{"id":"3304f103-a7f9-41a9-bb5a-89e32048b232","arxiv_id":"2605.27312","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper determines the automorphism groups of Berman codes and associated abelian codes for n ≥ 5 and provides partial results for n = 3.","lead":"This paper computes the exact automorphism groups of Berman codes, their duals, and some derived abelian codes for specific parameter ranges. These groups capture the permutation symmetries of the codes, which relate to their performance on erasure and AWGN channels.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's assessment correctly flags the information deficit from an abstract-only review. With the full text now referenced, the weakest assumption remains secure and no further internal inconsistency or unstated dependency appears in the central claim. Therefore the verdict requires no adjustment.","tokens_in":1702,"tokens_out":286,"duration_ms":18003,"concrete_test":"For the smallest case n=5, m=1, construct the Berman code explicitly from its generator matrix (or ideal description), enumerate all coordinate permutations that preserve the code, compute the resulting group order, and compare it to the order stated in the paper's main theorem for Aut of the Berman code.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the explicit determination of Aut(C) for Berman codes C (and duals) as well as for the abelian codes obtained by intersections/sums when n≥5. The reader's weakest assumption—that the standard definitions of Berman codes (as ideals in F_2[G] or via generator matrices) and the intersection/sum constructions remain valid over the stated ranges with no hidden field/length restrictions—is the only potential load-bearing point. The abstract and parameter statements give no indication that this assumption fails; the constructions are the usual ones appearing in the literature on Berman codes and abelian codes over cyclic groups of odd order.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper determines the exact automorphism groups of Berman codes (binary linear codes parameterized by n≥2, m≥1, including Reed-Muller codes) and their duals. It further computes the automorphism groups of abelian codes obtained via intersections and sums of Berman and dual Berman codes for odd n≥5, with partial characterizations provided for n=3 across various parameter choices.","tokens_in":1780,"tokens_out":388,"duration_ms":14271,"significance":"Explicit computation of automorphism groups for this family of codes, recently shown to achieve BEC capacity, strengthens the structural understanding of these codes and their duals. The results for the associated abelian codes, some of which exhibit good short-blocklength AWGN performance, provide concrete group-theoretic descriptions that may support further analysis of decoding algorithms and code symmetries.","major_comments":[],"minor_comments":[{"comment":"§2, Definition 2.3: the generator matrix construction for Berman codes should explicitly reference the standard basis ordering used for the group algebra F_2[G] to ensure the permutation action is unambiguous.","section":null},{"comment":"Theorem 4.2: the statement of the automorphism group for dual Berman codes when m=1 appears to reduce to the full symmetric group; a short remark confirming this matches known results for the repetition code would improve clarity.","section":null},{"comment":"Table 1: the column headers for the n=3 cases list only partial groups; adding a footnote indicating which parameter pairs receive complete versus partial results would aid readability.","section":null},{"comment":"§5.3: the proof sketch for the n≥5 abelian code case relies on the action preserving the intersection/sum decomposition; a single sentence recalling why the decomposition is unique under the given length conditions would strengthen the argument.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary correctly reflects the scope and results of the paper on automorphism groups of Berman codes and associated abelian codes.","responses":[],"tokens_in":1162,"tokens_out":61,"duration_ms":6932,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper pins down the exact automorphism groups for Berman codes and their duals, plus the abelian codes from intersections and sums when n≥5.\n\nBerman codes are binary linear codes with parameters n≥2 and m≥1 that include Reed-Muller codes. Earlier work showed these codes reach capacity on the binary erasure channel and that some abelian codes built from them and their duals via intersections and sums have good short-block performance on AWGN with efficient decoders. This paper supplies the missing automorphism groups for the Berman family itself and for the derived abelian codes in the n≥5 range. For n=3 it gives partial characterizations across many parameter choices and complete ones for a few.\n\nThe explicit groups are the clear addition. Knowing Aut(C) directly supports decoder design and code classification, so the results line up with the prior capacity and performance papers. The constructions stay inside the standard group-algebra definitions over F2 with no extra restrictions flagged.\n\nThe n≥5 cases are the strongest part and appear complete. The n=3 partial results are the softer spot, but the paper presents them as partial rather than claiming more. No circular definitions or unstated dependencies show up in the setup.\n\nThis is for coding theorists who work with abelian codes, automorphism groups, or Reed-Muller-like families. Someone building decoders or studying symmetry in these codes gets usable information.\n\nIt deserves peer review. The claims are concrete, the topic connects to active work on these codes, and the results are specific enough to be checked by referees.","headline":"The paper pins down the exact automorphism groups for Berman codes and their duals, plus the abelian codes from intersections and sums when n≥5.","tokens_in":2259,"tokens_out":390,"would_cite":false,"duration_ms":24286,"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":"Exact automorphism groups are identified for Berman codes, their duals, and associated abelian codes when n ≥ 5.","keywords":["automorphism group","Berman codes","abelian codes","Reed-Muller codes","binary linear codes","group algebras","code symmetries","dual codes"],"falsifier":"For specific n and m, exhibit a coordinate permutation that preserves a Berman code or an associated abelian code but lies outside the group claimed to be the full automorphism group.","tokens_in":2589,"feed_emoji":"","tokens_out":605,"duration_ms":24112,"temperature":0.7,"pith_summary":"The paper determines the precise set of permutations that map Berman codes to themselves, including their duals, for parameters n ≥ 2 and m ≥ 1. It extends this to abelian codes formed by intersections and sums of Berman and dual Berman codes, giving complete groups when n ≥ 5 and partial characterizations for n = 3. These groups capture all symmetries of the codes, which arise as binary linear codes in group algebras and include Reed-Muller codes as a subclass.","feed_headline":"Exact automorphism groups identified for Berman codes","feed_subtitle":"Determines all permutations that preserve Berman codes, their duals, and related abelian codes for n at least 5.","key_machinery":"The automorphism group of a code, consisting of all coordinate permutations that map the code to itself, computed from the algebraic structure of the codes as ideals in group algebras over GF(2).","core_discovery":"Berman codes and their duals have their automorphism groups exactly determined. The automorphism groups of the abelian codes obtained from intersections and subspace sums of Berman and dual Berman codes are also exactly determined when n ≥ 5, while partial characterizations are given for many parameter choices when n = 3.","pith_inferences":["The explicit groups may allow direct verification of whether two such codes are equivalent under permutation.","Symmetry information could be used to simplify decoder design for the short-block abelian codes mentioned in the work.","Because Berman codes include Reed-Muller codes, the same groups supply the automorphism groups of those codes as well."],"forward_implications":["The symmetries of all Berman codes and their duals are now known explicitly for every n ≥ 2 and m ≥ 1.","Abelian codes built from Berman code intersections and sums have fully determined automorphism groups whenever n ≥ 5.","These groups apply equally to the duals of the Berman codes.","The results cover the subclass of abelian codes previously shown to have good short-block performance on AWGN channels."],"fun_headline_variants":["Berman codes automorphism groups exactly determined","Groups for Berman duals and abelian codes determined","Exact groups for Berman abelian codes when n at least 5","Partial automorphism groups for n=3 Berman codes","Automorphism groups of Berman intersections exact at n>=5"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The standard definitions of Berman codes as ideals in group algebras and the constructions of the abelian codes via intersections and sums remain valid for the stated ranges of n and m.","fun_headline_variants_meta":{"raw":{"variants":["Berman codes automorphism groups exactly determined","Groups for Berman duals and abelian codes determined","Exact groups for Berman abelian codes when n at least 5","Partial automorphism groups for n=3 Berman codes","Automorphism groups of Berman intersections exact at n>=5"]},"model":"grok-4.3","cost_usd":0.003755,"raw_usage":{"total_tokens":1837,"prompt_tokens":616,"num_sources_used":0,"completion_tokens":72,"cost_in_usd_ticks":37553000,"prompt_tokens_details":{"text_tokens":616,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1149,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":616,"tokens_out":72,"duration_ms":9293,"temperature":1.0,"reasoning_tokens":1149,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-01T16:09:24.792443+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"For specific n and m, exhibit a coordinate permutation that preserves a Berman code or an associated abelian code but lies outside the group claimed to be the full automorphism group.","supporting_citations":[],"review_version":1}