A Note on Linear Complementary Pairs of Group Codes

Javier de la Cruz; Martino Borello; Wolfgang Willems

arxiv: 1907.07506 · v1 · pith:KLBZLWILnew · submitted 2019-07-17 · 💻 cs.IT · math.CO· math.IT

A Note on Linear Complementary Pairs of Group Codes

Martino Borello , Javier de la Cruz , Wolfgang Willems This is my paper

Pith reviewed 2026-05-24 20:05 UTC · model grok-4.3

classification 💻 cs.IT math.COmath.IT

keywords linear complementary pairsgroup codestwo-sided idealsgroup algebraspermutation equivalencedual codescyclic codes

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The pith

For linear complementary pairs of two-sided ideals in a group algebra, one code uniquely determines its complement and the dual is permutation equivalent to it.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that when a linear complementary pair consists of two-sided ideals in a group algebra, the complement to one code is unique. It also proves that the dual of the second code is permutation equivalent to the first. The argument relies only on the algebraic structure of the ideals and avoids lengthy calculations. This recovers earlier findings on multidimensional cyclic codes as direct consequences. A reader would care because the result makes it easier to identify and work with such pairs in algebraic coding constructions.

Core claim

If (C, D) is a linear complementary pair where both C and D are two-sided ideals in a group algebra, then D is the unique such complement to C, and furthermore the dual code of D is permutation equivalent to C.

What carries the argument

The two-sided ideal structure in the group algebra, which supplies the relations that force uniqueness of the complement and the permutation equivalence of the dual.

If this is right

Results previously shown for nD cyclic codes follow immediately as special cases.
Verification of the uniqueness and equivalence properties requires only basic group-algebra facts rather than extended polynomial manipulations.
The same conclusions apply to any finite group algebra that admits two-sided ideals forming complementary pairs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same ideal-structure argument might simplify analysis of complementary pairs in other algebraic code families that possess a suitable ring or module structure.
Classification of all such pairs could become feasible by enumerating ideals rather than searching arbitrary linear subspaces.

Load-bearing premise

The codes must be two-sided ideals in the group algebra and must form a linear complementary pair.

What would settle it

An explicit pair of two-sided ideals in some group algebra that form a linear complementary pair but where the complement is not unique or the dual fails to be permutation equivalent to the original code.

read the original abstract

We give a short and elementary proof of the fact that for a linear complementary pair $(C,D)$, where $C$ and $D$ are $2$-sided ideals in a group algebra, $D$ is uniquely determined by $C$ and the dual code $D^\perp$ is permutation equivalent to $C$. This includes earlier results of Carlet et al. and G\"uneri et al. on nD cyclic codes which have been proved by subtle and lengthy calculations in the space of polynomials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Short elementary proof unifies known facts on complementary 2-sided ideal pairs in group algebras but adds no new theorem.

read the letter

This note proves that for a linear complementary pair of 2-sided ideals C and D in a group algebra, D is fixed once C is chosen and D perp is permutation equivalent to C. The argument uses only standard facts about ideals and the group algebra structure, which replaces the lengthy polynomial calculations in the earlier papers on nD cyclic codes by Carlet et al. and Güneri et al. That simplification is the real contribution here and it works cleanly for the stated setting. The proof is direct and avoids any fitting or extra assumptions beyond the ideal property. The central claim is not new, however. The paper states outright that it recovers those prior results, so the value is in the shorter route rather than a fresh statement. That is a modest but honest scope for a note. No load-bearing gaps appear in the logic as described, and the citations to the earlier work are accurate. This is aimed at researchers already working with group codes and complementary pairs who might want a cleaner reference to cite or build from. It is narrow but the algebraic approach is sound. I would send it to peer review as a useful cleanup of the literature.

Referee Report

0 major / 0 minor

Summary. The manuscript gives a short elementary proof that for a linear complementary pair (C, D) of 2-sided ideals in a group algebra, D is uniquely determined by C and the dual D^perp is permutation equivalent to C. The argument uses standard properties of group algebras and ideals to recover and generalize earlier results on nD cyclic codes that had required lengthy polynomial calculations.

Significance. If the result holds, the paper supplies a clean algebraic derivation that replaces case-by-case computations with the ideal structure of the group algebra. This yields a more conceptual and potentially extensible treatment of linear complementary pairs, while explicitly recovering the prior theorems of Carlet et al. and Güneri et al. as special cases.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation to accept the manuscript. The summary correctly identifies the contribution of the short elementary proof and its relation to prior work on nD cyclic codes.

Circularity Check

0 steps flagged

No circularity; derivation follows directly from group algebra ideal properties

full rationale

The paper claims a short elementary proof that for 2-sided ideals C and D forming a linear complementary pair in a group algebra, D is uniquely determined by C and D^perp is permutation equivalent to C. This follows from standard algebraic structure of group algebras and ideals, without any fitted parameters, self-definitional reductions, or load-bearing self-citations. The result generalizes prior work on nD cyclic codes via direct consequences of the ideal property rather than re-deriving or renaming inputs. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard definition of 2-sided ideals in group algebras and the definition of linear complementary pairs; no free parameters, new entities, or ad-hoc axioms are introduced.

axioms (1)

domain assumption 2-sided ideals in group algebras over finite fields satisfy the usual module and ideal properties used in coding theory
Invoked to obtain uniqueness and duality equivalence from the complementary pair condition.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

If C ⊕ D = KG where C and D are 2-sided ideals in KG, then D is uniquely determined by C and D⊥ is permutation equivalent to C.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma B. If D = eKG with e² = e ∈ KG, then D⊥ = (1 − ê)KG.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.