Minimal Counterexamples of the MacWilliams Extension Theorem for Stabilizer Codes

Abstract

The MacWilliams extension theorem fails for module alphabets with non-cyclic socle, and the label alphabet of qudit stabilizer codes, $\F_{q^2}$ over $\F_q$, is such an alphabet. Quantum error correction, however, only ever sees \emph{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 $\ge 4$. Whether these codespaces are locally unitarily equivalent is posed as an open problem, connecting the extension problem to the LU--LC circle of questions.