n
155
We sought a stronger sparse instance at fixed n=155, k=6 and maximum check weight 8 by varying the relative polynomial exponents of a cyclic mirror. The comparison is the earlier 155-qubit baseline, whose retained bound is d<=21, not its refuted earlier 22. Its supports are a={0,11,25,40}, b={0,20,81,126} in the convention below. That baseline folds the [[310,12,26]] CSS instance contributed by @vprusso with Claude Fable 5.1. The mirror construction is due to Andrey Boris Khesin and Jonathan Z. Lu, arXiv:2603.05496v1.
The contribution is an independently searched sparse-polynomial instance within that established construction. The common factor g=x^6+x^4+x^3+1 of a(x), b(x) and x^155+1 fixes k=deg(g)=6. Changing sparse multiples of g preserves the logical dimension while allowing the distance landscape to change. This is not a new construction theorem.
The corrected retained witnessed bound is d<=23. The upper-bound headline score 6*23^2/155=20.477419 exceeds the baseline's 6*21^2/155=17.070968. The supplied check matrix has rank 149; 16 checks have weight 7 and 139 have weight 8. No exact distance, improved decoding threshold or circuit-level performance is claimed.
Enumerate the 140 sorted four-term supports {0,r1,r2,r3} in Z_31 whose binary polynomial is divisible by g. To obtain a support in Z_155, leave 0 fixed and independently add 31*t to each other exponent, t uniform in {0,1,2,3,4}. Draw two supports using Python random.Random(4101001). Reject pairs for which deg gcd(a,b,x^155+1) differs from 6. Remove repeats under independent support translations, a common unit multiplier modulo 155, and exchange of a,b; exclude the baseline's orbit. These translations use the inverse of 2, so this quotient is restricted to odd block lengths.
This yielded 80 new 155-qubit candidates after 82 draws. The baseline plus all 80 received 300 direct Pauli RIS trials, seed 4101001, pair depth 8. The submitted pair was new candidate index 16, counting from zero. The 47 cases with initial bound at least 27 received 2,000 direct trials, seed 4101021, pair depth 20. Eight selected leads received 100,000 native doubled trials per CSS side; four leads received 20,000 direct trials and 200,000 exact-three-bit-embedding trials per side. Finalists were selected adaptively; this is not an exhaustive search or an unbiased distance survey.
A parallel shorter-length exploration screened 100 analogous candidates on Z_93 (seed 4101002); those are distinct experiments, not extra trials on this matrix. All returned logicals were validated and saved with the shared kit.
Direct searches use the unchanged trusted Pauli RIS routine. Native searches use the unchanged trusted CSS accelerator; their trial counts are per CSS side. The results below are returned source Pauli weights, after validating both the embedded logical and its mapping back to the source.
| Method | Trials | Seed | Pair depth / threads | Returned weight | | --- | ---: | ---: | --- | ---: | | Direct Pauli | 300 | 4101001 | 8 / Python | 34 | | Direct Pauli | 2,000 | 4101021 | 20 / Python | 30 | | Native doubled | 100,000/side | 4101041 | 20 / 2 | 26 | | Direct Pauli | 20,000 | 4101031 | 20 / Python | 25 | | Native exact three-bit embedding | 200,000/side | 4101031 | 20 / 2 | 27 | | Native doubled | 1,000,000/side | 4101051 | 20 / 2 | 25 | | Native exact three-bit embedding | 1,000,000/side | 10412026 | 8 / 8 | 25 | | Native exact three-bit embedding | 1,000,000/side | 4101051 | 20 / 2 | 25 |
The same 1,000,000/side three-bit search with seed 10412026, pair depth 8 and eight threads was independently rerun on the baseline: it returned a weight-21 Pauli logical, versus 25 on this candidate. The embedded Hamming weights were respectively 24 and 27; those are not source Pauli distances. Both then-retained claims were reached at this matched budget. This supports a comparative heuristic result, not a lower-bound proof for either code.
The additional matched seed 4101051 runs at 1,000,000/side, pair depth 20 and two threads returned 24 on the baseline with both embeddings, versus 25 here. They did not reattain the baseline's known 21 and are inconclusive at that seed; they never raised its retained bound.
The earlier local gate passes used seeds 532862037 (claim 25) and 973489408 (claim 24), each with an 8,000-trial ceiling under a wall-clock cap. They were subsequently superseded by two deeper independent CI runs:
| CI run | Seed | Completed accelerated trials | Refutation | | --- | ---: | ---: | --- | | 36861550907 | 575463646 | 6,540,000 | 25 to 24 | | 36867434625 | 1546517486 | 6,540,000 | 24 to 23 |
Both are genuine distance refutations. The earlier direct and million-trial results above remain historical observations; neither 25 nor 24 is retained. The two CI seeds used the unchanged trusted symplectic-doubling refutation routine (pair depth 8, four native threads), not the three-bit embedding.
The submitted JSON carries the second CI witness: X on {13,14,16,23,55,58,66,73,101,117} and Z on {4,8,14,26,38,46,58,64,66,70,72,76,88,109,135,147}. The union has 23 qubits; the three overlaps count once. Independent trusted checks confirm zero syndrome and nontriviality outside the stabilizer row space. The generator matrix and fingerprint f9e6f7cd2793720b are unchanged. The first CI witness is also preserved in the previous committed revision.
The retained candidate/baseline bounds are now 23/21. The matched-budget observations above do not establish an improvement in true distance. These remain witnessed upper bounds, not lower-bound certificates.
The corrected claim passed the unchanged full local candidate gate with seed 156064700 against upstream 1a1d84ba9447dc5f11e39a95a5e60ba08fee74f2. It found no exact/WL duplicate or dominator. Its random-search ceiling was 8,000 trials under a wall-clock cap, not a measured completed-trial count. The comparison with the baseline is still flagged as distance-only.
An additional unchanged CI-native doubled refutation pass used independent seed 2026100141, pair depth 8 and four native threads. It completed all 6,540,000 trials in 396.418 seconds without returning a validated logical below 23. This bounded non-refutation is not a proof of exact distance.
An exhaustive check of common unit multipliers, independent translations and a/b exchange excludes the baseline's obvious affine orbit. Exact stabilizer fingerprints and supplied-check support signatures differ. The baseline's supplied weight spectrum is one weight-6, fourteen weight-7 and 140 weight-8 checks; this instance has the spectrum given above. This distinguishes those supplied check systems under qubit/check permutations and local Cliffords. Arbitrary changes of stabilizer generator basis together with permutations/local Cliffords have not been exhausted.
The paper's nearly exhaustive abelian search through 300 qubits specifically describes three-plus-three supports; these use four-plus-four. Its family and gauge ideas are prior work. The repository gate and a separate bounded source/open-submission audit found no matching instance. This does not prove literature novelty; provenance.novelty remains unknown.
Early witnessed bounds were often much too high: another 155-qubit pair fell from 36 to 22, another from 35 to 21, and another from 30 to 18 under deeper searches. The baseline itself returned 27 at the 300-trial screen despite its known weight-21 witness. Earlier/larger values were never restored after a lighter logical was found. No reproduction-only candidate was treated as an original parameter improvement.
GPT-6 Astra in Codex for @mrvee-qC-bee, with independent strategy and submission-compliance agents. The repository stabilizer submission builder, shared staging kit, unchanged GF(2) routines, Pauli predicates, RIS engines and full candidate gate were used. The campaign took tens of minutes of parallel laptop computation; the per-candidate search budgets are above.
For S=(A|B), the doubled CSS matrices are HX=(A|B), HZ=(B|A). An X-side vector (u,v) maps to Pauli (u,v), and a Z-side vector maps to (v,u). For the exact three-bit embedding use HX=[A,0,B;I,I,I], HZ=[B,A+B,A]. An X vector (u,v,w) maps to (u+v,v+w); a Z logical has v=u+w and maps to (w,u). All operations are over F2. Source Pauli weight counts the union of X and Z supports. The trusted predicates check every returned mapping.
Number qubits and checks 0,...,154. Row i has X on i+{0,38,143,147} and Z on -i-{0,68,113,120}, modulo 155. Overlap means Y. This complete recipe regenerates the exact ordered checks in codes/155-6-23.json:
import json
import numpy as np
from pathlib import Path
n = 155
a, b = [0,38,143,147], [0,68,113,120]
A = np.zeros((n,n), dtype=np.int8)
B = A.copy()
for i in range(n):
A[i, [(i+x)%n for x in a]] = 1
B[i, [(-i-x)%n for x in b]] = 1
checks = [{"X": np.flatnonzero(A[i]).tolist(),
"Z": np.flatnonzero(B[i]).tolist()} for i in range(n)]
doc = json.loads(Path("codes/155-6-23.json").read_text())
assert checks == doc["checks"]["S"]