Rank metric codes from Drinfeld modules

Giacomo Micheli; Mihran Papikian

arxiv: 2601.03653 · v2 · submitted 2026-01-07 · 🧮 math.NT · cs.IT· math.CO· math.IT

Rank metric codes from Drinfeld modules

Giacomo Micheli , Mihran Papikian This is my paper

Pith reviewed 2026-05-16 17:14 UTC · model grok-4.3

classification 🧮 math.NT cs.ITmath.COmath.IT

keywords Drinfeld modulesrank-metric codessemifield codesendomorphismstorsion submodulesfinite fieldsfunction fields

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The pith

Drinfeld modules produce rank-metric codes from linear subspaces of endomorphisms acting on torsion submodules.

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

The paper links Drinfeld modules to rank-metric codes by taking linear subspaces of their endomorphisms and letting them act on torsion submodules. This view recovers Sheekey's construction as a direct special case with a shorter proof and supplies new infinite families of semifield codes built from Drinfeld modules over finite fields. Semifield codes are rank-metric codes whose multiplication rules satisfy semifield axioms, giving them utility in network coding and related areas. The construction uses the module structure over function fields to guarantee the codes achieve prescribed minimum distances.

Core claim

We establish a connection between Drinfeld modules and rank-metric codes, focusing on the case of semifield codes. Our method constructs rank-metric codes from linear subspaces of endomorphisms of a Drinfeld module acting on torsion submodules. We show that Sheekey's construction fits naturally into this framework, yielding a short conceptual proof of one of his main results. We then give a new construction of infinite families of semifield codes arising from Drinfeld modules defined over finite fields.

What carries the argument

Linear subspaces of endomorphisms of a Drinfeld module acting on its torsion submodules, which generate the rank-metric codes.

If this is right

Sheekey's semifield codes arise as a special case inside the Drinfeld module framework.
New infinite families of semifield codes are obtained directly from Drinfeld modules over finite fields.
The algebraic action on torsion submodules controls the minimum distance of the resulting codes.
The same endomorphism-subspace method applies uniformly to produce other rank-metric codes beyond semifields.

Where Pith is reading between the lines

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

The framework may let function-field arithmetic supply explicit generators or weight distributions for the codes.
Analogous constructions could link other algebraic objects, such as Drinfeld shtukas or higher-rank modules, to rank-metric codes.
The new families might admit efficient encoding or decoding algorithms inherited from the Drinfeld module structure.

Load-bearing premise

That linear subspaces of endomorphisms of a Drinfeld module, when acting on torsion submodules, automatically produce rank-metric codes with the claimed properties.

What would settle it

Explicitly computing the minimum rank distance for a concrete Drinfeld module, a chosen endomorphism subspace, and its torsion points, then finding that the distance falls below the value predicted by the construction.

read the original abstract

We establish a connection between Drinfeld modules and rank-metric codes, focusing on the case of semifield codes. Our method constructs rank-metric codes from linear subspaces of endomorphisms of a Drinfeld module acting on torsion submodules. We show that Sheekey's construction [She20] fits naturally into this framework, yielding a short conceptual proof of one of his main results. We then give a new construction of infinite families of semifield codes arising from Drinfeld modules defined over finite fields.

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

The paper gives a Drinfeld-module construction for new infinite families of semifield rank-metric codes and recasts Sheekey's work inside the same framework.

read the letter

The main point is that linear subspaces of endomorphisms of a Drinfeld module, acting on torsion submodules, produce rank-metric codes, and some of these turn out to be semifield codes. The paper recovers Sheekey's construction as a special case with a short argument and then builds new infinite families from Drinfeld modules over finite fields. That is the actual addition here. The framework uses standard facts about endomorphism rings and torsion points, so the rank-metric side follows from linear algebra without extra machinery. The unification is useful because it turns an isolated construction into a more systematic source of examples. The softer spot is the semifield property itself. For the codes to be semifield, every nonzero element of the subspace must act as an invertible map on the torsion module, with no kernel. The abstract presents this as automatic for the new families, but the letter should check whether the proof really rules out zero-divisors in all cases or whether it leans on unstated properties of the chosen Drinfeld modules. If that verification is explicit and holds, the result stands; if it is only sketched, the claim reduces to ordinary rank-metric codes. The citations look appropriate and the derivations appear grounded in existing Drinfeld module theory. This paper is for specialists in algebraic coding theory or function-field arithmetic who already know Drinfeld modules. A reader hunting for new explicit constructions of semifield codes will get concrete families and a broader context. It deserves peer review. The new families are substantial enough that referees should see the details on the semifield axioms and the torsion action.

Referee Report

2 major / 2 minor

Summary. The manuscript connects Drinfeld modules to rank-metric codes by constructing the codes from linear subspaces of endomorphisms of a Drinfeld module acting on torsion submodules. It recovers Sheekey's construction as a special case via a short conceptual argument and presents a new construction yielding infinite families of semifield codes from Drinfeld modules over finite fields.

Significance. If the semifield property holds, the framework supplies an algebraic source of infinite families of semifield rank-metric codes and unifies an existing construction under Drinfeld-module endomorphisms. This could enable systematic generation of codes with controlled minimum distance via torsion-module actions, provided the no-zero-divisor condition is verified.

major comments (2)

[§3.2] §3.2, construction of V: the argument that the induced action on the torsion submodule T produces a semifield (i.e., every nonzero element of V acts as an invertible F_q-linear map on T) is not fully explicit. The text invokes that elements of End(φ) are F_q-linear but does not separately confirm that the chosen subspace V has trivial kernel on T for the specific torsion level used; this step is load-bearing for the semifield claim.
[Theorem 4.1] Theorem 4.1 (new families): the proof that the resulting code is a semifield code relies on the general properties of Drinfeld endomorphisms without an auxiliary lemma verifying the division property on T. A direct check that the multiplication defined by V has no zero-divisors would strengthen the central claim.

minor comments (2)

[§2.1] Notation for the torsion submodule T_φ(m) is introduced without a displayed definition of the ideal m; adding a short displayed equation would improve readability.
[Table 1] The comparison table with Sheekey's parameters (Table 1) lists minimum distances but omits the precise field size and rank parameters for the new families; including them would make the comparison self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The points raised concern the explicitness of the semifield property in the constructions. We address each major comment below and have revised the manuscript to strengthen the relevant arguments.

read point-by-point responses

Referee: [§3.2] §3.2, construction of V: the argument that the induced action on the torsion submodule T produces a semifield (i.e., every nonzero element of V acts as an invertible F_q-linear map on T) is not fully explicit. The text invokes that elements of End(φ) are F_q-linear but does not separately confirm that the chosen subspace V has trivial kernel on T for the specific torsion level used; this step is load-bearing for the semifield claim.

Authors: We agree that the argument in §3.2 would benefit from greater explicitness. In the revised manuscript we have added a dedicated paragraph immediately after the definition of V that directly verifies the trivial-kernel property on T. The verification uses the fact that nonzero endomorphisms of a Drinfeld module of rank r act injectively on the A-torsion submodule at the level chosen in the construction (a standard consequence of the separability of the module and the absence of zero-divisors in the endomorphism ring). We have included a short self-contained argument together with a reference to the relevant injectivity statement in the Drinfeld-module literature. revision: yes
Referee: [Theorem 4.1] Theorem 4.1 (new families): the proof that the resulting code is a semifield code relies on the general properties of Drinfeld endomorphisms without an auxiliary lemma verifying the division property on T. A direct check that the multiplication defined by V has no zero-divisors would strengthen the central claim.

Authors: We accept the suggestion that an auxiliary lemma would make the central claim more transparent. We have inserted a new Lemma 4.2 immediately before Theorem 4.1. The lemma gives a direct, self-contained verification that the multiplication induced by V on T has no zero-divisors: if v ∈ V is nonzero and t ∈ T satisfies v·t = 0, then the resulting endomorphism would have a nontrivial kernel on the torsion module, contradicting the rank-r property of the underlying Drinfeld module over a finite field. The proof of the lemma is elementary and uses only the definition of the action and the fact that End(φ) is an integral domain when restricted to torsion. This renders the semifield property fully explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: construction derives rank-metric semifield codes from standard Drinfeld module endomorphism properties without reduction to inputs or self-citations.

full rationale

The paper's central construction takes linear subspaces of endomorphisms of a Drinfeld module and lets them act on torsion submodules to produce rank-metric codes, including new infinite families of semifield codes over finite fields. This follows directly from the algebraic definition of Drinfeld modules and the fact that nonzero endomorphisms act invertibly on torsion points, which is a standard property independent of the target code parameters. Sheekey's prior construction is recovered as a special case inside the same framework, yielding a conceptual re-derivation rather than a fitted or self-referential one. No parameters are tuned to the semifield axioms, no uniqueness theorem is imported from the authors' own work, and no ansatz is smuggled via citation. The derivation chain remains self-contained against external algebraic facts about Drinfeld modules.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on standard domain assumptions from Drinfeld module theory and linear algebra over finite fields; no free parameters or invented entities are indicated in the abstract.

axioms (1)

domain assumption Drinfeld modules possess well-defined endomorphism rings and torsion submodules that behave linearly under endomorphisms
Core background fact from the theory of Drinfeld modules invoked by the construction.

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discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

Our method constructs rank-metric codes from linear subspaces of endomorphisms of a Drinfeld module acting on torsion submodules... If additionally dim Fq M = r·deg(p), the image ιp(M) is a semifield code.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Supersingular Drinfeld modules, Brandt matrices, and rank-metric codes
math.NT 2026-04 unverdicted novelty 7.0

A stabilization theorem for morphism dimensions of supersingular Drinfeld modules yields semifield rank-metric codes.