{"id":"69ab4f85-f063-4ccf-9cb8-77eb3ad825a9","arxiv_id":"2606.22069","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Derives generator polynomials for hulls and sums of separable constacyclic codes over a product ring and constructs new quantum codes from them.","lead":"The paper derives generator polynomials for the Euclidean and Hermitian duals, hulls, and sums of separable constacyclic codes over the ring S = F_q × (F_q + v F_q) along with their Gray images. It then uses these to construct new quantum error-correcting codes claimed to have better parameters than prior constructions.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"Gray images of hulls/sums must satisfy CSS/Hermitian self-orthogonality for the claimed QECC constructions to hold","rationale":"The reader's weakest assumption is precisely the load-bearing step; the algebraic derivations of generator polynomials are standard in constacyclic-code literature and the novelty claim rests on the final quantum parameters being both correct and superior. No other internal inconsistency is visible from the abstract and claim structure.","tokens_in":1698,"tokens_out":386,"duration_ms":17960,"concrete_test":"Pick the smallest explicit example in the paper (smallest q odd prime power, smallest n, explicit α, β), compute the hull generator polynomial over S by the stated formula, apply the Gray map to obtain the image code over F_q, then directly compute its Euclidean (or Hermitian) dual and check whether the image lies inside its dual; if the inclusion fails or the minimum distance differs from the claimed [[N,K,D]] by more than 1, the quantum-code claim does not hold for that instance.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper derives generator polynomials for Euclidean/Hermitian duals, hulls and sums of separable constacyclic codes over S = F_q × (F_q + v F_q) (v² = v) and their Gray images. The two methods for producing QECCs then apply standard CSS or Hermitian constructions to these Gray images. This step implicitly requires that the Gray map (typically a coordinate-wise map from the ring to F_q²) preserves the relevant inner-product relations so that the image code C satisfies C ⊆ C^⊥ (or the Hermitian analogue) with the stated dimension and distance. If the map fails to preserve orthogonality for the specific ring elements or constacyclic multipliers, the resulting objects are not quantum codes or do not achieve the listed parameters.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript derives generator polynomials for the Euclidean and Hermitian duals of separable constacyclic codes over the ring S = F_q × (F_q + v F_q) (v² = v, q odd prime power), along with the generator polynomials of their Gray images. It then obtains the generator polynomials for the Euclidean and Hermitian hulls and sums of these codes and their Gray images. Finally, it proposes two methods to construct quantum error-correcting codes from the hulls and sums, claiming to produce new QECCs with improved parameters over existing constructions.","tokens_in":1873,"tokens_out":540,"duration_ms":14802,"significance":"If the derivations are correct and the Gray images of the hulls/sums satisfy the self-orthogonality conditions needed for the CSS and Hermitian quantum constructions, the work supplies explicit algebraic constructions for new families of quantum codes. Explicit generator polynomials and the focus on separable constacyclic codes over this ring extension are standard tools that can yield reproducible families; credit is due for attempting to produce parameter improvements via hulls and sums.","major_comments":[{"comment":"The quantum-code claims rest on the Gray images satisfying C ⊆ C^⊥ (or the Hermitian analogue) with the stated dimension and distance. The manuscript must explicitly verify that the coordinate-wise Gray map from S to F_q² preserves the relevant Euclidean/Hermitian inner-product relations for the constacyclic multipliers appearing in the generator polynomials; without this step the CSS/Hermitian constructions do not apply and the listed parameter improvements cannot be claimed (see the construction of the two methods in the final section).","section":"final section on quantum constructions"},{"comment":"The assertion that the new QECCs 'outperform the existing ones in terms of parameters' requires concrete tables or explicit comparisons against known bounds or tables of best-known quantum codes; the abstract supplies no such verification, and any such tables must be checked for correctness of the distance and dimension calculations.","section":"final section on quantum constructions"}],"minor_comments":[{"comment":"All generator polynomials should be accompanied by short derivation sketches or references to the standard results on constacyclic codes over product rings that are being extended.","section":null},{"comment":"Notation for the ring S and the Gray map should be introduced once and used consistently; ensure that the definition of the Gray map appears before its first use in the dual/hull statements.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below.","responses":[{"response":"We agree that an explicit verification is required. The manuscript derives the relevant generator polynomials but does not contain a dedicated proof that the Gray map preserves the Euclidean and Hermitian inner products for the constacyclic multipliers in question. In the revised manuscript we will insert a lemma establishing this preservation property, thereby justifying the application of the CSS and Hermitian constructions to the Gray images of the hulls and sums.","revision_made":"yes","referee_comment":"[final section on quantum constructions] The quantum-code claims rest on the Gray images satisfying C ⊆ C^⊥ (or the Hermitian analogue) with the stated dimension and distance. The manuscript must explicitly verify that the coordinate-wise Gray map from S to F_q² preserves the relevant Euclidean/Hermitian inner-product relations for the constacyclic multipliers appearing in the generator polynomials; without this step the CSS/Hermitian constructions do not apply and the listed parameter improvements cannot be claimed (see the construction of the two methods in the final section)."},{"response":"We accept that concrete parameter comparisons are necessary to substantiate the improvement claim. The revised manuscript will include tables that list the parameters of the new quantum codes, compare them with existing constructions from the literature, and reference known quantum-code bounds; all distance and dimension calculations will be double-checked before inclusion.","revision_made":"yes","referee_comment":"[final section on quantum constructions] The assertion that the new QECCs 'outperform the existing ones in terms of parameters' requires concrete tables or explicit comparisons against known bounds or tables of best-known quantum codes; the abstract supplies no such verification, and any such tables must be checked for correctness of the distance and dimension calculations."}],"tokens_in":1381,"tokens_out":402,"duration_ms":14296,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main contribution is the derivation of generator polynomials for the Euclidean and Hermitian duals of separable constacyclic codes over S = F_q × (F_q + v F_q), followed by the same for their hulls and sums, and then for the Gray images of all of those. The authors close by applying the usual CSS and Hermitian constructions to the images to produce quantum codes they say beat known tables.\n\nThey handle the algebraic bookkeeping cleanly. Working out the explicit generators for the hulls and sums over this specific ring extends the existing literature on constacyclic codes in a straightforward way, and recording the Gray images of those objects gives a usable description. That part of the work is concrete and reproducible in principle.\n\nThe soft spot is the transition to quantum codes. The constructions require that the Gray images of the hulls and sums remain self-orthogonal (or Hermitian self-orthogonal) with the claimed distances. The abstract asserts that the images satisfy the needed conditions and yield better parameters, but supplies no examples or verification that the map preserves the relevant inner-product relations for these constacyclic multipliers. If that step does not hold exactly as stated, the listed quantum codes do not exist or do not improve on the tables.\n\nThis is for people who build quantum codes from classical constacyclic families and want explicit generators or updated parameter lists. A reader already working in that corner can extract the polynomials and check the Gray-map step themselves. The paper deserves a serious referee because the algebraic claims are specific enough to be checked and the topic is active, even though the quantum-code improvements need direct confirmation.","headline":"The paper gives explicit generator polynomials for the duals, hulls, and sums of separable constacyclic codes over this product ring together with their Gray images, then uses those to claim new quantum codes with improved parameters.","tokens_in":2365,"tokens_out":414,"would_cite":false,"duration_ms":16349,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Hulls and sums of separable constacyclic codes over the ring S = F_q × (F_q + v F_q) produce new quantum error-correcting codes with improved parameters.","keywords":["constacyclic codes","hulls","sums","Gray images","quantum error-correcting codes","separable codes","product rings"],"falsifier":"A concrete separable constacyclic code over S whose Gray-image hull or sum produces a quantum code whose minimum distance or dimension falls short of the values listed in the paper's tables or violates a known quantum bound.","tokens_in":2583,"feed_emoji":"","tokens_out":608,"duration_ms":12534,"temperature":0.7,"pith_summary":"The paper derives explicit generator polynomials for the Euclidean and Hermitian duals, hulls, and sums of separable constacyclic codes over the ring S, together with the generator polynomials of their Gray images. It then presents two construction methods that turn these hulls and sums into quantum error-correcting codes. The resulting codes are stated to have better parameters than previously known constructions. A reader would care because improved quantum codes directly affect the distance and dimension achievable in quantum error correction.","feed_headline":"Hulls of separable constacyclic codes give new quantum codes","feed_subtitle":"Explicit polynomials for hulls and sums over F_q × (F_q + v F_q) produce QECCs with better parameters than prior tables.","key_machinery":"The generator polynomials of the Euclidean and Hermitian hulls and sums (and their Gray images), which determine the quantum code parameters via the two proposed construction methods.","core_discovery":"The generator polynomials of the Euclidean and Hermitian hulls and sums of separable constacyclic codes over S and of their Gray images are given explicitly; two methods are proposed that convert these objects into quantum error-correcting codes whose parameters exceed those of existing tables.","pith_inferences":["The same hull-and-sum approach could be tested on non-separable constacyclic codes over the same ring to see whether the improvement persists.","If the Gray-image step preserves the hull structure for other rings of the form F_q × R, the methods might extend beyond the specific ring S studied here."],"forward_implications":["The two methods convert any separable constacyclic code over S whose hull or sum meets the required conditions into a quantum code.","The explicit generator polynomials allow direct computation of the parameters of the resulting quantum codes.","New quantum codes are obtained whose parameters improve on those recorded in prior tables for the same length and dimension."],"fun_headline_variants":["Hulls and sums of separable constacyclic codes give quantum codes","Constacyclic separable hulls and sums over Fq x Fq+vFq","Hulls sums of separable constacyclic codes and Gray images","Euclidean Hermitian hulls and sums for constacyclic codes","Generator polynomials of separable constacyclic code hulls"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The Gray images of the hulls and sums satisfy the length and dimension conditions needed for the CSS or Hermitian quantum constructions, and the claimed parameter improvements are accurate against existing bounds.","fun_headline_variants_meta":{"raw":{"variants":["Hulls and sums of separable constacyclic codes give quantum codes","Constacyclic separable hulls and sums over Fq x Fq+vFq","Hulls sums of separable constacyclic codes and Gray images","Euclidean Hermitian hulls and sums for constacyclic codes","Generator polynomials of separable constacyclic code hulls"]},"model":"grok-4.3","cost_usd":0.008975,"raw_usage":{"total_tokens":3921,"prompt_tokens":608,"num_sources_used":0,"completion_tokens":87,"cost_in_usd_ticks":89753000,"prompt_tokens_details":{"text_tokens":608,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3226,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":608,"tokens_out":87,"duration_ms":22946,"temperature":1.0,"reasoning_tokens":3226,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T11:11:29.948825+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete separable constacyclic code over S whose Gray-image hull or sum produces a quantum code whose minimum distance or dimension falls short of the values listed in the paper's tables or violates a known quantum bound.","supporting_citations":[],"review_version":1}