{"id":"dfbec5b6-71a8-4ece-991b-a1e76acf9eda","arxiv_id":"2605.24519","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An ILP method yields new triorthogonal codes with prescribed dual distance not necessarily from triply-even classical codes, and a GRAND-based decoder performs well on high-distance instances over dephasing.","lead":"The paper derives an integer linear programming formulation combining triorthogonality constraints with MacWilliams identities to construct even-weight triorthogonal quantum CSS codes with a target dual minimum distance, and evaluates decoding performance of high-distance codes from the doubling construction over the dephasing channel. Smart readers might care because these codes enable transversal T gates for magic-state distillation, a bottleneck step toward universal fault","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Krawtchouk-MacWilliams conditions on dual weight distribution may not be sufficient to guarantee ILP solutions yield valid triorthogonal matrices","rationale":"The reader's weakest assumption is precisely the sufficiency step that converts ILP feasibility into guaranteed existence of a valid matrix. Because the modeling gap between enumerator constraints and explicit matrix realizability is internal to the argument and directly affects the central existence criterion, it is the single most load-bearing point. A direct reconstruction-and-verification check on reported solutions would settle whether the gap is real.","tokens_in":1747,"tokens_out":389,"duration_ms":37475,"concrete_test":"Solve the ILP for one of the parameter sets reported in the paper; for each feasible solution, attempt to recover or construct the corresponding generator matrix (or its row space), then compute the true dual minimum distance by exhaustive search or software (e.g., Magma) on all vectors of weight < target d; if any feasible ILP point yields a matrix whose dual distance falls short, the sufficiency claim fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The existence criterion formulates an ILP whose feasible solutions are claimed to correspond to even-weight triorthogonal generator matrices with prescribed dual minimum distance. Triorthogonality supplies linear constraints on row overlaps. MacWilliams identities supply additional linear equations (via Krawtchouk polynomials) that the weight enumerator must obey once the dual weights below the target distance are set to zero. These together are asserted to be sufficient. However, the weight-enumerator variables are only loosely coupled to the explicit matrix entries; nothing in the listed constraints enforces that a given integer solution for the enumerator is realizable by some matrix obeying the triorthogonality equations. Consequently an ILP-feasible point may satisfy all stated linear relations yet fail to correspond to any actual triorthogonal matrix whose dual has the claimed distance.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper derives an existence criterion for even-weight triorthogonal generator matrices with a target dual minimum distance by combining triorthogonality constraints with MacWilliams identities expressed via Krawtchouk-polynomial conditions on the dual weight distribution; this yields an integer linear programming (ILP) formulation. The work reports discovery of new nontrivial triorthogonal codes not necessarily arising from classical triply-even codes and evaluates decoding performance (bounded-distance, BP+OSD, and GRAND) of high-distance codes from the doubling construction over the dephasing channel.","tokens_in":1921,"tokens_out":552,"duration_ms":22619,"significance":"If the ILP solutions are guaranteed to produce valid matrices, the criterion supplies a systematic, non-ad-hoc method for constructing triorthogonal CSS codes with controlled dual distance, extending beyond the classical triply-even route and thereby supporting magic-state distillation. The decoding comparisons supply concrete performance data for these codes. The combination of linear-programming techniques with quantum coding constraints is a clear methodological contribution.","major_comments":[{"comment":"Existence criterion section: the manuscript asserts that the triorthogonality linear constraints together with the Krawtchouk-polynomial MacWilliams conditions are jointly sufficient for every feasible ILP solution to correspond to a realizable even-weight triorthogonal matrix whose dual has the prescribed minimum distance. The weight-enumerator variables, however, remain only loosely coupled to the explicit row-overlap and row-weight variables; no argument is supplied showing that an integer solution for the enumerator is always realizable by a matrix satisfying the overlap equations. This sufficiency gap is load-bearing for the central claim.","section":"existence criterion"},{"comment":"Abstract and construction results: the claim of having found 'new nontrivial triorthogonal codes' is stated without exhibiting any explicit generator matrices, their dimensions, or verification that the obtained ILP solutions satisfy all row-weight and overlap conditions simultaneously. Concrete examples would directly test whether the ILP produces valid codes and would allow readers to assess the practical reach of the criterion.","section":"abstract"}],"minor_comments":[{"comment":"The abstract lists three decoders but does not indicate which channel model parameters or code distances were used in the numerical comparisons; adding a brief statement of the simulation settings would improve readability.","section":"abstract"},{"comment":"Notation for the dual weight distribution and the Krawtchouk polynomials could be introduced once in a dedicated preliminary subsection rather than inline, to aid readers unfamiliar with the MacWilliams route.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful and constructive review. The comments identify important points regarding the rigor of the existence criterion and the presentation of concrete results. We address each major comment below and will revise the manuscript to strengthen these aspects.","responses":[{"response":"We thank the referee for identifying this gap. The ILP formulation incorporates the triorthogonality constraints on row weights and pairwise/triple overlaps together with the MacWilliams identities expressed via Krawtchouk polynomials on the dual weight distribution. While feasible integer solutions are expected to yield valid matrices, the manuscript does not supply an explicit argument proving that every solution to the enumerator variables is realizable by a matrix satisfying the overlap equations. In the revised version we will add a dedicated subsection establishing this sufficiency, either by a direct realizability proof or by exhibiting explicit matrix constructions for all ILP solutions reported. This will make the central claim fully rigorous.","revision_made":"yes","referee_comment":"[existence criterion] Existence criterion section: the manuscript asserts that the triorthogonality linear constraints together with the Krawtchouk-polynomial MacWilliams conditions are jointly sufficient for every feasible ILP solution to correspond to a realizable even-weight triorthogonal matrix whose dual has the prescribed minimum distance. The weight-enumerator variables, however, remain only loosely coupled to the explicit row-overlap and row-weight variables; no argument is supplied showing that an integer solution for the enumerator is always realizable by a matrix satisfying the overlap equations. This sufficiency gap is load-bearing for the central claim."},{"response":"We agree that explicit examples are necessary to substantiate the claim of new nontrivial triorthogonal codes. Although the abstract and results section report the discovery of such codes via the ILP, the submitted manuscript does not display the generator matrices, their dimensions, or direct verification of the overlap conditions. In the revision we will include concrete generator matrices for the new codes, together with their parameters and explicit checks confirming that all triorthogonality and dual-distance conditions are satisfied. These examples will allow readers to verify the ILP outputs and evaluate the method's reach.","revision_made":"yes","referee_comment":"[abstract] Abstract and construction results: the claim of having found 'new nontrivial triorthogonal codes' is stated without exhibiting any explicit generator matrices, their dimensions, or verification that the obtained ILP solutions satisfy all row-weight and overlap conditions simultaneously. Concrete examples would directly test whether the ILP produces valid codes and would allow readers to assess the practical reach of the criterion."}],"tokens_in":1430,"tokens_out":545,"duration_ms":32268,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is a fresh ILP that adds MacWilliams/Krawtchouk conditions on the dual weight distribution to the usual triorthogonality overlap rules, then uses it to search for even-weight generator matrices with a target dual distance. They also compare three decoders on the dephasing channel for codes built by doubling.\n\nThe construction side is the real addition. Standard triorthogonal matrices come from triply-even classical codes, but this formulation finds others. The decoder numbers are concrete and directly usable for anyone implementing these codes. Bounded-distance, BP+OSD, and the adapted GRAND all get reported, which is the kind of practical data that helps when choosing an implementation.\n\nThe soft spot is exactly the one the stress test flags. The ILP variables for the weight enumerator are only indirectly tied to the row vectors and their pairwise/triple overlaps. Nothing in the listed constraints forces every feasible enumerator to be realizable by an actual matrix that meets the triorthogonality equations. The authors state they obtained new nontrivial codes, so at least some solutions are valid, but without a small worked example showing the matrix, the enumerator, and all overlap checks side by side, it is hard to judge how tight the formulation really is.\n\nThis paper is for people working on magic-state distillation and CSS code constructions. It deserves a serious referee because the ILP angle is new, the claimed codes are new, and the decoder results are reproducible enough to check. Minor revisions on the verification examples would make it stronger, but the core work is worth the time.","headline":"New ILP for triorthogonal codes works in practice for the examples they found, but the loose link between weight enumerator and actual matrix rows needs explicit checks.","tokens_in":2407,"tokens_out":395,"would_cite":false,"duration_ms":20369,"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":"An integer linear program constructs even-weight triorthogonal matrices whose duals meet a target minimum distance by enforcing both overlap rules and Krawtchouk-polynomial conditions from the MacWilliams identities.","keywords":["triorthogonal codes","quantum CSS codes","magic-state distillation","integer linear programming","MacWilliams identities","Krawtchouk polynomials","dephasing channel","quantum decoding"],"falsifier":"An explicit matrix produced by the integer program that satisfies all listed linear constraints yet whose dual code has minimum distance strictly smaller than the target value, or that fails to be triorthogonal.","tokens_in":2666,"feed_emoji":"","tokens_out":741,"duration_ms":20947,"temperature":0.7,"pith_summary":"The paper establishes an existence criterion that turns the search for even-weight triorthogonal generator matrices into an integer linear programming problem. The formulation joins the linear constraints that enforce pairwise and triple-wise row overlaps with additional linear conditions on the dual weight distribution, derived from MacWilliams identities via Krawtchouk polynomials. If a feasible solution exists, the resulting matrix is guaranteed to be triorthogonal and to produce a dual code whose minimum distance meets the prescribed target. The same framework yields previously unknown triorthogonal codes that need not arise from classical triply-even codes. The work also evaluates decoding of high-distance codes obtained by the doubling construction over the dephasing channel, comparing bounded-distance, belief-propagation, and GRAND-based decoders.","feed_headline":"Integer program yields triorthogonal matrices with target dual distance","feed_subtitle":"Krawtchouk conditions plus overlap constraints produce new even-weight codes and enable decoder comparisons over dephasing noise","key_machinery":"Integer linear programming formulation that augments the triorthogonality linear constraints with Krawtchouk-polynomial conditions on the dual weight distribution obtained from the MacWilliams identities.","core_discovery":"Triorthogonality constraints together with Krawtchouk-polynomial conditions on the dual weight distribution, when solved as an integer linear program, produce even-weight triorthogonal generator matrices whose duals achieve any chosen minimum distance; the method yields new nontrivial codes and enables concrete decoding comparisons for doubled high-distance instances over the dephasing channel.","pith_inferences":["The same linear-programming approach could be tested on other CSS-code families that require simultaneous linear constraints and weight-distribution targets.","If the GRAND decoder remains strong, it could be combined with the new triorthogonal matrices to improve magic-state distillation thresholds.","The existence criterion might be relaxed or tightened by replacing the Krawtchouk conditions with direct minimum-distance constraints when the block length remains modest."],"forward_implications":["New triorthogonal codes exist that are not generated by classical triply-even codes.","The doubling construction produces high-distance triorthogonal codes whose decoding performance over the dephasing channel can be compared across bounded-distance, belief-propagation, and GRAND decoders.","The GRAND-based decoder adapted to the quantum setting emerges as a competitive option for these codes.","The integer-linear-programming existence criterion can be applied to search for even-weight triorthogonal matrices at larger block lengths."],"fun_headline_variants":["ILP builds triorthogonal codes targeting dual distance","New even-weight triorthogonal codes via Krawtchouk ILP","Triorthogonal codes from dual distance integer programming","Decoder eval for high-distance triorthogonal over dephasing","Krawtchouk conditions yield new triorthogonal matrices in ILP"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The Krawtchouk-polynomial conditions derived from the MacWilliams identities, when added to the triorthogonality constraints, are enough to ensure that every feasible integer solution corresponds to a valid triorthogonal matrix whose dual has the target minimum distance.","fun_headline_variants_meta":{"raw":{"variants":["ILP builds triorthogonal codes targeting dual distance","New even-weight triorthogonal codes via Krawtchouk ILP","Triorthogonal codes from dual distance integer programming","Decoder eval for high-distance triorthogonal over dephasing","Krawtchouk conditions yield new triorthogonal matrices in ILP"]},"model":"grok-4.3","cost_usd":0.004469,"raw_usage":{"total_tokens":2221,"prompt_tokens":652,"num_sources_used":0,"completion_tokens":80,"cost_in_usd_ticks":44687000,"prompt_tokens_details":{"text_tokens":652,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1489,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":652,"tokens_out":80,"duration_ms":15131,"temperature":1.0,"reasoning_tokens":1489,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T13:13:01.303982+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit matrix produced by the integer program that satisfies all listed linear constraints yet whose dual code has minimum distance strictly smaller than the target value, or that fails to be triorthogonal.","supporting_citations":[],"review_version":1}