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REVIEW 5 major objections 4 minor 29 references

Geometry of the symplectic group and optimal EAQECC codes

T0 review · 5 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper develops a symplectic-geometric additive-code formalism for entanglement-assisted quantum codes and uses it to construct Singleton-saturating families and resolve an open optimality question.

desk verdict The main EAQECC families are invalid: a weight enumerator in Theorem 3 is false, leaving only the formalism restatement and a possibly correct non-existence example. read the letter →

arxiv 2501.15465 v2 pith:CGJOHCPF submitted 2025-01-26 quant-ph cs.ITmath.IT

Ruihu Li , Yuezhen Ren , Chaofeng Guan , Yang Liu This is my paper · ORCID
classification quant-phcs.ITmath.IT MSC 81P7094B0594B65
keywords additivecodesquantumentanglement-assistedsymplecticgeometryoptimalEA-SingletonboundquaternaryEAstabilizer
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces an additive-code version of the entanglement-assisted stabilizer formalism, grounded in the geometry of the symplectic group, and uses it to construct families of binary entanglement-assisted quantum error-correcting codes (EAQECCs) that saturate the EA-Singleton bound. It also settles an open optimality question by proving that no [[5,2,4;3]] code exists, so the known [[5,2,3;3]] code is optimal. The framework converts any quaternary additive code with known trace radical into an EAQECC and its dual, giving a systematic path from classical code tables to quantum codes and explaining how some EAQMDS codes can have arbitrarily large length even for binary alphabets.

What carries the argument

The load-bearing object is the isometry φ((a|b)) = ωa + ϖb from the 2n-dimensional binary symplectic space to F4^n. It carries a subspace of symplectic type (m, c)[s] to an (n, 2^m) additive code of type (m, c)[t], and it carries the symplectic radical V ∩ $V^{{⊥s}}$ to the trace radical R(C) = C ∩ $C^{{⊥t}}$, whose dimension is m − 2c. Theorem 2 is the engine: it says such a C EA-stabilizes [[n, n − c − l, d; c]] with d = min{wt(g) : g ∈ $C^{{⊥t}}$ \ R(C)}, and that $C^{{⊥t}}$ EA-stabilizes the dual code. The constructions in Theorem 3 then choose explicit generator matrices whose weight enumerators are simple enough to read off d directly.

What would settle it

Compute the weight enumerator of the [5,3,3]_4 code generated by G_{3,n} with n=5. The row sum r_1+r_2+r_3=(1,1,ϖ,1,ϖ) has weight 5, whereas the asserted enumerator W(t)=1+$15z^{4}$+$48z^{3}$ contains no $z^{5}$ term; verifying this one vector would show the enumerator claim for that case fails.

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Extended reading notes

Core claim

The central claim is that the EA stabilizer formalism can be re-framed as a statement about additive codes over F4. Given an (n, 2^m) additive code C of type (m, c)[t] with trace radical R(C) = C ∩ $C^{{⊥t}}$ of dimension l, C EA-stabilizes an EAQECC with parameters [[n, n − c − l, d_ea; c]], where d_ea is the smallest symplectic weight in $C^{{⊥t}}$ \ R(C), and the trace dual $C^{{⊥t}}$ stabilizes the dual EAQECC with the roles of C and $C^{{⊥t}}$ exchanged. This reduces code construction to choosing additive codes with controlled trace radicals, and the paper uses it to produce the four families of Theorem 3 and to prove the non-existence of a [[5,2,4;3]] code, thereby showing the known [[5,2,3;3]] code is optimal.

Load-bearing premise

The distance claims for the new families rest on the asserted weight enumerators of the constructed additive codes; if one of those enumerators is incorrect for some n, the corresponding minimum distance is not established by this argument.

Editorial extensions

If this is right

  • Every quaternary additive code C with known trace radical yields an EAQECC and its dual with the parameters in Theorem 2, so existing tables of additive codes become construction engines for EAQECCs.
  • The families in Theorem 3 give binary EAQECCs with [[n,1,n−1;n−3]] and [[n,1,n−2;n−5]] parameters that saturate the EA-Singleton bound; the first family is EAQMDS.
  • The proof that no [[5,2,4;3]] code exists fixes the minimum distance of the n=5, k=2, c=3 case at d=3, resolving the open question raised in [20].
  • The claim that binary EAQMDS codes can have arbitrarily large n contradicts the previously expected n ≤ q^2 + d − 2 type constraint, so the examples bear on the general existence question for EAQMDS codes.
  • Because the formalism gives the EA-normalizer explicitly as the trace dual, it also supplies the data needed to design encoding and decoding circuits for the constructed codes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper, the weight-enumerator route suggests that any additive code whose weight distribution is concentrated on two values will automatically normalize a high-distance EAQECC, so the search for new families can focus on two-weight additive codes.
  • The non-existence argument for [[5,2,4;3]] uses the classification of MDS additive codes; the same strategy could decide optimality for other small EAQECC parameters once analogous classifications exist.
  • The paper shows n can be arbitrarily large for binary EAQMDS codes of distance growing with n; whether this extends to larger alphabets or to EAQMDS codes with more logical qubits is left open.
  • The formalism is stated for binary EAQECCs, but the same additive-code translation should transfer to non-binary alphabets by replacing F4 with other finite fields and the trace inner product with the appropriate trace form.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 4 minor

Summary. The paper develops an additive-code formulation of the entanglement-assisted stabilizer formalism via the symplectic geometry of F_2^{2n} and the trace inner product over F4, and uses it to construct four families of binary EAQECCs that are claimed to saturate the EA-Singleton bound. It also claims to settle the optimality of the [[5,2,3;3]] code by proving the nonexistence of [[5,2,4;3]], and to disprove a conjecture on the parameters of EAQMDS codes. The central constructions are proved by asserting weight enumerators of the relevant additive codes and then applying Theorem 2 to obtain the EA code parameters.

Significance. If correct, the additive-code reformulation would provide a convenient bridge between symplectic geometry and EAQECC parameters, and the constructed families, especially the [[n,1,n-2;n-5]] and [[n,1,n-2s;n-4s-1]] families, would be new parameter records saturating the EA-Singleton bound. The optimality proof for [[5,2,3;3]] would close an open problem listed in the literature. However, the paper's advertised disproof of the EAQMDS conjecture is not actually presented, and the main construction proofs rely on weight enumerators that are asserted without derivation, one of which is demonstrably false. As written, the central results are therefore not established.

major comments (5)
  1. [Section 4, Theorem 3 item 2] The weight enumerator W(t)=1+15z^4+48z^{n-2} claimed for Cn is false. For the three rows r1,r2,r3 of G3,n, the F4-linear combination r1+r2+r3 equals (1,1,ϖ,1,ϖ,1,...,1) with Hamming weight 5+2m=n. For n=5 this codeword has weight 5, but the asserted enumerator has no z^5 term; the actual enumerator of the [5,3,3] code includes a z^5 term. Consequently the minimum-distance computation that yields d_ea=n-2 is unsupported, and the claimed [[n,1,n-2;n-5]] family is not proved by the manuscript's argument.
  2. [Section 4, Theorem 3 items 3 and 4] The weight enumerators for the codes in items 3 and 4 are not computed; they are written as WR(t)=1+a_4z^4+...+a_{8s}z^{8s} (or with 8s+4) plus a single high-weight term, with the coefficients a_i left unspecified. The sentence 'It is easy to verify' provides no derivation, and in light of the false enumerator in item 2, these asserted enumerators cannot be taken on faith. The claimed minimum distances for these families are therefore not established.
  3. [Introduction (Problem (b)) and Conclusion] The paper advertises a disproof of the conjecture that EAQMDS codes satisfy a length constraint analogous to n ≤ q^2+d-2. The introduction states that the paper will show this conjecture is incorrect, and the conclusion repeats that a conjecture has been disproven, but no theorem or example in the body states the conjecture precisely or demonstrates a code that violates it. This is a promised central contribution that is missing from the manuscript.
  4. [Example 2] The nonexistence proof for [[5,2,4;3]] rests entirely on the cited classification in [27] that the only (5,2^4,4) additive MDS code has type (4,0)[t]. This type assertion is load-bearing for the optimality claim, but the paper does not reproduce the classification or provide the generator matrix of the unique code so the reader can verify the type under the paper's own definitions. As written, the proof is a black-box citation for a critical step.
  5. [Section 4, application of Theorem 2] The proofs never explicitly identify which code is the EA-stabilizer and which is the normalizer when applying Theorem 2. For example, in item 1 the code Cn=[n,2]4 has additive type (m,c)=(4,1); if Cn were the stabilizer, Theorem 2 would give k=n+c-m=n-3 and c=1, not the claimed [[n,1,n-1;n-3]]. The claimed parameters follow when the stabilizer is Cn⊥t, with Cn serving as the normalizer, but the text says only 'Cn normalizes' and does not define this switch. This ambiguity makes the proof difficult to verify and should be corrected.
minor comments (4)
  1. [Title] The title reads 'optimal EAQECC codes'; this is redundant since EAQECC already contains 'codes'.
  2. [Section 4, item 1] The statement 'G2,n generates a Cn=[n,2,4] linear code' should be qualified for n≥6, since for n=4 the same construction yields a [4,2,3] code, as stated earlier in the same paragraph.
  3. [Section 4, item 1 odd case] The text says G4,5 generates a (5,2^4,3) code and then considers n=5+2m>5; the reader is left to infer how the (5,2^4,3) base case relates to the weight enumerator stated for the larger n, and this connection should be made explicit.
  4. [Throughout] There are typographical inconsistencies in notation, such as the use of both ϖ and its ASCII rendering, and the generator matrices contain apparent formatting artifacts (e.g., the '11110' row in G2,4) that should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central construction is a direct application of the externally established EA stabilizer formalism to explicit generator matrices with asserted weight enumerators; self-citations are ancillary.

full rationale

The paper's derivation chain is not circular. Theorem 2 is explicitly presented as a restatement of the known EA stabilizer formalism: the text says its equivalent symplectic formalism can be found in [17] and 'We can restate Theorem 1 as Theorem 2', and Theorem 1 is attributed to [9,15], all external sources. The map from F2^{2n} to F4^n is taken from [4] (Calderbank–Rains–Shor–Sloane). The constructions in Theorem 3 proceed by explicit generator matrices and then assert weight enumerators; the minimum distances are read off from those enumerators via Theorem 2. No parameter is fitted to data, and no target parameter is built into the definition of the input code. The claim in Example 2 that no [[5,2,4;3]] exists rests on the external classification [27], not on a self-citation. The self-citations that do appear (e.g., refs [26],[28] for quaternary additive codes and the concluding sentence 'Based on [25–28], we have constructed over 60 optimal EAQECCs … which will be presented in [29]') are either published and independently checkable or are deferred future work; they do not carry the proof of Theorem 3. A potential concern raised by the reader—that the asserted weight enumerator W(t)=1+15z^4+48z^{n−2} in Theorem 3 item 2 may be inconsistent with an explicit codeword—would be a mathematical correctness defect, not a circularity: it would mean the claimed distance is unproved, not that the conclusion is equivalent to its own input. Similarly, the additivity of the formalism as a translation of known results raises novelty, not circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new free parameters or invented entities. It relies on standard symplectic geometry, the known GF(4) correspondence, the established EA stabilizer formalism, and an external uniqueness result for a small additive MDS code.

assumptions (3)
  • standard math Standard symplectic geometry classification of subspaces of F2^{2n} by type (m,c)
    Used in Section 2 to assign radical dimensions and dual types; taken from [24].
  • domain assumption Equivalence of EA stabilizer formalism and additive codes over GF(4)
    Theorem 2 restates Theorem 1 from [9] and [15] via the map phi from [4]; the paper relies on this equivalence.
  • domain assumption Uniqueness of the (5,2^4,4) additive MDS code and its type
    Used in Example 2 to rule out [[5,2,4;3]]; cited from [27] without proof in this paper.

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Cite this review

Pith. "Pith review of Geometry of the symplectic group and optimal EAQECC codes." pith.science (2026). https://pith.science/paper/CGJOHCPF

@misc{pith2026250115465,
  author       = {Pith},
  title        = {Pith review of: Geometry of the symplectic group and optimal EAQECC codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CGJOHCPF}},
  note         = {Machine review of arXiv:2501.15465}
}
read the original abstract

A new type of link between geometry of symplectic group and entanglement-assisted (EA) quantum error-correcting codes (EAQECCs) is presented. Relations of symplectic subspaces and quaternary additive codes concerning parameters of EAQECCs are described. Thus, parameters of EA stabilizer codes are revealed in the nomenclature of additive codes. Our techniques enable us solve some open problems about optimal EAQECCs and entanglement-assisted quantum minimum distance separable (EAQMDS) codes, and are also useful for designing encoding and decoding quantum circuit of EA stabilizer codes.

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Reference graph

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