bibkey: mahmoud2026macwilliams authors: Ali Assem Mahmoud year: 2026 title: “Minimal Counterexamples of the MacWilliams Extension Theorem for Stabilizer Codes” doi: 10.48550/arXiv.2607.26214 url: https://arxiv.org/abs/2607.26214v1 claim: “Theorem 5.3 gives the weight-isometric, monomially inequivalent pairs of [[5,2]] codes S_A = <ZZZZI, XXIIX, IIXXX>, S_B = <ZZIZX, XIZXI, IXXYI> and of [[6,3]] codes S+ = <XXIIXX, IIXXXX, ZZZZIX>, S- = <XXZXII, ZZXYII, IZZZXX>; Question 5.4 asks whether the codespaces of S_A and S_B, or those of S+ and S-, are mapped to one another by some product unitary U1 x … x Un composed with a qubit permutation.” strata_touched:

• D5/S3/Quantum/Information/StabilizerPairLocalUnitaryInequivalence license: citation-only triage: anchor

Minimal Counterexamples of the MacWilliams Extension Theorem for Stabilizer Codes

A. A. Mahmoud, “Minimal Counterexamples of the MacWilliams Extension Theorem for Stabilizer Codes”, arXiv:2607.26214v1 (2026-07-30; primary quant-ph, cross-listed to math-ph).

The paper asks whether every weight-preserving isomorphism of stabilizer groups is implemented by local Cliffords and a qudit permutation, and constructs counterexamples at the smallest lengths. Lemma 3.2 shows that the number t of qubits on which the code projector acts as the identity is invariant under product unitaries and permutations, and Theorem 3.3 (iii) uses it to separate the codespaces of the basic [[q+1, q−1]]_q family. Theorem 5.3 gives two full-support pairs on which t vanishes:

• (ii) the [[5,2]] codes S_A = ⟨ZZZZI, XXIIX, IIXXX⟩ and S_B = ⟨ZZIZX, XIZXI, IXXYI⟩;
• (iii) the [[6,3]] codes S+ = ⟨XXIIXX, IIXXXX, ZZZZIX⟩ and S− = ⟨XXZXII, ZZXYII, IZZZXX⟩.

After Theorem 5.3 the paper states:

For the pairs in Theorem 5.3(ii),(iii) the invariant t vanishes on both sides, and we know of no invariant separating the two codespaces under general local unitaries: Question 5.4 (Extension analogue of LU–LC). Are the codespaces of SA and SB of Theorem 5.3(ii)—or those of S+ and S− of Theorem 5.3(iii)—mapped to one another by some product unitary U1⊗···⊗Un composed with a qubit permutation? More generally: do there exist weight-isometric stabilizer codes that are locally unitarily equivalent but not locally Clifford equivalent—or is local-unitary equivalence of weight-isometric pairs always witnessed monomially?

and §6.2 repeats it as open problem 1, calling the [[5,2]] pair “the minimal open instance”.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2607.26214
• URL: https://arxiv.org/abs/2607.26214v1 (the only version; full text retrieved 2026-09-29).
• Location: Lemma 3.2 and Theorem 3.3 in §3; Theorem 5.3 and Question 5.4 in §5; open problem 1 in §6.2.