← back to the stabilizer board
[[145,32,6]] d ≤stabilizer
n
145
k
32
d
6
kd²/n
7.945
w
10

Share this result

Distance

a general stabilizer code has no X and Z sides: d is the minimum Pauli weight of a nontrivial logical operator (a Y factor counts one qubit), and the witness is one Pauli operator that commutes with every generator and is not a product of them
d 6 · witness Pauli weight 6 (claimed upper_bound)
witness operator (Pauli string, 6 qubits)
XIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIZIIIIIIIIIIIIIIIIIIIXIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII X: [0, 16, 88, 91] Z: [58, 68]
certificate none yet · distance stands as a self-certified upper bound (d ≤); the Pauli-weight certifier is not built yet, so stabilizer entries cannot earn d= for now

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth S 4 (shortest cycle of the generator Tanner graph; longer is friendlier to belief propagation)
check weights S 7–10 (mean 9.828)
qubit degrees S 6–8 (mean 7.862)
trapping sets S (1,6)×1 (2,4)×6 (3,5)×2 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for S
(1,6): 1 (1,7): 18 (1,8): 126 (2,4): 6 (2,5): 11 (2,6): 53 (2,7): 35 (2,8): 150 (2,9): 41 (2,10): 287 (2,11): 156 (2,12): 739 (2,13): 226 (2,14): 1022 (3,5): 2 (3,6): 12 (3,7): 41 (3,8): 167 (3,9): 287 (3,10): 1352 (3,11): 992 (3,12): 4019 (3,13): 2065 (3,14): 8008 (3,15): 3083 (3,16): 12541 (3,17): 4559 (3,18): 16657 (3,19): 3259 (3,20): 11553 (3,21): 41 (3,22): 157

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction symplectic-halved CPM pair-partition code (arXiv:2609.30069 Prop 4 recipe), (J,L,P)=(4,10,29), folded stabilizer
model Mimo-V2.6-Flash (claimed, not verified)
builds on arXiv:2609.30069, arXiv:2607.14091
date 2026-09-28
notes Checked against the live board: not equivalent to any existing entry (exact-duplicate and weight/locality-equivalent checks both null in the trusted gate). Recipe follows the symplectic-halving construction of arXiv:2609.30069 Prop. 4; the authors' data repository was empty when searched, so this is an independent draw of the published recipe, not one of their tabulated instances. The related pair-partition CPM family already on the board is arXiv:2607.14091. The CSS parent of this fold, parameters [[290,64,6]], is submitted as a separate PR.
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

[[145,32,6]] — symplectic-halved CPM pair-partition stabilizer code

Direction & hypothesis

Target: the stabilizer board, which had no entries — any verified stabilizer code advances it. Structural idea: Prop. 4 of arXiv:2609.30069 halves a CSS pair-partition CPM code into a non-CSS stabilizer code with exactly half the logicals. Halving a (J,L,P)=(4,10,29) CSS parent from this session's screen gives [[145,32,6]] with k = 64/2 = 32, matching the paper's scaling. Efficiency kd^2/n = 7.945.

What was searched

The sweep solves the recipe's off-diagonal pair-partition equations over F_P for P in {19, 29, 31, 47}, draws sigma-pair-free matchings from the solution space, screens every draw for equal columns in H_X/H_Z (they force weight-2 logicals), and writes all draw matrices to disk. Each draw was screened with RIS at thousands of trials, wall-clock caps up to 90 s. Draw index 6 at P=29 was the first whose CSS parent and halved fold both screened at d <= 6 while keeping k at 64 and 32.

Evidence trail

Confirmation ladder, all witness-backed upper bounds, not exact certificates:

  • Draw-6 screen: RIS upper bound d <= 6 for the fold (Pauli side) and for the parent (both sides).
  • Trusted gate verify/validate_candidate.py: passed true, board_advancing true, empty dominator list, no weight/locality-equivalent duplicate; an 8000-trial refutation search found no logical lighter than 6.
  • CLI witness search (20000 RIS trials plus a 2,000,000-trial accelerator pass) recovered a weight-6 logical; verify/qldpc_verify.py re-checks it against the matrices in the submission.

Final claim: d <= 6 with a weight-6 logical witness embedded in codes/145-32-6.json. Not certified exact.

Dead ends

  • Equal-column draws, 87% of all draws: H_X/H_Z pick up repeated columns and a weight-2 kernel vector appears, d <= 2. Screened out before selection.
  • The no-4-/6-cycle screen, when switched on: 0 of 164 matchings survive. Any matching pairing {ell, sigma(ell)} forces a repeated row difference in the block layout (a structural 4-cycle), and cycles4(E)-freedom failed in 61 of 61 equal-column-clean draws. The screen exists but stayed off; girth is not part of the gate.
  • P=47 lift: k came out 100 rather than the ~109 of the record family, pushing the board threshold to d > 14 while the screen only reached 4. Dropped.
  • (4,12),P=31 parent at n=372: the board already holds [[372,130,16]] from the pair-partition CPM family of arXiv:2607.14091, so at check weight 12 the threshold is d > 17. Dropped.
  • A nullity report printed rows minus rank instead of variables minus rank (logging only, fixed).

Tools

Mimo-V2.6-Flash (provenance.model) under the opencode agent harness; NumPy plus the repository's css/gf2 field arithmetic and RIS routines; trusted gates verify/validate_candidate.py and verify/check_prose.py. Minutes of local CPU, no paid compute.

Reproduction

Exact exponents (rows i=0..3, columns ell=0..9) over Z_29 for the CSS parent:

E = [[2,20,21,27,6,21,13,12,6,14],[11,22,23,0,15,23,12,8,8,19],[5,26,8,1,3,8,26,5,22,1],[12,23,24,24,16,21,3,12,25,17]]

sigma(ell) = (ell+5) mod 10 and D[j][ell] = -E[j][sigma(ell)] mod 29. Parent blocks: HX[i*29+r, ell*29+c] = 1 iff c = r - E[i][ell] mod 29; HZ[j*29+r, ell*29+c] = 1 iff c = r - D[j][ell] mod 29. This reconstructs the submitted parent matrices bit for bit. The fold is the halving of that parent: under pi(ell, t) = (sigma(ell), eta*t) with eta = -1, reorder the L*P columns so each pair {q, pi(q)} is adjacent; H_X restricted to that order is (A | B), and S = (A | B) is the stabilizer code (the same reordering sends H_Z to (B | A), the identity the construction guarantees). Isotropy A B^T + B A^T = 0 holds and k_fold = k_parent/2 = 32.

Stabilizer generators

generators 116 (max weight 10; 116 mixed X/Z) (one binary symplectic matrix S = (A | B); generator i is X on A_i and Z on B_i, Y where both)
generators (116, Pauli strings on 145 qubits)
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symplectic rows (A | B) (116, sparse supports)
X: [27, 38, 66, 89, 139] Z: [21, 42, 70, 93, 130] X: [28, 39, 67, 90, 140] Z: [20, 41, 69, 92, 129] X: [0, 40, 68, 91, 141] Z: [19, 40, 68, 91, 128] X: [1, 41, 69, 92, 142] Z: [18, 39, 67, 90, 127] X: [2, 42, 70, 93, 143] Z: [17, 38, 66, 89, 126] X: [3, 43, 71, 94, 144] Z: [16, 37, 65, 88, 125] X: [4, 44, 72, 95, 116] Z: [15, 36, 64, 87, 124] X: [5, 45, 73, 96, 117] Z: [14, 35, 63, 115, 123] X: [6, 46, 74, 97, 118] Z: [13, 34, 62, 114, 122] X: [7, 47, 75, 98, 119] Z: [12, 33, 61, 113, 121] X: [8, 48, 76, 99, 120] Z: [11, 32, 60, 112, 120] X: [9, 49, 77, 100, 121] Z: [10, 31, 59, 111, 119] X: [10, 50, 78, 101, 122] Z: [9, 30, 58, 110, 118] X: [11, 51, 79, 102, 123] Z: [8, 29, 86, 109, 117] X: [12, 52, 80, 103, 124] Z: [7, 57, 85, 108, 116] X: [13, 53, 81, 104, 125] Z: [6, 56, 84, 107, 144] X: [14, 54, 82, 105, 126] Z: [5, 55, 83, 106, 143] X: [15, 55, 83, 106, 127] Z: [4, 54, 82, 105, 142] X: [16, 56, 84, 107, 128] Z: [3, 53, 81, 104, 141] X: [17, 57, 85, 108, 129] Z: [2, 52, 80, 103, 140] X: [18, 29, 86, 109, 130] Z: [1, 51, 79, 102, 139] X: [19, 30, 58, 110, 131] Z: [0, 50, 78, 101, 138] X: [20, 31, 59, 111, 132] Z: [28, 49, 77, 100, 137] X: [21, 32, 60, 112, 133] Z: [27, 48, 76, 99, 136] X: [22, 33, 61, 113, 134] Z: [26, 47, 75, 98, 135] X: [23, 34, 62, 114, 135] Z: [25, 46, 74, 97, 134] X: [24, 35, 63, 115, 136] Z: [24, 45, 73, 96, 133] X: [25, 36, 64, 87, 137] Z: [23, 44, 72, 95, 132] X: [26, 37, 65, 88, 138] Z: [22, 43, 71, 94, 131] X: [18, 36, 64, 87, 130] Z: [23, 41, 66, 95, 135] X: [19, 37, 65, 88, 131] Z: [22, 40, 65, 94, 134] X: [20, 38, 66, 89, 132] Z: [21, 39, 64, 93, 133] X: [21, 39, 67, 90, 133] Z: [20, 38, 63, 92, 132] X: [22, 40, 68, 91, 134] Z: [19, 37, 62, 91, 131] X: [23, 41, 69, 92, 135] Z: [18, 36, 61, 90, 130] X: [24, 42, 70, 93, 136] Z: [17, 35, 60, 89, 129] X: [25, 43, 71, 94, 137] Z: [16, 34, 59, 88, 128] X: [26, 44, 72, 95, 138] Z: [15, 33, 58, 87, 127] X: [27, 45, 73, 96, 139] Z: [14, 32, 86, 115, 126] X: [28, 46, 74, 97, 140] Z: [13, 31, 85, 114, 125] X: [0, 47, 75, 98, 141] Z: [12, 30, 84, 113, 124] X: [1, 48, 76, 99, 142] Z: [11, 29, 83, 112, 123] X: [2, 49, 77, 100, 143] Z: [10, 57, 82, 111, 122] X: [3, 50, 78, 101, 144] Z: [9, 56, 81, 110, 121] X: [4, 51, 79, 102, 116] Z: [8, 55, 80, 109, 120] X: [5, 52, 80, 103, 117] Z: [7, 54, 79, 108, 119] X: [6, 53, 81, 104, 118] Z: [6, 53, 78, 107, 118] X: [7, 54, 82, 105, 119] Z: [5, 52, 77, 106, 117] X: [8, 55, 83, 106, 120] Z: [4, 51, 76, 105, 116] X: [9, 56, 84, 107, 121] Z: [3, 50, 75, 104, 144] X: [10, 57, 85, 108, 122] Z: [2, 49, 74, 103, 143] X: [11, 29, 86, 109, 123] Z: [1, 48, 73, 102, 142] X: [12, 30, 58, 110, 124] Z: [0, 47, 72, 101, 141] X: [13, 31, 59, 111, 125] Z: [28, 46, 71, 100, 140] X: [14, 32, 60, 112, 126] Z: [27, 45, 70, 99, 139] X: [15, 33, 61, 113, 127] Z: [26, 44, 69, 98, 138] X: [16, 34, 62, 114, 128] Z: [25, 43, 68, 97, 137] X: [17, 35, 63, 115, 129] Z: [24, 42, 67, 96, 136] X: [24, 32, 79, 115, 142] Z: [8, 55, 63, 109, 117] X: [25, 33, 80, 87, 143] Z: [7, 54, 62, 108, 116] X: [26, 34, 81, 88, 144] Z: [6, 53, 61, 107, 144] X: [27, 35, 82, 89, 116] Z: [5, 52, 60, 106, 143] X: [28, 36, 83, 90, 117] Z: [4, 51, 59, 105, 142] X: [0, 37, 84, 91, 118] Z: [3, 50, 58, 104, 141] X: [1, 38, 85, 92, 119] Z: [2, 49, 86, 103, 140] X: [2, 39, 86, 93, 120] Z: [1, 48, 85, 102, 139] X: [3, 40, 58, 94, 121] Z: [0, 47, 84, 101, 138] X: [4, 41, 59, 95, 122] Z: [28, 46, 83, 100, 137] X: [5, 42, 60, 96, 123] Z: [27, 45, 82, 99, 136] X: [6, 43, 61, 97, 124] Z: [26, 44, 81, 98, 135] X: [7, 44, 62, 98, 125] Z: [25, 43, 80, 97, 134] X: [8, 45, 63, 99, 126] Z: [24, 42, 79, 96, 133] X: [9, 46, 64, 100, 127] Z: [23, 41, 78, 95, 132] X: [10, 47, 65, 101, 128] Z: [22, 40, 77, 94, 131] X: [11, 48, 66, 102, 129] Z: [21, 39, 76, 93, 130] X: [12, 49, 67, 103, 130] Z: [20, 38, 75, 92, 129] X: [13, 50, 68, 104, 131] Z: [19, 37, 74, 91, 128] X: [14, 51, 69, 105, 132] Z: [18, 36, 73, 90, 127] X: [15, 52, 70, 106, 133] Z: [17, 35, 72, 89, 126] X: [16, 53, 71, 107, 134] Z: [16, 34, 71, 88, 125] X: [17, 54, 72, 108, 135] Z: [15, 33, 70, 87, 124] X: [18, 55, 73, 109, 136] Z: [14, 32, 69, 115, 123] X: [19, 56, 74, 110, 137] Z: [13, 31, 68, 114, 122] X: [20, 57, 75, 111, 138] Z: [12, 30, 67, 113, 121] X: [21, 29, 76, 112, 139] Z: [11, 29, 66, 112, 120] X: [22, 30, 77, 113, 140] Z: [10, 57, 65, 111, 119] X: [23, 31, 78, 114, 141] Z: [9, 56, 64, 110, 118] X: [17, 35, 63, 92, 129] Z: [21, 32, 70, 112, 133] X: [18, 36, 64, 93, 130] Z: [20, 31, 69, 111, 132] X: [19, 37, 65, 94, 131] Z: [19, 30, 68, 110, 131] X: [20, 38, 66, 95, 132] Z: [18, 29, 67, 109, 130] X: [21, 39, 67, 96, 133] Z: [17, 57, 66, 108, 129] X: [22, 40, 68, 97, 134] Z: [16, 56, 65, 107, 128] X: [23, 41, 69, 98, 135] Z: [15, 55, 64, 106, 127] X: [24, 42, 70, 99, 136] Z: [14, 54, 63, 105, 126] X: [25, 43, 71, 100, 137] Z: [13, 53, 62, 104, 125] X: [26, 44, 72, 101, 138] Z: [12, 52, 61, 103, 124] X: [27, 45, 73, 102, 139] Z: [11, 51, 60, 102, 123] X: [28, 46, 74, 103, 140] Z: [10, 50, 59, 101, 122] X: [0, 47, 75, 104, 141] Z: [9, 49, 58, 100, 121] X: [1, 48, 76, 105, 142] Z: [8, 48, 86, 99, 120] X: [2, 49, 77, 106, 143] Z: [7, 47, 85, 98, 119] X: [3, 50, 78, 107, 144] Z: [6, 46, 84, 97, 118] X: [4, 51, 79, 108, 116] Z: [5, 45, 83, 96, 117] X: [5, 52, 80, 109, 117] Z: [4, 44, 82, 95, 116] X: [6, 53, 81, 110, 118] Z: [3, 43, 81, 94, 144] X: [7, 54, 82, 111, 119] Z: [2, 42, 80, 93, 143] X: [8, 55, 83, 112, 120] Z: [1, 41, 79, 92, 142] X: [9, 56, 84, 113, 121] Z: [0, 40, 78, 91, 141] X: [10, 57, 85, 114, 122] Z: [28, 39, 77, 90, 140] X: [11, 29, 86, 115, 123] Z: [27, 38, 76, 89, 139] X: [12, 30, 58, 87, 124] Z: [26, 37, 75, 88, 138] X: [13, 31, 59, 88, 125] Z: [25, 36, 74, 87, 137] X: [14, 32, 60, 89, 126] Z: [24, 35, 73, 115, 136] X: [15, 33, 61, 90, 127] Z: [23, 34, 72, 114, 135] X: [16, 34, 62, 91, 128] Z: [22, 33, 71, 113, 134]
Code ID 145-32-6 · download JSON · raw on GitHub