2026/07/28 by Ali Assem Mahmoud
Mathematics · Physics and Astronomy · #acm:16D10 #acm:81P70 #acm:94B05 #math-ph #math.MP #msc:16D10 #msc:81P70 #msc:94B05 #quant-ph
arxiv created 2026/07/28 · arxiv updated 2026/07/30
The MacWilliams extension theorem fails for module alphabets with non-cyclic socle, and the label alphabet of qudit stabilizer codes, \Fq2 over \Fq, is such an alphabet. Quantum error correction, however, only ever sees self-orthogonal additive codes, and whether that rigidity rescues the theorem---equivalently, whether every weight-preserving isomorphism of stabilizer groups is implemented by local Cliffords and a qudit permutation---was asked by Gluesing-Luerssen and Pllaha and answered negatively by Pllaha for particular qubit codes. We develop the negative answer systematically and at the smallest possible scales. For every prime power q we construct a pair of [[q+1,q-1]]q stabilizer codes and a weight-preserving isomorphism between them extending to no monomial transformation; the codespaces are inequivalent even under arbitrary local unitaries combined with permutations, though they share Shor--Laflamme enumerators. Self-orthogonality is automatic here, by two elementary lemmas which also show that Dyshko's threshold-length counterexamples were already self-orthogonal, unremarked. For qubits we prove by exhaustive search that length 3 is minimal and the counterexample essentially unique. Dropping the ``idle qudit'' invariant that detects these, we find the minimal full-support lengths: 4 for a non-extendable isometry, 5 for a weight-isometric pair that is not monomially equivalent, realized by explicit [[5,2]] codes; at length 6 all nontrivial stabilizer elements can have weight ≥ 4. Whether these codespaces are locally unitarily equivalent is posed as an open problem, connecting the extension problem to the LU--LC circle of questions.