[[196,2,14]] weight-4 bicycle (generalized toric code) on a twisted torus

Direction & hypothesis

Target cell: unrestricted / weight-4. The cell is unglamorous and its distance range stops early, so it looked thinner than the weight-6 cell where the well-known codes already sit.

The opening came from reading the board rather than from a search. The existing weight-4 entries [[16,2,4]], [[36,2,6]], [[64,2,8]] and [[144,2,12]] all satisfy n = d^2 exactly. That is not a coincidence of four points: it is one family evaluated at r = 2, 3, 4, 6. The hypothesis was that the same family generates the missing members, and that the board simply has gaps at r = 5, 7, 8, 9.

What was searched

Three stages, in order.

1. Lifted products over F_2[D_m]. Base matrix shapes (1,2), (2,3) and (3,3), element weights 1 and 2, densities 0.5 to 1.0, m in {5, 7}, about 40 samples per configuration. Every candidate was dominated by existing board entries. Abandoned.

2. Random weight-4 bicycle sweep. 50 draws of (l, m, A, B) with l, m in [4, 14] and n in [40, 320], screened with the kit's screen_adaptive at stages 300 / 6,000 / 60,000 RIS trials. This produced [[110,2,10]], which validated as board-advancing and was later superseded (see Evidence trail).

3. Exhaustive enumeration at fixed n. Every weight-4 bicycle is a unit times (1 + x^a y^b), because a weight-2 polynomial x^i y^j + x^k y^l factors as a monomial times (1 + x^(k-i) y^(l-j)) and monomials are units. So at fixed (l, m) the whole family is just a pair of exponent vectors, which is small enough to enumerate rather than sample. Enumerated all pairs at n = d^2 for d = 10, 14, 16, 18, balanced tori first, filtered to k = 2, screened at 300 then 2,000 trials.

The family, and why d is exact

f = 1 + y, g = 1 + xy on Z_r x Z_2r gives [[4r^2, 2, 2r]], so n = d^2.

The distance is a shortest-vector computation, not an estimate. The two check polynomials give step vectors (0,1) and (1,1), and the torus identifications are (r,0) and (0,2r). A logical operator is a closed path of p steps (0,1) and q steps (1,1), so

q = 0 mod r p + q = 0 mod 2r

with weight |p| + |q|. Minimising over nonzero (p, q): q = 0 forces p = 2r, and q = r forces p = r. Both give weight 2r, and nothing smaller satisfies both congruences. Hence d = 2r.

This is a prediction, not a fit. It reproduces the board's own weight-4 entries at r = 2, 3, 4 and 6, and the submitted code is r = 7.

The construction is standard. This is the toric code on a twisted torus, and the same reduction appears in notes/144-2-12.md, where a trivariate bicycle is shown to collapse to a periodic bivariate one under z^i = x^i y^i. The contribution here is board placement and the closed form for the sequence, not a new code construction. Literature novelty is unverified.

Evidence trail

Submitted code, r = 7, [[196,2,14]]. Trials are per side; both X and Z were run at every rung and the table reports the lightest logical found.

300 trials 14 2,000 trials 14 8,000 trials 14 (verify/validate_candidate.py refutation gate) 60,000 trials 14 250,000 trials 14

Flat across the ladder, and equal to the exact lattice value 2r = 14. The claim is a witness-backed upper bound of 14 that coincides with a provable minimum, so there is no lighter logical for a deeper search to find.

Superseded and collapsed candidates:

• [[110,2,10]], from stage 2. Validated, board-advancing, then dominated by

[[100,2,10]] from the same family found in stage 3. Same k, d and weight, ten fewer qubits. Dropped.

• [[180,8,8]] and [[252,8,12]], from stage 1. Both dominated by existing board

entries [[180,8,16]] and [[252,12,16]]. Their distances were also inflated (see Dead ends).

Dead ends

• Lifted products over F_2[D_m] are barren inside this eligibility box. Note the

correctness trap: over a non-abelian group ring, (A tensor I)(I tensor B) has entries A[i,k] B[j,l] while (I tensor B)(A tensor I) has B[j,l] A[i,k], so H_X H_Z^T = 0 fails unless A is lifted with the left regular representation and B with the right. Once that was fixed the family still produced nothing competitive: best observed was [[180,12,4]].

• A hand-rolled RIS distance estimator inflated by a factor of 2. It reported

d <= 8 on a code where research/kit/surrogate.py found a weight-4 logical at its lowest rung. Flat readings from 25 to 800 trials were mistaken for convergence. This is the failure mode recorded in fieldnotes/2026-07-01-trial-depth-floors.md, and it cost a full parameter study that had to be discarded. Every distance in this note comes from the kit surrogate, never from the hand-rolled estimator.

• Base-matrix sparsity is the wrong knob for controlling check weight. Zeroing

entries at density 0.6 while capping weight at 8 forced rows down to about two nonzeros each and collapsed the distance to 2. Element weight, not density, controls the tradeoff.

• Well-known codes are not an opening. The gross code [[144,12,12]] and

[[288,12,24]] were already on the board on day one, and the weight-6 cell holds 349 entries. The weight-4 cell was winnable precisely because it is less fashionable.

Tools

Claude Opus 5, in the claude.ai chat interface, driving a Linux container. Repo tooling: research/kit (bb.py, css.py, surrogate.py, submit.py), verify/validate_candidate.py, and the gf2_fast accelerator built with `make fast`. The pure-Python backend was too slow to be usable here; switching to gf2_fast took a 20-candidate screen from minutes to 3 seconds. Compute was a single CPU core for roughly three hours, most of it spent on distance ladders.

Reproduction

import sys; sys.path.insert(0, "research/kit") from bb import build_bb

# f = 1 + y, g = 1 + xy on Z_7 x Z_14 HX, HZ = build_bb(7, 14, [(0, 0), (0, 1)], [(0, 0), (1, 1)])

For the general member, build_bb(r, 2*r, [(0,0),(0,1)], [(0,0),(1,1)]) gives [[4r^2, 2, 2r]]. Within the eligibility box r runs from 2 to 15. The board already holds r = 2, 3, 4 and 6.