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On the Automorphism Groups of Berman Codes and associated Abelian Codes

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Pith's one-line read Exact automorphism groups are identified for Berman codes, their duals, and associated abelian codes when n ≥ 5.

desk verdict 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. read the letter →

arxiv 2605.27312 v1 pith:QR56CVOA submitted 2026-05-26 cs.IT math.GRmath.IT

Harshvardhan Pandey , Prasad Krishnan This is my paper · ORCID
classification cs.ITmath.GRmath.IT
keywords automorphismgroupBermancodesabelianReed-Mullerbinarylinearalgebrascodesymmetriesdual
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 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.

What carries the argument

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).

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

0 major / 4 minor

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.

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.

minor comments (4)
  1. §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.
  2. 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.
  3. 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.
  4. §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.

Simulated Author's Rebuttal

0 responses · 0 unresolved

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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper's central claims concern explicit computation of automorphism groups for Berman codes (defined via standard ideals in F_2[G] or generator matrices) and their intersections/sums, for n≥5. These rest on algebraic group actions and code properties without any self-definitional reduction, fitted-parameter predictions, or load-bearing self-citations that collapse the result to its inputs. The abstract and parameter statements reference prior literature on Berman codes only for context, not as the sole justification for the automorphism results. The derivation chain is therefore self-contained against external algebraic benchmarks.

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

No free parameters, axioms, or invented entities are mentioned or required by the abstract description.

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Pith. "Pith review of On the Automorphism Groups of Berman Codes and associated Abelian Codes." pith.science (2026). https://pith.science/paper/QR56CVOA

@misc{pith2026260527312,
  author       = {Pith},
  title        = {Pith review of: On the Automorphism Groups of Berman Codes and associated Abelian Codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QR56CVOA}},
  note         = {Machine review of arXiv:2605.27312}
}
abstract

The automorphism group of a code is the group of permutations that map a code to itself. Berman codes are a class of binary linear codes characterized by two integer parameters $n\geq 2$ and $m\geq 1$, and this class includes the Reed-Muller codes as well. The class of Berman codes and their duals were recently shown to achieve the capacity of the binary erasure channel. A number of abelian codes that arise from the intersection and subspace sums of Berman and Dual Berman codes were also identified recently, for odd $n\geq 3$. A subclass of these abelian codes was shown to have good short block-length performance for AWGN channels, with efficient decoding algorithms. In this work, we identify the exact automorphism group for Berman codes and their duals. Further, we find the exact automorphism group for the above mentioned abelian codes, when $n\geq 5$. In the case of such abelian codes with $n=3$, we present partial characterizations of the automorphism groups for a large collection of parameter choices, and complete characterizations for a few.

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