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arxiv: 2606.18371 · v1 · pith:BJDI7NR7new · submitted 2026-06-16 · 💻 cs.IT · math.IT

A skew polynomial framework for constructing division algebras and linear maximum rank distance codes

Susanne Pumpluen This is my paper

Pith reviewed 2026-06-26 22:17 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords skew polynomialsdivision algebrasmaximum rank distance codesMRD codessemifieldstwisted fieldscoding theoryfinite fields
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The pith

Skew polynomials over fields yield non-unital division algebras and linear MRD codes that generalize several known constructions.

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

The paper develops a method to build division algebras and linear maximum rank distance matrix codes directly from skew polynomials over fields. These algebras generalize twisted cyclic pre-semifields, Jha-Johnson semifields, and Albert's generalized twisted fields. The codes extend earlier MRD constructions by providing a common source for them. Explicit criteria determine when the resulting structures form division algebras or attain maximum rank, and some invariants of the algebras are computed. The approach also allows comparison of isotopic division algebras appearing in the literature.

Core claim

Skew polynomials over fields produce non-unital division algebras generalizing Sheekey's twisted cyclic pre-semifields and Albert's generalized twisted fields, together with linear MRD codes that generalize those of Lobillo, Santonastaso and Sheekey. Criteria are supplied for the algebras to be division algebras and for the codes to have maximum rank, and invariants of the algebras are computed.

What carries the argument

Skew polynomials over fields, together with the stated criteria that ensure the output structures are division algebras or attain maximum rank.

If this is right

  • New families of non-unital division algebras arise whenever suitable skew polynomials are selected.
  • Linear MRD codes arise by the same process and include all previous examples from Lobillo et al. as special cases.
  • Isotopy classes of division algebras from different classical constructions become comparable inside the single framework.
  • Invariants of the constructed algebras can be read off from the underlying skew polynomial data.

Where Pith is reading between the lines

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

  • The same polynomials may generate semifields usable in projective planes or other finite geometries.
  • The framework could be tested on skew polynomials over function fields to produce infinite families.
  • Different choices of the base field might reveal whether the generalized constructions are exhaustive for certain parameters.

Load-bearing premise

The chosen skew polynomials and base fields must satisfy the paper's criteria for the output to be a division algebra or to reach maximum rank.

What would settle it

An explicit skew polynomial and field satisfying the stated criteria yet producing either a non-division algebra or an MRD code whose rank falls short of the maximum.

read the original abstract

We construct division algebras and linear maximum rank distance (MRD) matrix codes using skew polynomials over fields. The non-unital division algebras we obtain generalize several prominent constructions: Sheekey's twisted cyclic pre-semifields, i.e. the pre-semifields associated with Jha-Johnson semifields and the semifields associated with Albert's generalized twisted fields. Our linear MRD codes generalize the constructions of Lobillo, Santonastaso and Sheekey. We present criteria for these algebras to be division algebras, respectively, for when these codes have maximum rank, and compare isotopic division algebras that appear throughout recent and classical literature. We compute some of their invariants.

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 / 2 minor

Summary. The manuscript develops a framework using skew polynomials over fields to construct non-unital division algebras and linear maximum rank distance (MRD) matrix codes. The algebras are claimed to generalize Sheekey's twisted cyclic pre-semifields (including those associated with Jha-Johnson semifields and Albert's generalized twisted fields), while the MRD codes generalize the Lobillo-Santonastaso-Sheekey constructions. Criteria are presented for the division algebra property and for maximum rank, isotopic comparisons across literature are made, and selected invariants are computed.

Significance. If the criteria are shown to be necessary and sufficient and the specializations recover the cited constructions without hidden restrictions, the work supplies a common algebraic setting for several prominent families in semifield and MRD-code theory, potentially enabling systematic generation of new examples and invariant calculations.

major comments (2)
  1. [Criteria and specialization sections] The central claim that the framework recovers the cited constructions as special cases (Sheekey twisted cyclic, Jha-Johnson, Albert generalized twisted fields, Lobillo et al. MRD codes) is load-bearing; the manuscript must exhibit explicit parameter choices or ring homomorphisms that realize each family inside the skew-polynomial setting, rather than only stating that the criteria are satisfied.
  2. [Criteria for division algebras / MRD property] The criteria for a skew polynomial to yield a division algebra or an MRD code are stated but their verification for the chosen families appears to rely on the same algebraic properties used to define the criteria; an independent check (e.g., direct substitution into the rank or zero-divisor condition for a concrete automorphism and degree) is required to confirm the criteria are not tautological for the advertised examples.
minor comments (2)
  1. Notation for the skew polynomial ring and the associated multiplication should be introduced once with a clear reference to the underlying field automorphism and its order.
  2. The comparison of isotopic division algebras would benefit from a table listing the isotopism classes recovered from each cited construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Criteria and specialization sections] The central claim that the framework recovers the cited constructions as special cases (Sheekey twisted cyclic, Jha-Johnson, Albert generalized twisted fields, Lobillo et al. MRD codes) is load-bearing; the manuscript must exhibit explicit parameter choices or ring homomorphisms that realize each family inside the skew-polynomial setting, rather than only stating that the criteria are satisfied.

    Authors: We agree that explicit parameter choices are required to fully substantiate the recovery of the cited families as special cases. In the revised manuscript we will add a dedicated subsection that lists, for each construction, the concrete automorphism σ, the degree n, the specific coefficients of the skew polynomial, and (where relevant) the ring homomorphism realizing the embedding into the skew-polynomial quotient. This will make the specializations fully explicit and verifiable. revision: yes

  2. Referee: [Criteria for division algebras / MRD property] The criteria for a skew polynomial to yield a division algebra or an MRD code are stated but their verification for the chosen families appears to rely on the same algebraic properties used to define the criteria; an independent check (e.g., direct substitution into the rank or zero-divisor condition for a concrete automorphism and degree) is required to confirm the criteria are not tautological for the advertised examples.

    Authors: The criteria are obtained from the general theory of skew polynomials, yet we recognize the value of an independent verification for the examples. In the revision we will include, for at least one representative from each family, a direct computation of the matrix rank (or explicit check for zero-divisors) using the original definitions and a concrete automorphism, performed separately from the statement of the criterion. This will confirm that the criteria apply non-trivially. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rely on independent algebraic criteria

full rationale

The paper states criteria for the skew polynomials to yield division algebras or MRD codes and claims generalizations of prior constructions (Sheekey, Jha-Johnson, Albert, Lobillo et al.). No quoted step reduces a prediction or central claim to a fitted input, self-definition, or load-bearing self-citation chain. The criteria are presented as external conditions on the polynomials and fields; their verification is described as depending on properties of the underlying objects rather than being tautological. The derivation chain is therefore self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard mathematical axioms from ring theory and field theory. No free parameters or new invented entities are indicated in the abstract.

axioms (1)
  • standard math Fields and skew polynomial rings over them are standard algebraic objects with known properties.
    The paper builds upon established theory of skew polynomials.

pith-pipeline@v0.9.1-grok · 5632 in / 1246 out tokens · 53638 ms · 2026-06-26T22:17:05.554574+00:00 · methodology

discussion (0)

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Reference graph

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