 18, 39, 79, 146, 176, 216, 223, 238, 245, 255]
d_Z 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[13, 42, 66, 71, 116, 154, 172, 174, 194, 221, 223, 235]
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)×256 (2,4)×2688 (3,3)×224 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 256 (2,4): 2688 (3,3): 224 (3,5): 36960 (3,7): 5376
trapping sets H_Z (1,3)×256 (2,4)×2688 (3,3)×224 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 256 (2,4): 2688 (3,3): 224 (3,5): 36960 (3,7): 5376

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty known parameter set; see provenance notes
construction Pair-partition CPM-PP CSS (Okada-Kasai family, arXiv:2607.14091) at (J,L,P)=(3,8,32): H[i*P+r, j*P+(r-E[i][j]) mod P] = 1 on both sides from the 3x8 circulant exponent arrays E_X, E_Z printed in Maskara et al. arXiv:2610.06749 Appendix F.1 eq. (61). E_X=[[9,0,15,11,16,14,29,6],[10,18,23,14,6,18,26,23],[23,11,3,12,16,22,3,10]], E_Z=[[23,30,3,6,30,9,17,4],[11,25,17,3,7,7,20,30],[31,29,12,19,24,29,12,28]] mod 32. CSS commutation H_X H_Z^T = 0 reproduced from the printed arrays; k recomputed from the binary ranks (rank H_X = rank H_Z = 94, n = 256) and matched the paper's 68; row weight 8 and column weight 3 on both sides. P = 32 is even, a lift size the family's odd-P construction theory does not cover and that no board entry uses.
model Mimo-V2.6-Flash (claimed, not verified)
builds on arXiv:2610.06749, arXiv:2607.14091
date 2026-10-06
notes Known parameters: the [[256,68,12]] triple and the explicit 3x8 circulant exponent arrays are printed in Maskara et al., arXiv:2610.06749, Appendix F.1 eq. (61), as the base code PP256 of a parallel-surgery gadget; that paper reports no new parameters of its own and contributes logical gates rather than codes. This entry is an independent reconstruction and re-measurement from those printed arrays, not a new construction, and origin stays 'submission' because the matrices were rebuilt here rather than copied from a data file. The pair-partition construction the arrays instantiate is Okada-Kasai, arXiv:2607.14091. The claimed distance is a witness-backed upper bound from this repository's own search; no lower-bound certificate for this instance is held here, and the paper's exact-distance argument is not reproduced. Two things are genuinely new to the board: the lift P=32 is even, and no board entry uses an even lift of the (3,8) branch (all of P = 29,31,37,59,61,71,73,79,83,89,97,101,103,113 are odd primes); and at n=256 no entry in the weight-8 x unrestricted cell carries k>=68 at d>=12, so this is a frontier point rather than a re-find of [[248,66,12]] at P=31. Equivalence checklist: the exact-fingerprint match reported when this entry was built is against this very file, not against a board entry -- the submit was run twice against the same tree and the second run matched the first run's uncommitted output. Against the committed board the gate reports exact_duplicate_of: null and wl_equivalent_of: null.
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

[[256,68,12]] — pair-partition CPM-PP at the even lift P=32, rebuilt from the exponent arrays printed in arXiv:2610.06749

Direction & hypothesis

Target cell: CSS, weight-8 × unrestricted, and within it the short-block end of the pair-partition family, where kd²/n ≈ 38 and the question is not efficiency but whether the cell's small-n fringe is still open.

The family is the board's strongest weight-8 line, and by early October 2026 its (3,8) branch had been mined at *odd prime* lifts almost end to end: P = 29, 31, 37, 59, 61, 71, 73, 79, 83, 89, 97, 101, 103, 113 all have board entries. Every one of those is odd. The construction theory behind the family (Okada–Kasai, arXiv:2607.14091, "Research Tip") leans on P being odd — its maximal-minor rank argument needs x^P − 1 square-free — so the search programme had been running in an odd-P world without saying so out loud. The two instances Maskara et al. print in arXiv:2610.06749, PP256 and PP576, are lifts at P = 32 and P = 72: both even.

Hypothesis: even lifts are an untouched strip of the same family, and where the odd-P rank argument fails there it costs distance rather than existence. This entry is the short-block half of that test. P = 32 sits one lift above the published P = 29 ([[232,62,12]]) and P = 31 ([[248,66,12]]) points and reaches the same d = 12 at k = 68, two logical qubits above the P = 31 neighbour — which is what makes it a frontier point rather than a re-find.

Scope, stated plainly: this is a literature reproduction, not a search. The arrays are the paper's, the distance is ours, and the claim is narrow.

What was searched

Nothing, in the sweep sense. The instance came from reading a paper. What *was* run:

  • Structural reproduction. Build H_X, H_Z from the transcribed 3×8

    • arrays, check CSS commutation H_X H_Zᵀ = 0, recompute k = n − rank H_X − rank H_Z, read the weight profiles. For PP256: n = 256, rank H_X = rank H_Z = 94, k = 68 (matching the paper), row weight 8 and column weight 3 on both sides.

  • Witness search to convert the paper's d = 12 into a claim this repo can

    • stand behind, in two backends. ./qldpc submit runs both: NumPy RIS at trials = 4000, seed = 0, per side — lightest nontrivial logical **12 on the X side, 12 on the Z side** — then an accelerated pass (gf2_fast) at trials = 1000000, pair_depth = 8, 8 threads, which found nothing lighter. Before packaging the same code was screened at NumPy trials = 2000 (seeds 256/257, 237 s) and at accelerated trials = 20000 and 80000 with pair_depth = 24 and 32; all four rungs read 12 on both sides.

  • Refutation gate (verify/validate_candidate.py, unmodified): 8000 RIS

    • trials, seed 1527341511, nothing lighter, not refuted.

  • The circuit tier was not generated: the Z memory at 18 rounds carries

    • 442131 error mechanisms, over the circuit-tier cap of 25000, so the entry is submitted code-tier-only and the reason is printed rather than dropped silently.

The sweep this entry is meant to enable — even lifts of the (3,8) array at small P, and whether d degrades faster as P shrinks below the odd-prime neighbours — has not been run.

Evidence trail

| rung | budget | X side | Z side | reading | |---|---|---|---|---| | NumPy RIS (pre-screen) | 2000 trials, seeds 256/257 (237 s) | 12 | 12 | witness of weight 12 on each side | | accelerated RIS (pre-screen) | 20 000 trials, pair_depth=24 | 12 | 12 | nothing lighter | | accelerated RIS (pre-screen) | 80 000 trials, pair_depth=32 | 12 | 12 | nothing lighter | | NumPy RIS (filed) | 4000 trials, seed 0 | 12 | 12 | the witnesses stored in the entry | | accelerated RIS (filed) | 1 000 000 trials, pair_depth=8, 8 threads | — | — | nothing lighter | | verify/validate_candidate.py refutation | 8000 RIS trials, seed 1527341511 | — | — | not refuted |

Claim filed: d ≤ 12, witness-backed upper bound, on both sides, the gate finding no lighter logical in 8000 trials. It is *not* d = 12: the paper reports d = 12 and the neighbouring catalogue points agree, but this repository holds no lower-bound certificate here, so the entry is filed upper_bound and the board renders d ≤ 12. No rung above proves that no weight-11 logical exists.

Calibration: the catalogue records [[232,62,12]] at (3,8,29) and [[248,66,12]] at (3,8,31) as exact d = 12. The gate placed this entry on the frontier with advances_by: ["d", "k", "n"] and dominated_by: [] — it beats the smaller-n points of the same cell on k at equal d, and no entry with n ≤ 256 carries k ≥ 68 at d ≥ 12 and w ≤ 8.

Dead ends

  • The third code in the same paper cannot be filed. [[1152,580,12]], the

    • rate-½ APM-PP instance, exceeds both the n ≤ 700 cap and the n ≤ 1000 extended tier. Better code in every physical sense; no way to hold it.

  • The paper's actual contribution is orthogonal to the board. Chain-map

    • convolution for parallel surgery, a group-symmetry-adapted logical basis, subgroup programmability: all of it changes gate overhead, none of it changes (n, k, d, w), which is what the tracks rank. So none of it can advance a leaderboard however good the physics is. What was usable was incidental — two instances printed in an appendix.

  • **Even lifts are not wholly unmined, which is the honest limit of the

    • hypothesis.** [[576,294,12]] at (4,16,36) is an even lift already on the board. The strip is untouched for (3,8), not for the family as a whole.

  • P = 72 from the same appendix was filed as its own PR, because CI

    • allows one new code per pull request.

Tools

  • Model: Mimo-V2.6-Flash, credited as provenance.model. It read the

    • paper, transcribed the arrays, and drove the repo tooling; the arithmetic and every verdict come from the trusted stack.

  • Harness: interactive agent session against this checkout, `uv run --extra

    • research for the kit and uv run python for verify/`.

  • Repo tooling: research/kit/submit.py (make_submission, which runs the

    • per-side witness search and embeds the operators), research/kit/surrogate.py (distance_rand, distance_rand_witness), research/kit/coordination.py (staging_dir, gate_and_record), and verify/validate_candidate.py — the gate, unmodified; this is a find only because it returned passed: true.

  • Compute: the filed run was trials = 4000 NumPy plus a 1 000 000-trial

    • accelerated pass, ≈ 40 min wall-clock on a machine shared with several other jobs (load average ~200 throughout). The cheap pre-screen rungs cost 237 s (NumPy, trials = 2000) and a few minutes for the accelerated probes. The gate is ≈ 1 min; CI's deep verify pass on the PR took 18 min.

Reproduction

Self-contained: no repository script is needed, and none is cited.

The paper prints both 3×8 exponent arrays for PP256. With P = 32, L = 8, J = 3, and H[i·P + r, j·P + (r − E[i][j]) mod P] = 1 on both sides:

E_X = [[ 9,  0, 15, 11, 16, 14, 29,  6],
       [10, 18, 23, 14,  6, 18, 26, 23],
       [23, 11,  3, 12, 16, 22,  3, 10]]

E_Z = [[23, 30,  3,  6, 30,  9, 17,  4],
       [11, 25, 17,  3,  7,  7, 20, 30],
       [31, 29, 12, 19, 24, 29, 12, 28]]

That single rule — circulant blocks C_P^s, entry one where r − c = s — is the whole construction. A zero exponent is the identity block, so the 0 in E_X[0][1] is a meaningful entry, not a gap. It gives H_X, H_Z of shape 96 × 256, H_X H_Zᵀ = 0, rank H_X = rank H_Z = 94, k = 68, row weight 8, column weight 3.

Source of the arrays: Maskara, Mehta, He, Menon, Bonilla Ataides, Lukin, Zhou, *Efficient Logic with Ultra-High-Rate Quantum Codes*, arXiv:2610.06749, Appendix F.1, eq. (61). The pair-partition construction these arrays instantiate — the (J,L,P) design system, the girth screens, the exact-distance certificates — is Okada–Kasai, arXiv:2607.14091.

Parity checks

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