[[13,1,5]] twisted XZZX toric code

Direction & hypothesis

Twisted XZZX codes are the family where a short Pauli string buys distance out of proportion to blocklength, and the Error Correction Zoo (https://errorcorrectionzoo.org) states both a three-parameter claim and the one-line generator for a small member of it. The question this note answers is whether that published point already sits on the board's frontier: if a known twisted code records, the cell is thin and worth searching; if every known twisted code in a cell is dominated, that cell needs a genuinely better construction.

The source is the zoo entry stab_13_1_5 (https://errorcorrectionzoo.org/c/stab_13_1_5): thirteen cyclic permutations of the XZZX-type string XIZZIXIIIIIII, a small twisted XZZX toric code from Kovalev, Dumer and Pryadko (arXiv:1108.5490, Example 11 and Fig. 3). The tableau is non-CSS — the generator mixes X and Z — so the code is typed as a general stabilizer code.

What was searched

A crawl of the zoo index filtered to entries whose hierarchy path passes through a QLDPC concept returned 681 quantum entries; 190 are primary qLDPC nodes, of which 35 state a three-parameter [[n,k,d]] claim. Each claim was bracketed against a snapshot of the live board before any matrix was built — exact match on (n,k,d), dominated in every cell, records in some cells, or records in all of them. Only claims that could advance a cell were built, and this one was in the "records in all cells" bucket. No search over constructions was run: the generators came from the zoo.

Evidence trail

The verifier's Pauli-weight witness search (20,000 random information-set trials, then the 2,000,000-trial accelerator pass on the symplectic doubling) returned a witness of Pauli weight 5, so the claim is d <= 5, a witness-backed upper bound that matches the zoo's stated distance. Structural verification passed: all thirteen generators mutually commuting (rank 12, so k = 1), max check weight 4, witness valid.

The trusted gate verify/validate_candidate.py on the built document returned passed: true with the label "advances the weight-4 x unrestricted stabilizer board", no exact duplicate and no WL-equivalent entry. Literature novelty is reported as unverified: the parameters are published; the claim is the frontier, not novelty.

Dead ends

Of the 35 stated claims, 20 were already on the board as exact parameters and 9 were dominated in every cell they could land in. Two were conditional: the [[14,3,3]] rhombic dodecahedron code advances only in its non-CSS form (a CSS realization would sit in the weight-4 cell where [[12,3,3]] already records), and the [[30,8,3]] Bring code has weight-5 generators, which puts it in the weight-6 class where [[25,9,3]], [[30,10,3]] and [[30,8,4]] dominate it, so it was not built.

Tools

Model Mimo-V2.6-Flash; the repository CLI for the build, the witness search and the submission document, and the repository's verify/ stack for the gate. Approximate compute: under a minute of CPU.

Reproduction

One generator, cyclically shifted thirteen times:

gen = X I Z Z I X I I I I I I I (positions 0, 2, 3, 5 carry the weight) row_i = gen[i:] + gen[:i] for i = 0 .. 12

Each row splits into an X half and a Z half; the CLI takes an .npz with keys a and b, or a single key s holding A | B row-major. n = 13, rank S = 12 so k = 1, max check weight 4, distance witness 5. The invocation that produced codes/13-1-5.json was ./qldpc submit with --authors @MathysRennela --model "Mimo-V2.6-Flash" --family topological.

Cite the source: "([[13,1,5]] twisted toric code)", The Error Correction Zoo (V. V. Albert & P. Faist, eds.), https://errorcorrectionzoo.org/c/stab_13_1_5, arXiv:2606.11484; construction from A. A. Kovalev, I. Dumer and L. P. Pryadko, "Design of additive quantum codes via the code-word-stabilized framework", arXiv:1108.5490.