The quantum extension property for full-support stabilizer codes: the minimal lengths at stabilizer dimensions 4 and 5, distance-3 counterexamples, and a positive answer for CSS codes
Abstract
The MacWilliams extension theorem fails over the label alphabet A=𝔽₂² of qubit Pauli operators, and Mahmoud has recently located the smallest scales at which it fails for stabilizer codes. Writing k for the 𝔽₂-dimension of the stabilizer, he determines, for k ≤ 3, the minimal length of a full-support non-extendable weight-preserving isometry (it is 4) and the minimal length of a full-support weight-isometric pair of codes that are not even monomially equivalent (it is 5), and asks for k ≥ 4. We answer that question at k=4 and k=5: the two minimal lengths are (4,5) at k=4 and (5,6) at k=5, so over the four stabilizer dimensions now known the isometry length is max(4,k), the pair length is max(5,k+1), and the gap of one between them persists. We answer a second question of his negatively: distance >2 does not restore the extension property. The perfect [[5,1,3]] code carries a weight-preserving automorphism of its stabilizer group implemented by no local Clifford circuit together with a qubit permutation; since every nonzero element of that stabilizer has weight 4, weight preservation is a vacuous hypothesis there, and we therefore also exhibit a pair of [[6,1]] codes of stabilizer distance at least 3 with a non-extendable weight-preserving isometry whose weight vector takes four distinct values, so that the hypothesis is doing work. A third question we answer positively: for CSS codes with CSS-preserving isometries the extension property holds, at every length, because the class incidence matrix is invertible over ℚ — certified by an explicit integral matrix carrying it to a nonzero multiple of the identity, at k=3 and k=4. Every existence statement below is machine-checked against the definitions themselves, the failure of extension being tested against all 6ⁿ n! monomial transformations, 33 592 320 of them at n=6; the matching lower bounds at k=5 rest on an exhaustive orbit census over 63 082 768 parameterized data and are labelled throughout as computations rather than as formal proofs.
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Source snapshot 2026-08-30 15:34 UTC
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Claim ledger
Stated results
QF1candidate2026-08-30
open problem 4 at k = 4: the minimal full-support lengths are 4 (for a non-extendable isometry) and 5 (for a monomially inequivalent pair) – the gap persists, with the same values the source proves for k = 3
QF2candidate2026-08-30
open problem 4 at k = 5: the minimal full-support non-extendable isometry length is 5, i.e. the floor n = k, so the answer stops being 4 and becomes k
QF3candidate2026-08-30
open problem 3, answered negatively: distance > 2 does NOT restore the extension property – the perfect [[5,1,3]] code has a weight-preserving automorphism implemented by no local Clifford circuit and qubit permutation
QF4candidate2026-08-30
open problem 6, answered positively for the CSS subclass: the class incidence matrix is invertible over Q, so the weight vector determines the multiset of coordinate kernels and every CSS-preserving weight-preserving isometry extends
QF5prose2026-08-30
the CSS positive answer at every length and dimension: for all kX, kZ, n, a weight-preserving CSS-preserving isometry between CSS stabilizer codes on a common splitting extends to a monomial transformation
This ledger entry is reported in prose and is not bound to a Lean theorem.QF6known data2026-08-30
the compute-first control: the source's own k = 3 census reproduced from its definitions, and the finite content of its kernel reduction at k = 3, 4, 5
QF7routine2026-08-30
negative controls: the model, the monomial-extension test, each hypothesis, and each search are all non-vacuous in both directions
QF8measurement2026-08-30
the census beyond Lean's reach: exact GL(k,2)-orbit counts by union-find under the elementary transvections, and the cost of the whole search
This ledger entry is reported in prose and is not bound to a Lean theorem.QF9candidate2026-08-30
open problem 4 at k = 5, the pair half: the minimal full-support length carrying a monomially inequivalent weight-isometric pair of stabilizer dimension 5 is exactly 6, so the gap between the two minimal lengths persists at k = 5 and the answer there is (5, 6) against (4, 5) at k = 3 and k = 4
QF10candidate2026-08-30
open problem 3 with the vacuity escape closed: a [[6,1,3]] stabilizer code and a second of distance >= 3, with a weight-preserving isometry that extends to no monomial transformation and whose weight vector is NOT constant, so weight preservation is a real hypothesis – distance > 2 does not restore the extension property even then
QF11measurement2026-08-30
the k = 5, n = 6 census: 63082768 full-support data, 43278108 weight vectors, 14436204 counterexample classes, exactly 937440 of them monomially inequivalent, and 13516 all of whose codes have stabilizer distance >= 3
This ledger entry is reported in prose and is not bound to a Lean theorem.QF12candidate2026-08-30
open problem 6 with the CSS-preserving hypothesis dropped: the extension property fails for CSS codes, and at the unrestricted minimal lengths – 4 for a non-extendable weight-preserving isometry between two CSS stabilizer codes, 5 for a monomially inequivalent weight-isometric CSS pair – so the CSS restriction on its own buys nothing and the whole of QF4/QF5's positive answer rests on CSS-preserving
QF13measurement2026-08-30
the monomially-inequivalent column of the CSS filter: how many counterexample weight classes contain two codes in different GL(k,2) orbits, with and without the CSS restriction, at k = 3, 4 and n <= 6
This ledger entry is reported in prose and is not bound to a Lean theorem.QF14routine2026-08-30
open problem 6, the graph-code sub-question, answered negatively at the isometry level: the four-qubit linear cluster state C_(P₄) has a weight-preserving automorphism of its stabilizer group – transpose the two leaf generators – implemented by no local Clifford circuit and qubit permutation; minimal length 4; and the published counterexample of Gluesing-Luerssen–Pllaha is checked here to BE that cluster state
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Founded 2026-08-30 from arXiv:2607.26214 v1 (Ali Assem Mahmoud, *Minimal Counterexamples of the MacWilliams Extension Theorem for Stabilizer Codes*, announced 2026-07-28), on rows R1–R3 of journal/2026-08-30-scout-qec.md. The family answers three of that paper's seven numbered open problems.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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