Gnu Code¶
A gnu code encodes a logical qubit in parity-separated, binomially weighted Dicke states whose excitation weights are spaced by gap g across m=gnu symmetric physical qubits.
Core Idea¶
A gnu code is Ouyang's particular permutation-invariant quantum code: \(m=gnu\) physical qubits support Dicke states at excitation weights \(0,g,\ldots,ng\); square-root binomial amplitudes and even/odd ladder indices form logical zero and one. Permutation symmetry is shared with other PI codes; the g-spaced binomial/parity construction gives this family its identity.[^ref-5db685b2942e]
Scope of Application¶
The nine-qubit \((3,3,1)\) example exactly corrects one arbitrary qubit error. The distinct nine-qubit \((2,4,9/8)\) example approximately corrects one spontaneous-decay error under Ouyang's small-damping conditions, with a nonzero worst-case error bound. A later paper studies shifted gnu codes and deletion correction; those guarantees must not be assigned to every original unshifted code.[ref-5db685b2942e][ref-ee6f70b680f7]
Clarity¶
\(g\) spaces occupied Dicke weights, \(n\) counts ladder intervals, and \(u=m/(gn)\geq1\) scales length. A Dicke state is a symmetric fixed-excitation state, not an entire code. Symmetry alone does not prove a channel-specific recovery claim.
Manages Complexity¶
The PI codewords organize many computational-basis strings by a small set of excitation weights. The simplification helps combinatorial error analysis, but parameter choice, qubit overhead, preparation and a specified recovery theorem remain necessary.[^ref-5db685b2942e]
Abstract Reasoning¶
Each Dicke component survives qubit relabeling. Even and odd ladder supports are disjoint logical states; gap and binomial weighting are used in Ouyang's correction proofs. A stronger target error weight can require more physical qubits: the original paper's two-arbitrary-error example has 25 rather than nine.[^ref-5db685b2942e]
Knowledge Transfer¶
The construction varies across parameters and error channels only with its Dicke ladder and parity structure intact. A generic PI code or a shifted deletion variant is related but not identical. The live Error-Correcting Code is the strict genus; a narrower quantum-code intermediate remains a future question.
[^ref-5db685b2942e]: Ouyang, Permutation-invariant quantum codes, original full paper. [^ref-ee6f70b680f7]: Ouyang, Permutation-invariant coding for quantum deletion channels, shifted variant.
Relationships to Other Abstractions¶
Current abstraction Gnu Code Domain-specific
Parents (1) — more general patterns this builds on
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Gnu Code is a kind of Error-Correcting Code Domain-specific
A gnu code is an error-correcting code with a specific symmetric Dicke-state ladder and parity split.
Hierarchy path (1) — routes to 1 parentless root
- Gnu Code → Error-Correcting Code → Encoding And Decoding → Transformation → Function (Mapping)
Neighborhood in Abstraction Space¶
Gnu Code sits in a sparse region of the domain-specific corpus (68th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Quantum States & Information Measures (25 abstractions)
Nearest neighbors
- Steane Code — 0.85
- Quantum Walk — 0.84
- Even code — 0.84
- Hamming Scheme — 0.84
- Clifford gate — 0.83
Computed from structural-signature embeddings · 2026-10-08