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[[234,78,3]] d =
n
234
k
78
d
3
kd²/n
3.0
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 3 · witness weight 3 (claimed exact)
witness operator (support, 3 qubits)
[18, 57, 96]
d_Z 3 · witness weight 3 (claimed exact)
witness operator (support, 3 qubits)
[132, 171, 210]
certificate exact, d = 3 · scipy/HiGHS cutoff IP
X: no logical < 3 exists; Z: no logical < 3 exists

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 6 · H_Z 6
qubit degrees H_X 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)
trapping sets H_X (1,2)×117 (2,2)×351 (3,0)×195 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 117 (1,4): 117 (2,2): 351 (2,4): 702 (2,6): 234 (3,0): 195 (3,2): 1170 (3,4): 2925 (3,6): 2691 (3,8): 468
trapping sets H_Z (1,2)×117 (2,2)×351 (3,0)×195 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 117 (1,4): 117 (2,2): 351 (2,4): 702 (2,6): 234 (3,0): 195 (3,2): 1170 (3,4): 2925 (3,6): 2691 (3,8): 468

Construction & provenance

authors @cbjuan
provenance submitted through the challenge
novelty novelty not audited
construction Periodic bivariate-bicycle code on the group algebra Z_9 x Z_13 (n = 2*l*m = 234). Convention: x = S_9 tensor I_13, y = I_9 tensor S_13 (cyclic shifts), qubit index = i*m + j; A = 1 + x6y0, B = 1 + x2y2 + x5y2 + x6y0; H_X = [A|B], H_Z = [B^T|A^T]. Found by an automated evolutionary/mutation search over (l, m, A, B) exponent sets targeting the unrestricted x weight-6 (or weight-8) board cells, screened by a randomized lightest-logical surrogate and promoted through increasing trial budgets.
date 2026-09-16
notes Internal autoresearch candidate id 6423 (canonical key 842ade0cea9868db725ecc26), search lane 'w6-gap', proposal kernel 'shared-binomial-factor'. Initial NumPy lightest-logical surrogate: trials=8000, seed=196597815. Confirmed with an accelerated bit-packed random-information-set search at rung '60k' (60000 trials, seed=196597832, pair_depth=16), tightest witness on side X at weight 3. Distance also taken through an internal exact-certification pipeline (MILP, one solve per logical-basis row per side, the same method as this repository's own verify/certify.py). Result: side_status X=exact, Z=exact (solver wall time 59.6s, CPU time 58.6s, summed over all per-row MILP solves on both sides). Per CONTRIBUTING.md's confidence-tier policy, this 'exact' claim is self-declared and will display as an upper bound on the board until a maintainer independently re-runs verify/certify.py and confirms it.
family bivariate bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[234,78,3]] — periodic bivariate-bicycle on Z_9 x Z_13

Direction & hypothesis

Target: the unrestricted x weight ≤ 6 board cell, via a periodic bivariate-bicycle (BB) code on Z_9 x Z_13, found by an automated continuous evolutionary/mutation search over BB exponent sets (l, m, A, B) (lane w6-gap, proposal kernel shared-binomial-factor). This submission independently rebuilds and re-verifies the resulting [[234,78,3]] candidate using only this repository's own trusted tools before submitting.

What was searched

The source campaign's search over Z_9 x Z_13 exponent pairs used lane w6-gap with proposal kernel shared-binomial-factor (internal candidate id 6423); screening was by a randomized lightest-logical surrogate promoted through increasing trial budgets, the same style of screen described in research/AUTORESEARCH.md step 2-3 (CSS commutation, k, then a randomized distance upper bound) but run in the external campaign above. NumPy surrogate stage: trials=8000, seed=196597815. Bit-packed accelerated confirmation stage: rung 60k, 60000 trials, seed=196597832, pair_depth=16, tightest witness on side X at weight 3.

Evidence trail

  • Rebuilt (H_X, H_Z) from the polynomials below using this repository's own research/kit/bb.py:build_bb, and confirmed the reconstruction reproduces the campaign's own check supports exactly (same X/Z check sets, as sets of qubit supports) before packaging.
  • research/kit/css.py:verify_css / compute_k -- CSS commutation holds, k = 78 exactly (GF(2) rank), n = 234, max check weight 6 on X and 6 on Z.
  • Verified locally with this repository's own trustless gate, verify/qldpc_verify.py (verify(doc, refute=True)): schema-valid, connected Tanner graph and stabilizer group, CSS commutation, both witnesses valid (in the kernel of the opposite side's checks, outside the rowspace of their own), and the refutation search found nothing lighter.
  • Distance was also taken through an internal exact-certification pipeline (one MILP per logical-basis row per side, the same method as this repository's own verify/certify.py), run as part of an internal autoresearch campaign on branch certification/periodic-bb-exact-64core of this same repository. That branch is not merged to main at the time of this submission, so nothing here relies on it being externally checkable. Both sides closed exact (no logical lighter than the claimed weight exists on either side); this is reported only as the basis for the exact confidence self-declared below -- per CONTRIBUTING.md's confidence-tier policy this claim will display as an upper bound on the board until a maintainer independently re-runs verify/certify.py and confirms it.
  • Final claim: exact, d = 3.

Dead ends

Not applicable -- this submission mines a single already-vetted candidate from an external campaign's sweep rather than running a new search in this repository; the campaign's own dead ends (screened-and-collapsed candidates) are not part of this submission's own evidence trail.

Tools

This candidate's discovery and screening ran as an internal autoresearch campaign on branch certification/periodic-bb-exact-64core of this same repository: an automated (non-LLM) exponent-set mutation search with MILP-based exact certification. That branch is not merged to main at the time of this submission, so nothing here relies on it being externally checkable. No provenance.model is set because no generative model produced this code. This submission's independent reconstruction, re-verification, and packaging in this repository was done separately by Claude Sonnet 5, via Claude Code, operating on this repository's research/kit and verify/ modules only (no edits to verify/). Compute: seconds to a few minutes for the reconstruction and verifier pass.

Reproduction

A = 1 + x^6y^0, B = 1 + x^2y^2 + x^5y^2 + x^6y^0 on Z_9 x Z_13 (n = 2·l·m = 234):

from bb import build_bb
HX, HZ = build_bb(l=9, m=13, A_terms=[(0,0), (6,0)], B_terms=[(0,0), (2,2), (5,2), (6,0)])

Parity checks

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