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arxiv: 2604.15158 · v2 · submitted 2026-04-16 · 🧮 math.RA

Projector additive group codes

Javier de la Cruz This is my paper

Pith reviewed 2026-05-10 08:43 UTC · model grok-4.3

classification 🧮 math.RA
keywords additive group codesprojector codesgroup algebrasprojective modulessemisimple algebrastrace dualityself-dual codes
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The pith

Additive left group codes over group algebras arise as images of FG-linear projectors, covering all such codes in semisimple cases and direct summands otherwise.

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

The paper defines projector additive left group codes as the images of FG-linear projectors acting on the group algebra KG, and restricted versions as images of submodules under those projectors. This approach is introduced because idempotents in KG fail to give a broad enough setting for additive codes. The construction yields projective left FG-submodules, so that every additive left group code equals a projector code when the algebra is semisimple, while non-semisimple cases recover exactly the direct summands of KG. The work also ties trace dualities to adjoint projectors and gives conditions for LCD and self-dual codes.

Core claim

Projector additive left group codes are a particular class of projective left FG-submodules of KG. Consequently, in the semisimple case every additive left group code arises in this way, whereas in the non-semisimple case the projector construction captures precisely the direct summands of KG as left FG-modules, and hence a natural subclass of projective left FG-submodules. The paper further relates trace-Euclidean and trace-Hermitian duality to adjoint projectors, establishes criteria for the LCD and self-dual cases, studies the Murray-von Neumann equivalence of projectors, and interprets quotients by orthogonal codes in terms of module duals.

What carries the argument

FG-linear projectors on KG, whose images define the codes and serve as the algebraic mechanism replacing idempotents to generate projective submodules.

Load-bearing premise

Idempotent elements of KG do not yield a sufficiently general and natural algebraic framework for the study of additive left group codes.

What would settle it

An explicit additive left group code over a semisimple group algebra that cannot be realized as the image of any FG-linear projector on KG.

read the original abstract

Let $F=\mathbb{F}_q$ and let $K=\mathbb{F}_{q^m}$ be a finite extension. An additive left group code is a left $FG$-submodule of the group algebra $KG$. In this paper, we introduce projector additive left group codes and restricted projector additive left group codes as additive counterparts of idempotent group codes in the classical theory of group codes. More precisely, they are defined, respectively, as images of $FG$-linear projectors on $KG$ and as images of left $FG$-submodules under such projectors. This perspective is motivated by the fact that idempotent elements of $KG$ do not yield a sufficiently general and natural algebraic framework for the study of additive left group codes. Projector additive left group codes are a particular class of projective left $FG$-submodules of $KG$. Consequently, in the semisimple case every additive left group code arises in this way, whereas in the non-semisimple case the projector construction captures precisely the direct summands of $KG$ as left $FG$-modules, and hence a natural subclass of projective left $FG$-submodules. We further relate trace-Euclidean and trace-Hermitian duality to adjoint projectors, establish criteria for the LCD and self-dual cases, study the Murray--von Neumann equivalence of projectors, and interpret quotients by orthogonal codes in terms of module duals.

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.

Referee Report

2 major / 3 minor

Summary. The paper introduces projector additive left group codes as images of FG-linear projectors on the group algebra KG (with F=𝔽_q, K=𝔽_{q^m}, G a finite group) and restricted projector additive left group codes as images of left FG-submodules under such projectors. These are positioned as a particular class of projective left FG-submodules of KG. The central claims are that, in the semisimple case, every additive left group code arises this way, while in the non-semisimple case the construction captures precisely the direct summands of KG as left FG-modules. Additional results relate trace-Euclidean and trace-Hermitian duality to adjoint projectors, give criteria for LCD and self-dual cases, study Murray-von Neumann equivalence of projectors, and interpret quotients by orthogonal codes via module duals.

Significance. If the results hold, the work supplies a module-theoretic framework for additive group codes that properly generalizes the classical idempotent construction and directly ties the codes to projective modules and direct summands. This is especially useful for non-semisimple group algebras, where not every submodule is a direct summand. The connections to duality, LCD/self-dual criteria, and Murray-von Neumann equivalence provide concrete tools for classification and construction that align with standard ring and module theory.

major comments (2)
  1. [§3, Theorem 3.4] §3, Theorem 3.4: the proof that every additive left group code is a projector code in the semisimple case relies on the Artin-Wedderburn decomposition of KG; it would be strengthened by an explicit reference to the fact that every module over a semisimple artinian ring is a direct summand of a free module, together with a short verification that the projector can be chosen FG-linear.
  2. [§4, Definition 4.1 and Proposition 4.5] §4, Definition 4.1 and Proposition 4.5: the adjoint projector construction for trace-Hermitian duality assumes the existence of a non-degenerate trace form; the paper should clarify whether this form is the standard one induced by the field trace Tr_{K/F} or a more general bilinear form, as the latter would affect the self-dual criteria in Theorem 4.8.
minor comments (3)
  1. [§2] The notation for the group algebra is introduced as KG but later switches between FG-submodules and left FG-modules without a uniform convention; a single sentence in §2 fixing the left/right convention would improve readability.
  2. Figure 1 (if present) comparing idempotent and projector codes lacks a caption explaining the arrows; adding one would clarify the diagram.
  3. The reference list omits the standard text on group algebras and coding theory (e.g., a citation to Passman or to the survey on group codes by the authors of the classical idempotent theory); adding 1-2 such references would place the new definitions in context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive suggestions. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3, Theorem 3.4] §3, Theorem 3.4: the proof that every additive left group code is a projector code in the semisimple case relies on the Artin-Wedderburn decomposition of KG; it would be strengthened by an explicit reference to the fact that every module over a semisimple artinian ring is a direct summand of a free module, together with a short verification that the projector can be chosen FG-linear.

    Authors: We agree that an explicit reference to the standard fact that every module over a semisimple artinian ring is projective (hence a direct summand of a free module) would strengthen the argument. In the revised manuscript we have added this reference together with a short verification that the projector arising from the Artin-Wedderburn decomposition can be chosen FG-linear, thereby clarifying the construction in Theorem 3.4. revision: yes

  2. Referee: [§4, Definition 4.1 and Proposition 4.5] §4, Definition 4.1 and Proposition 4.5: the adjoint projector construction for trace-Hermitian duality assumes the existence of a non-degenerate trace form; the paper should clarify whether this form is the standard one induced by the field trace Tr_{K/F} or a more general bilinear form, as the latter would affect the self-dual criteria in Theorem 4.8.

    Authors: We thank the referee for this observation. The non-degenerate trace form employed in Definition 4.1 and Proposition 4.5 is precisely the standard form induced by the field trace Tr_{K/F}. We have inserted an explicit clarification to this effect in the revised text and verified that the self-dual criteria stated in Theorem 4.8 continue to hold under this standard choice. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines projector additive left group codes directly as images of FG-linear projectors on KG and restricted versions as images of submodules under such projectors. These definitions rest on standard module-theoretic notions of projectors, direct summands, and projective modules over group algebras, which are external to the paper. The consequent statements about the semisimple case recovering all additive left group codes and the non-semisimple case capturing direct summands follow from classical facts on completely reducible modules over semisimple artinian rings, without any reduction of claims to self-referential equations, fitted parameters, or load-bearing self-citations. The motivation regarding idempotents is presented as context for the new definitions rather than a derived result. The derivation chain is self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on standard finite-field group algebra setup and module theory; the only invented entity is the new code class itself, introduced without independent evidence beyond the definitions.

axioms (2)
  • domain assumption F = F_q and K = F_{q^m} are finite fields with K a extension of F
    Standard setup for the group algebra KG in finite-field coding theory.
  • standard math KG is the group algebra over the group G
    Basic algebraic object in which the codes are defined as submodules.
invented entities (1)
  • projector additive left group code no independent evidence
    purpose: Additive counterpart to idempotent group codes via images of linear projectors
    Newly defined class whose properties are studied in the paper.

pith-pipeline@v0.9.0 · 5539 in / 1429 out tokens · 48288 ms · 2026-05-10T08:43:57.053807+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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