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[[290,64,6]] d ≤
n
290
k
64
d
6
kd²/n
7.945
w
10
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · w_X = 10, w_Z = 10 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[16, 91, 104, 206, 219, 222]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[69, 79, 156, 169, 239, 242]
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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 10 · H_Z 10
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×290 (2,2)×174 (3,4)×4089 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 290 (2,2): 174 (2,4): 638 (2,6): 3422 (3,4): 4089 (3,6): 21083 (3,8): 58464 (3,10): 4118
trapping sets H_Z (1,4)×290 (2,2)×174 (3,4)×4089 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 290 (2,2): 174 (2,4): 638 (2,6): 3422 (3,4): 4089 (3,6): 21083 (3,8): 58464 (3,10): 4118

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), CSS parent
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 stabilizer fold of this code, parameters [[145,32,6]], 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

[[290,64,6]] — symplectic-halved CPM pair-partition CSS code

Direction & hypothesis

Target: the CSS weight-9plus x unrestricted cell at n = 290, where the incumbent threshold was only d > 4 — a thin cell worth filling. Structural idea: this code is the exact CSS double of the [[145,32,6]] stabilizer fold from the same screen, so one sweep of the symplectic-halving recipe of arXiv:2609.30069 (Prop. 4) produces both a first stabilizer-board entry and this CSS entry. Efficiency kd^2/n = 7.945, check weight 10.

What was searched

(J,L,P) = (4,10,29), sigma(ell) = (ell+5) mod 10, nondegenerate shift gauge of the recipe, off-diagonal pairing equations solved for the exponent arrays with sigma-pair-free matchings; every draw screened for equal columns in H_X/H_Z (weight-2 collapse otherwise) with RIS at thousands of trials per draw, wall-clock caps up to 90 s. Lifts at P in {19, 31, 47} were screened too and discarded (see dead ends). Draw index 6 at P=29 gave k = 64 with both distance sides screening at d <= 6.

Evidence trail

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

  • Draw-6 screen: RIS upper bound d <= 6 on both X and Z 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) returned d_X <= 6 and d_Z <= 6; verify/qldpc_verify.py re-checks both witnesses against the matrices in the submission.

Final claim: d_X <= 6 and d_Z <= 6 with logical witnesses embedded in codes/290-64-6.json. Not certified exact.

Dead ends

  • Equal-column draws, 87% of all draws: weight-2 kernel vectors, d <= 2. Screened out.
  • No-4-/6-cycle filters switched on: 0 of 164 matchings survive — a matching pairing {ell, sigma(ell)} structurally forces a 4-cycle in the block layout, and cycles4(E)-freedom failed 61 of 61 times among equal-column-clean draws. Filters left optional and off; girth is not part of the gate.
  • P=47 lift: k = 100 instead of ~109, so the board threshold there needs d > 14 while the screen reached 4. Dropped.
  • (4,12),P=31 at n = 372: the board already holds [[372,130,16]] (pair-partition CPM family, arXiv:2607.14091), so at check weight 12 the threshold is d > 17. Dropped.

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 j=0..3, columns ell=0..9) over Z_29:

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. 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 matrices bit for bit (verified: rebuilt HX/HZ equal the submission's arrays exactly). The stabilizer fold of this parent is filed separately as [[145,32,6]]; this note needs nothing from that submission.

Parity checks

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