论文简报
cs.IT 2605.27312v1 值得读

On the Automorphism Groups of Berman Codes and associated Abelian Codes

Harshvardhan Pandey, Prasad Krishnan

发布日期：2026-05-26 17:23 相关性：1.0000 价值：0.7400 分类：cs.IT math.GR
arXiv 摘要 PDF 原文 网页正文

摘要

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.

相关性判断

high
相关方向
coding_theory information_theory communications group_theory
判断依据

Directly about automorphism groups of Berman codes, duals, and related abelian codes; this is core coding theory with explicit connections to capacity and decoding over communication channels.

价值判断

High relevance to cs.IT coding theory with exact automorphism classifications for Berman, dual Berman, and associated abelian code families. Structure analysis indicates substantial technical content, including graph automorphism reductions, wreath products, affine groups, and symmetric-group subgroup arguments. The result is specialized but useful because these code families are connected to capacity-achieving behavior and short-blocklength decoding performance.

核心问题与主要方法

核心问题

Determine the exact permutation automorphism groups of Berman codes, their duals, and related abelian codes

场景：Binary linear codes and DFT-defined abelian codes on length $n^m$ arrays, with odd $n$ for the abelian family and special handling of the $n=3$ regime

主要方法

For Berman and dual Berman codes, minimum-weight codeword supports are characterized as axis-aligned hypercubes in [n]^m; intersections of these hypercubes force any code automorphism to preserve lines. Line preservation turns a code automorphism into an automorphism of the Hamming graph on [n]^m, whose automorphism group is the product-action wreath product S_n wr S_m. For C(n,m,W) with odd n >= 5, the proof combines the known wreath-product subgroup with maximal-subgroup facts for S_{n^m} and parity, leaving only S_n wr S_m or the full symmetric group. Boundary weight sets correspond to trivial, repetition, single-parity-check, or full-space binary codes, explaining the full symmetric-group cases. For n = 3, affine containment is derived via maximal-subgroup classification; singleton-weight refinements use DFT-domain weight preservation and quadratic-form constraints over F_3.

关键贡献与后续阅读

关键贡献

Exact automorphism group identification for non-Reed-Muller Berman codes and their duals: S_n wr S_m for 1 <= r <= m-1 and full S_{n^m} in boundary orders. Exact classification for the associated DFT-defined abelian codes C(n,m,W) when odd n >= 5: wreath product for non-boundary W and full symmetric group for boundary W. A reduction strategy for Berman-code automorphisms through minimum-weight supports, axis-aligned hypercubes, line preservation, and Hamming-graph automorphisms. A maximal-subgroup-based classification route for abelian-code automorphisms, using the placement of S_n wr S_m inside S_{n^m}. Partial n = 3 theory showing non-boundary automorphism groups lie inside AGL(m,3), with sharper AO(m,3) or A~O(m,3) bounds and exact cases for singleton weight sets.

研究启发

For the n = 3 non-singleton non-boundary cases, how large is the gap between the AGL(m,3) upper bound and the actual automorphism group? Does the graph-automorphism reduction proposed in the conclusion yield practical generator computation for moderate m, and is code or benchmarking available beyond the cited m = 3 verification? Can these exact automorphism classifications be turned into concrete automorphism-ensemble or permutation decoding gains for the AWGN-performing BiD subclasses?

限制与不确定性

Impact may be concentrated in a narrow algebraic coding theory niche rather than broad communications practice. The n=3 abelian-code case is only partially characterized, which limits completeness. Assessment relies on relevance and structure metadata only, not a full paper review.

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