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This paper proves that the MacWilliams extension theorem fails for quantum stabilizer codes at every qudit dimension, and pins the failure at its smallest possible scales.

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2026-08-01 00:31 UTC pith:QKVCAABI

load-bearing objection Solid constructive counterexamples for the stabilizer MacWilliams question, with a genuine gap in the computational minimality proof that the author should fix before publication. the 2 major comments →

arxiv 2607.26214 v1 pith:QKVCAABI submitted 2026-07-28 quant-ph math-phmath.MP

Minimal Counterexamples of the MacWilliams Extension Theorem for Stabilizer Codes

Ali Assem Mahmoud This is my paper
classification quant-ph math-phmath.MP MSC 94B0581P7016D1020C20
keywords MacWilliams extension theoremstabilizer codesself-orthogonal additive codesweight-preserving isometrylocal Clifford equivalencequdit codescodespace invariantexhaustive search
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves that the MacWilliams extension theorem—the classical rigidity principle that a weight-preserving isomorphism of codes must be a coordinate permutation with scalar factors—fails for quantum stabilizer codes at every qudit dimension, and describes the smallest such failures. The author constructs, for each prime power q, a pair of [[q+1,q-1]]_q stabilizer codes and a weight-preserving additive isomorphism between them that extends to no monomial transformation; the corresponding code spaces are inequivalent even under arbitrary product unitaries plus permutations, while the Shor-Laflamme enumerators coincide. For qubits the paper shows by exhaustive search that length 3 is the smallest counterexample and that it is essentially unique. It also determines the minimal lengths under a 'no idle qudit' condition: 4 for a non-extendable isometry and 5 for a pair of codes not monomially equivalent at all. If correct, this means the rigidity that the extension theorem provides in classical coding theory is genuinely unavailable to quantum error correction, even for the self-orthogonal codes that define stabilizer codes.

Core claim

On the paper's own terms, the central discovery is that self-orthogonality—the condition that makes an additive code a stabilizer code—does not rescue the MacWilliams extension theorem. For every prime power q, the codes C+ = {(x,...,x,0)} and C- = {(lambda_0(x),...,lambda_q(x))} built from the q+1 lines of F_q^2 are both trace-symplectically self-orthogonal and have the same weight distribution, but the natural componentwise bijection preserves Hamming weights while extending to no monomial map. The obstruction is detected by an elementary codespace invariant: the number t of qudits on which the code acts trivially is unchanged by any product unitary and permutation, and C+ has t=1 while C-

What carries the argument

The pair is built from the projective line over F_q: one code repeats a single message q times and appends one zero coordinate; the other evaluates the message against q+1 rank-one functionals whose kernels are the q+1 lines. Three structures do the work: Lemma 3.1 (repeated blocks with multiplicity divisible by the field characteristic, and all coordinate maps of rank at most one, give self-orthogonality automatically), Lemma 3.2 (the number of idle qudits t is invariant under product unitaries and permutations), and the orbit criterion (an isometry extends monomially iff the multisets of coordinate-kernel orbits match). The exhaustive minimality claims rest on a classification of isometry

Load-bearing premise

The minimality, uniqueness, and full-support census claims all rely on the completeness of the exhaustive computer search over all self-orthogonal additive codes up to length 6 and all kernel-multiset configurations for stabilizer dimension 3; if that enumeration missed a class, the claimed smallest lengths could be too small.

What would settle it

Rerun the enumeration independently: for q=2, k=3, and each n up to 6, generate all kernel multisets without using the Lemma 5.2 parameterization, impose joint injectivity and self-orthogonality condition (1), and check whether exactly 168 weight-isometry classes at n=5 and the stated lengths appear. A single new n=4 or n=5 class would disprove Theorem 5.3; alternatively, a brute-force search over all 64·4! monomial maps at n=4 finding a monomial extension of f would disprove part (i).

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • QEP(q, q+1) fails for every prime power q: for every qudit dimension there are stabilizer codes with identical Shor-Laflamme enumerators that are inequivalent under local operations and permutations.
  • For qubits the failure is not exotic: at length 3 every counterexample is a monomial image of the single canonical pair, and none exists at length 1 or 2.
  • The obstruction is not an artifact of idle qudits: full-support counterexamples exist, with minimal lengths 4 for a non-extendable isometry and 5 for a monomially inequivalent pair of codes.
  • High stabilizer-element weights do not restore extension: at length 6 there is a pair with all nontrivial elements of weight at least 4.
  • The quantum MacWilliams identities transfer while the extension theorem does not; code equivalence in the stabilizer setting is strictly finer than enumerator equivalence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • We infer that the length-5 full-support pair is the natural minimal test case for whether the extension obstruction survives at the codespace level: a local-unitary equivalence between those two codespaces would show the obstruction is purely a label-level phenomenon, while its absence would extend the negative answer of the main theorem to full-support codes.
  • The automatic self-orthogonality lemmas suggest a general recipe: any weight-isometric pair built from coordinate maps of rank at most one, or from blocks repeated a multiple of the field characteristic, automatically lives on the self-orthogonal quadric; this may generate counterexamples in other self-orthogonal code families beyond the projective-line pair.
  • The enumerator equality in the main theorem shows weight distribution is far from a complete invariant for stabilizer codes; the census at length 3 hints that a complete classification of local-Clifford-with-permutation equivalence classes is tractable at small lengths, a program the paper only starts.
  • A testable extension would be to restrict to CSS codes or graph codes and ask whether the extension property is restored; the paper lists this as an open direction, and our reading is that the projective-line codes are not CSS, so a positive answer would carve out a meaningful subclass.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper studies a quantum analogue of the MacWilliams extension theorem: is every Hamming-weight-preserving isomorphism between self-orthogonal additive codes — i.e. stabilizer groups — implemented by local Clifford unitaries and a qudit permutation? The main construction (Theorem 3.3) gives, for every prime power q, a pair of [[q+1,q−1]]_q stabilizer codes of size q^2 on q+1 qudits, with identical Shor–Laflamme enumerators, and an additive weight-preserving bijection between them that extends to no monomial transformation. A simple invariant t (the number of idle qudits) proves the corresponding codespaces are inequivalent even under arbitrary product unitaries composed with permutations. For q=2 an exhaustive census shows length 3 is minimal and essentially unique (Theorem 4.1). Dropping the idle-qudit loophole, the paper gives explicit full-support pairs at lengths 4, 5, and 6 (Theorem 5.3), with minimal lengths claimed as 4 for a non-extendable isometry and 5 for a monomially inequivalent pair, together with an open LU–LC-type question.

Significance. The paper addresses a natural and previously open question at the right level of generality. The q-ary family is explicit, parameter-free, and satisfies a strong codespace-level obstruction; the enumerators match, so the failure is invisible to the Shor–Laflamme MacWilliams identities. The qubit minimality/census and the full-support examples are clean and reproducible: the authors supply scripts, direct enumerators, and independent structural checks (kernel-multiset criterion, vanishing-subcode invariant). If the computational layer in §5 is fully certified, this is a definitive small-scale resolution of the stabilizer version of the extension problem. My main reservation is that one lemma driving the exhaustive search is only sketched; this is localized and repairable.

major comments (2)
  1. [§5, Lemma 5.2] The first assertion of Lemma 5.2 — that every full-support, equal-length, weight-preserving difference δ=η+−η− for k=3 has the form δ(line F2v)=−Σ_{P∋v}ν(P), δ(P)=ν(P) with Σν(P)=0 — is the completeness basis for the exhaustive search described immediately afterward. The proof is only sketched as 'linear algebra on the indicator relations ... a short computation, which we record as a lemma since it drives the search.' This does not establish completeness, nor the integrality of the integer-lattice classification. Theorem 5.3(ii)'s negative statement at n=4 and the minimality of n=5 depend on this classification; if a generator were omitted, the search would miss realizable configurations. Please provide a full derivation of the relation lattice, or a formal/independently verified certificate of the enumeration, and state exactly which verification step certifies this.
  2. [Abstract; §5, Theorem 5.3(ii)] The abstract states unconditionally that the minimal full-support lengths are 4 for a non-extendable isometry and 5 for a non-monomially-equivalent pair. The first claim is unconditional, but the second is proved in Theorem 5.3(ii) only 'among counterexamples with k≤3'; the exhaustive search is parameterized by Lemma 5.2, which is specific to k=3. Nothing in the paper rules out a k≥4 full-support monomially inequivalent pair at n=4. Either prove the unconditional statement or qualify the abstract, the introduction, and any summary claims accordingly.
minor comments (3)
  1. [§5, Theorem 5.3(iii)] The phrase 'at length 6 all nontrivial stabilizer elements can have weight ≥4' is existential; consider rewording to 'there exists a pair at length 6 in which all nontrivial elements have weight ≥4' to avoid a momentary universal reading.
  2. [§1.4] The assertion that Dyshko's threshold-length pair coincides, up to monomial equivalence, with (C−,C+) is stated without proof. A short argument or a precise reference to the codeword structure would make the historical observation easier to verify.
  3. [§7] The verification section promises accompanying scripts but does not give version identifiers or a reproducibility statement. Since several negative search claims rely on these scripts, please include file names/checksums or a small certificate for the key enumerations, especially the Lemma 5.2 search.

Circularity Check

0 steps flagged

Independent explicit constructions; no circular reduction found. Full-support minimality depends on an unproved enumeration lemma, but that is a correctness risk, not circularity.

full rationale

The central derivation chain is self-contained and non-circular. Theorem 1.1 constructs explicit codes C+ and C- and an explicit isometry f, proves self-orthogonality by Lemma 3.1, weight preservation by direct counting (both sides have all nonzero weights equal to q), non-extension by the zero-coordinate count, and codespace inequivalence by the t-invariant of Lemma 3.2. No fitted parameter is renamed as a prediction, and no conclusion is defined in terms of its input. The exhaustive results in Theorems 4.1 and 5.3 are finite brute-force checks over explicitly bounded sets, e.g. 'we searched it exhaustively—over all kernel multisets, with joint injectivity and condition (1) imposed on both sides... for every n≤6'. The completeness direction of Lemma 5.2 is asserted with only a sketch ('a short computation, which we record as a lemma since it drives the search'), so the full-support minimal-length claims inherit a potential correctness gap: if that classification were incomplete, the search would miss cases. But this is an unverified computational premise, not a circular reduction—the conclusions are not true by construction. The author's self-citations [7,8] are background on cyclic-socle extension theory and annihilator weights; they are not load-bearing premises for Theorems 1.1–1.3. The paper also explicitly acknowledges prior constructions (Dyshko, Wood, Pllaha) rather than renaming them as new. Overall, no circularity is exhibited; the residual risk is correctness/completeness of the computational minimality layer, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

The paper's constructions are explicit and parameter-free: no numbers are fitted, no constants tuned, and the n≤6 searches are exhaustive enumerations of kernel multisets (no multiplicity cap) reduced by GL(3,2). There are no invented entities: the idle-qudit invariant t of Lemma 3.2 is a theorem-proven quantity (partial-trace argument), not a postulated mechanism. The axioms the claims rest on are the standard quantum-coding dictionary (CRSS/Ketkar correspondence; monomial = local-Clifford⋊pers from [14, Thm 5.8]; trace construction for prime powers; quantum MacWilliams identity), background module-theory results (Wood's cyclic-socle classification), and the in-paper Lemma 5.2 kernel-multiplicity classification, whose completeness is asserted with a compressed argument and carries the weight of the minimality census.

axioms (7)
  • domain assumption Stabilizer groups correspond bijectively (up to phases) to trace-symplectically self-orthogonal additive codes over A_q = F_{q^2} (CRSS correspondence; Ketkar et al. [16,17]).
    Used throughout §2.1 to translate the quantum extension question into a question about self-orthogonal additive codes; the whole paper operates inside this dictionary.
  • domain assumption Weight-preserving isomorphisms of stabilizer codes that extend to monomial maps of A_q^n are exactly those implemented by local Clifford operations and qudit permutations ([14, Thm 5.8], restated in §2.2).
    Bridges algebraic non-extension statements to the physical 'no local Clifford + permutation' claims; taken from prior work, not proved in this paper.
  • domain assumption For non-prime q, the generalized Pauli group uses the standard trace construction over F_q ([17]).
    Invoked in §2.1 so that Theorem 1.1 covers every prime power, not just primes.
  • standard math The Shor–Laflamme weight enumerator A determines the dual enumerator B via the quantum MacWilliams identity ([18]).
    Used in Theorem 3.3(iv) and Theorem 5.3 to move from equal A to equal B.
  • standard math Wood's classification: a finite module alphabet has the extension property iff it is pseudo-injective with cyclic socle ([3,4]).
    Background (§1.1, §6.1) motivating why F_q² as an F_q-module lacks the classical extension property; not used in the proofs of the new theorems.
  • standard math Lemma 5.2 completeness: for k=3, q=2, the δ parameterization exhausts all full-support equal-length weight-preserving isometry data.
    Proved in-paper by a compressed linear-algebra argument; completeness is the load-bearing part for the n≤6 exhaustive search and the minimality claims.
  • standard math Rains' theorem: automorphisms of linear stabilizer codes of distance > 2 are local Clifford ([19]).
    Cited in Remark 3.6 for contrast; not load-bearing for the theorems.

pith-pipeline@v1.3.0-alltime-deepseek · 13726 in / 39432 out tokens · 342822 ms · 2026-08-01T00:31:15.748130+00:00 · methodology

0 comments
read the original 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.

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Reference graph

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