n
93
This weight-8, unrestricted stabilizer entry encodes six logical qubits in 93 physical qubits. Its distance claim is a **witness-backed upper bound of 14**; the submitted kd²/n is 392/31 = 12.6451613. It offers a shorter six-logical-qubit block than the 155-qubit baseline, with a smaller retained distance bound. These different resource points do not establish a strict improvement in true distance or circuit performance.
The hypothesis was to change relative sparse-polynomial exponents while retaining a common factor that fixes k=6. The contribution is this independently searched instance and its reproducible evidence within the standard abelian mirror construction of Andrey Boris Khesin and Jonathan Z. Lu. The research baseline was the mirror fold of @vprusso's generalized-bicycle parent [[310,12,26]], produced using Claude Fable 5.1; its supports are in the pinned parent note. The present instance search and validation are by @mrvee-qC-bee with GPT-6 Astra. The construction is established; broader instance novelty remains unverified.
We screened 100 accepted sparse support pairs on Z_93, using four terms per polynomial and 300 direct Pauli RIS trials each, seed 4101002. The same campaign screened 80 new pairs on Z_155 plus its baseline, seed 4101001. The 93-qubit winner was accepted pair index 38, counting from zero.
The sampling pool consisted of the 140 supports {0,u,v,w} in Z_31 whose binary polynomial is divisible by f(x)=x^6+x^4+x^3+1. Nonzero exponents were lifted independently by adding 31 times a random element of {0,1,2} for n=93. We retained pairs with gcd(a(x),b(x),x^93+1)=f(x), and removed repetitions under independent support translations, common unit multipliers and A/B exchange. Sampling used Python Random with the screening seed. Eight proposals were rejected by the combined rank-factor and affine-dedup filters before 100 were accepted. This was a finite sample, not an exhaustive search of the family.
The final supports are a={0,8,25,36}, b={0,1,39,73}. The 93 supplied checks have binary symplectic rank 87: 16 have Pauli weight 7 and 77 have weight 8. No geometric layout is supplied.
The initial direct screen used pair depth 8; the later direct stages used pair depth 20. Native budgets are per CSS side. Each returned operator was mapped back, checked by the trusted Pauli predicate and retained; heavier returns never raised the bound.
| Stage | Seed and budget | Returned Pauli weight | | --- | --- | ---: | | Initial direct RIS | 4101002; 300 trials | 17 | | Shortlist direct RIS | 4101021; 2,000 trials | 15 | | Deeper direct RIS | 4101031; 20,000 trials | 14 | | Exact three-bit embedding | 4101031; 200,000 native trials/side; pair depth 20; 2 threads | 14 | | Independent three-bit run | 4101061; 1,000,000 native trials/side; pair depth 8; 4 threads | 14 | | Second independent three-bit run | 4101062; 1,000,000 native trials/side; pair depth 8; 4 threads | 14 |
The shortlist and deeper direct stages used the same seeds and budgets across the selected 93- and 155-qubit instances. The two million-trial runs both returned the same weight-14 witness; neither is a lower-bound certificate. The stored JSON contains a valid Pauli-weight-14 logical.
The unchanged full candidate gate passed against main at 0ad745c011ce7393c921a39cf35fe760090222c4. Its fresh direct refutation at seed 174219399 used a configured 6,220-trial ceiling, subject to its wall-clock cap, and found no lighter logical. The completed-trial count is not exposed. It reported fingerprint 8c2e1ae62bca119f, no exact or WL duplicate, and an advancing weight-8 unrestricted stabilizer point. This is the gate's parameter-frontier result, not a category-leading kd²/n claim.
A separate snapshot inspected every nondeleted changed code JSON in all 25 open PRs on 2026-10-01: 17 payloads were fetched at pinned heads, with no filename or n filter. None had n=93,k=6. The authors' numerical catalog at its pinned revision contained 8,336 parsed records but no group order 93. These checks do not prove literature novelty or general code inequivalence.
The proposed local-Clifford-to-CSS helper in PR 2599 at its reviewed revision found no local Clifford making every supplied generator pure X or Z. That test does not exclude a CSS description after changing the stabilizer generator basis. General local-Clifford/permutation equivalence remains unresolved.
The winner's initial bound 17 fell to 15 and then 14. Another selected 93-qubit instance also fell from 15 to 14 under the deeper search, so the short screen did not distinguish those retained distances. Across the 155-qubit branch, an attractive initial bound 35 fell to 21, and another bound 36 fell to 22. These collapses motivated the million-trial checks. The preserved common factor keeps k=6; this experiment sought useful sparse instances, not a new commutation theorem or a distance certificate.
GPT-6 Astra in Codex, with independent structural, source and submission compliance review. Searches used the unchanged trusted direct Pauli RIS and native GF(2) engines. Structural and witness checks used verify/qldpc_verify.py; the full admission gate used verify/validate_candidate.py. Search budgets and parallelism are listed above; this submission makes no decoding, circuit-error-rate or locality claim.
Number qubits and rows from 0 to 92. Generator i has X support {i+a mod 93 : a in {0,8,25,36}} and Z support {-i-b mod 93 : b in {0,1,39,73}}. An X/Z overlap is one Y and contributes one to Pauli weight. This exactly specifies the submitted matrix:
import json
with open("codes/93-6-14.json") as stream:
doc = json.load(stream)
rows = [
{"X": sorted((i+a) % 93 for a in (0,8,25,36)),
"Z": sorted((-i-b) % 93 for b in (0,1,39,73))}
for i in range(93)
]
assert rows == doc["checks"]["S"]
For source matrices S=(A|B), construct the CSS embedding H_X=[A,0,B; I,I,I], H_Z=[B,A+B,A]. Its X logical quotient maps by (u,v,w) -> (u+v,v+w); adding (t,t,t) does not change the mapped Pauli. Each nonidentity Pauli admits a one-bit lift, so the embedded X distance equals the source Pauli distance. A returned Z operator has form (u,u+w,w) and maps to (w,u). Always validate and score the mapped Pauli: embedded Hamming weight need not equal the returned representative's Pauli weight. This exact relationship does not make the random search exact.
Run the repository verifier on codes/93-6-14.json to check the structure and stored witness. To repeat a native stage, call the trusted distance_rand_witness on the two block matrices with the stated seed, trial budget, pair depth and thread count, then validate the mapped operator with the trusted Pauli predicate. The original candidate gate was run before adding the code to the board.