← Five-qubit perfect code

Entry facet

Representation ladder

R1 checks through R5 cross-model views, with canonicality and provenance.

  1. R1

    Checks

    Sparse Pauli generators in symplectic form, including stabilizer and gauge rows.

    • cyclic-mixed-pauli canonical

      The exact representative is generated by XZZX_ and three one-site cyclic translates. Pairwise commutation comes from an even number of local X/Z disagreements, not from a CSS separation into X-only and Z-only rows.

      References: laflamme1996perfect, gottesman1997stabilizer

  2. R2

    Structure

    CSS pairs, chain complexes, lattices, and other structure behind the checks.

    • binary-symplectic derived by five_one_three.build plus Pauli.symplectic 1

      Each Pauli row becomes a binary (x|z) vector. The symplectic products vanish, GF(2) row reduction has rank four, and the remaining normalizer quotient contains one canonical logical X/Z pair.

      References: gottesman1997stabilizer

  3. R3

    Dynamics

    Ordered measurement schedules and their instantaneous stabilizer groups.

    No representation at this rung is recorded for this entry.

  4. R4

    Circuit & detectors

    Named syndrome-extraction circuits and detector error models.

    No representation at this rung is recorded for this entry.

  5. R5

    Cross-model views

    Executable or illuminating algebraic, ZX, fermionic, tensor, and statistical dictionaries.

    • perfect-syndrome-packing derived by single-qubit symplectic syndrome enumeration 1

      Enumerating X, Y, and Z at every site uses every nonzero four-bit syndrome exactly once. This both defines the exact lookup decoder and explains why a second distinct single-site fault cannot share a syndrome.

      References: laflamme1996perfect, bennett1996mixed

    • additive-gf4 curated from gottesman1997stabilizer

      Packing each binary (x_i,z_i) pair into one GF(4) symbol exposes the additive-code classification. It is a dictionary for the same binary stabilizer subspace, not permission to call this entry a qudit family.

      References: gottesman1997stabilizer, rains1997distance

    • graph-code curated from schlingemann2002graph

      A graph-code or CWS presentation adds an encoder input and a graph-state dictionary. Logical claims transfer only after converting back to the exact tableau; a pentagon graph state by itself is not the code.

      References: schlingemann2002graph, cross2009codeword, khesin2025universalgraph