[[4,2,2]] rotated toric code (weight-4, 2D-local single-layer)

Direction & hypothesis

The board's least populated primary-track cell is weight-4 x local-2d-single (175 of 1,126 entries at the time of writing): the surface-code-like regime that most qLDPC families (weight >= 6 checks) cannot reach. The standard topological ladders are only seeded at a few sizes on the board (rotated surface [[L^2,1,L]] at L = 5,7,9,11; rotated toric [[L^2,2,L]] at L = 4,6,8), so the hypothesis was that the missing square sizes extend this cell's Pareto frontier on d, at both k = 1 (surface) and k = 2 (toric).

What was searched

Full enumeration of both rotated ladders under the board's admission caps (n <= 700, or the extended tier n <= 1000 with w <= 8 and d <= 40; w = 4 throughout): every square L with admissible(n, 4, L) true, plus every rectangular (L1, L2) configuration -- 1,270 configurations in total. Each was packaged with the kit's own submit.make_submission (RIS witness search) and passed through the trusted gate verify/validate_candidate.py. The gate's novelty verdict split the enumeration exactly along a structural line: squares advance, rectangles never (each [[L1*L2,1,L2]] is strictly dominated by its own square [[L2^2,1,L2]]: same k, same d, same weight, strictly fewer qubits). Only board-advancing survivors were kept; this PR is one of them.

Evidence trail

• Witness search (kit surrogate, RIS): <= 3,000 trials/side scaled with n;

the lightest logical found on each side has weight 2 (a row or column of plaquettes), matching the exact-by-construction distance.

• Trusted gate (verify/validate_candidate.py): verify ok, distance not

refuted (independent fresh-seed RIS search found nothing lighter), not a board duplicate (fingerprint and WL signature checked against the whole board), and board-advancing in the weight-4 x local-2d-single cell.

• Claim carried: witness-backed upper bound, d <= 2. The distance is exact

by construction (no logical lighter than a minimum non-contractible chain exists on the torus), but it is submitted as an upper bound; a maintainer can run verify/certify.py to upgrade it.

Dead ends

• Rectangular rotated surface/toric codes: all dominated by their own square

(see above) -- generated and gate-checked, then discarded.

• [[9,1,3]] (dominated by the seeded [[7,1,3]]) and [[8,2,2]] (dominated by

[[4,2,2]]): gate-flagged as not board-advancing, discarded.

• Rotated toric L = 10,12,14: parameter sets already on the board

([[100,2,10]] generalized-bicycle, [[144,2,12]] trivariate-bicycle, [[196,2,14]] bivariate-bicycle), skipped as parameter duplicates.

• An initial torus layout used a ring fold with a seam gap of 3 grid units,

which pushed the measured interaction radius past the single-layer cap for L >= 10 (demoting those codes to local-2d-bilayer). The seeded entries' fold convention (seam gap 1) keeps every plaquette diameter at 2*sqrt(2) for every L; that is what this submission uses.

• Unrotated surface [[2d^2-2d+1,1,d]] and 2D color codes were attempted for a

companion fill but their from-scratch constructions failed self-checks; deferred until built from published constructions.

Tools

GLM 5.3 Flash (Zed coding agent); repo kit only (research/kit/submit.py, verify/validate_candidate.py, qldpc_verify), numpy-only constructions. Constructions were verified bit-for-bit (check-support set equality) against the board's own seeded codes/25-1-5.json (L = 5,7,9) and codes/16-2-4.json (L = 4,6,8) before scaling to new sizes.

Reproduction

Rotated toric code [[L^2,2,L]], L = 2 (even):

• qubits q = (i, j) -> q = i*L + j on an L x L torus (periodic in both axes);
• for each (i, j) in {0..L-1}^2 the periodic 2x2 plaquette

{(i,j), (i,j+1 mod L), (i+1 mod L, j), (i+1 mod L, j+1 mod L)} is an X check when i+j is even and a Z check when i+j is odd;

• layout: ring fold per axis, i -> 2i for i < L/2 and i -> 2(L-i)-1 for

i >= L/2 (the seeded entries' convention; seam gap 1, so every plaquette has diameter 2*sqrt(2)), single layer. Distance: a non-contractible row or column of plaquettes is a weight-L logical; none lighter exists (cycle lengths of the torus).