[[5,1,3]] five-qubit perfect code

Direction & hypothesis

The board keeps two separate rankings: CSS codes, which are well covered, and general stabilizer codes, which the schema only admits as of version 0.4. The Error Correction Zoo (https://errorcorrectionzoo.org) publishes explicit stabilizer tableaux for many non-CSS codes, and a tableau is exactly what a submission needs, so the question this note answers is whether a published non-CSS code already sits on the stabilizer board's frontier — and therefore whether that board is worth searching at all.

The source is the zoo entry stab_5_1_3 (https://errorcorrectionzoo.org/c/stab_5_1_3), the five-qubit perfect code of Laflamme, Miquel, Paz and Zurek (quant-ph/9602019), whose four generators are the rows XZZXI, IXZZX, XIXZZ, ZXIXZ. None of the four is pure X or pure Z, so the code is genuinely non-CSS and cannot be typed as an H_X/H_Z pair.

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 it could land in, records in some cells, or records in all of them. Only claims that could advance a cell were built. This one was in the "records in all cells" bucket.

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 3, so the claim is d <= 3, a witness-backed upper bound rather than a certified distance. Structural verification passed: generators mutually commuting, 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 stabilizer entry in that cell has n' <= 5, k' >= 1, d' >= 3 at check weight <= 4. No exact duplicate and no WL-equivalent entry was found on either board. Literature novelty is reported as unverified — the parameters are a 1996 result; 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. Two were conditional: the [[14,3,3]] rhombic dodecahedron code advances only as a non-CSS entry (a CSS realization would land in the weight-4 cell where [[12,3,3]] already sits), 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. The Bring code was therefore 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

Four generators on five qubits, read directly from the zoo tableau:

g1 = X Z Z X I g3 = X I X Z Z g2 = I X Z Z X g4 = Z X I X Z

Split into the X and Z halves, S = (A | B), one row per generator:

A = 10010 01001 10100 01010 B = 01100 00110 00011 10001

The CLI takes an .npz holding keys a and b (or a single key s with A | B) and derives k = n - rank S. n = 5, k = 1, max check weight 4, distance witness 3. The invocation that produced codes/5-1-3.json was ./qldpc submit with `--authors @MathysRennela --model "Mimo-V2.6-Flash" --family other`.

Cite the source: "([[5,1,3]] perfect code)", The Error Correction Zoo (V. V. Albert & P. Faist, eds.), https://errorcorrectionzoo.org/c/stab_5_1_3, arXiv:2606.11484; original parameters from R. Laflamme, C. Miquel, J. P. Paz and W. H. Zurek, "Perfect Quantum Error Correction Code", quant-ph/9602019.