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

Technical Summary: A Skew Polynomial Framework for Constructing Division Algebras and Linear Maximum Rank Distance Codes

Problem Statement
The paper addresses the construction of non-unital division algebras and linear Maximum Rank Distance (MRD) matrix codes over fields. While previous works by Petit (1966), Sheekey (2016, 2019), and Lobillo, Santonastaso, and Sheekey (2026) have successfully utilized skew polynomial rings $R = K[t; \sigma]$ to generate such structures, these constructions were often restricted to specific parameters. Specifically, prior constructions typically limited the linear map $\rho$ used in the multiplication definition to be a ring automorphism (specifically a Galois automorphism) and fixed the index parameter $i_0$ to be 0. The paper seeks to generalize these frameworks to determine if broader classes of maps and indices yield new division algebras and MRD codes, and to establish rigorous criteria for when these generalized structures satisfy the division property or achieve maximum rank distance.

Methodology
The author employs a unified framework based on skew polynomial rings $R = K[t; \sigma]$, where $K/F$ is a cyclic Galois field extension of degree $n$ with Galois group generated by $\sigma$.

1. Algebra Construction:
   The core construction defines a multiplication on the vector space $R_m = \{g \in R \mid \deg(g) < m\}$. Given a monic irreducible skew polynomial $f \in R$ of degree $m$, a scalar $\nu \in K$, an integer $i_0 \in \{0, \dots, m-1\}$, and a bijective $F'$-linear map $\rho: K \to K$ (where $F' \supseteq F$), a new multiplication $\circ$ is defined for $b(t), c(t) \in R_m$:
   $$b(t) \circ c(t) = (b(t) + \nu\rho(b_{i_0})t^m)c(t) \mod_r f$$
   Here, the right-hand side denotes the remainder of the right division of the product by $f$. The resulting structure is denoted $S_f(K, i_0, \nu, \rho)$. This generalizes previous constructions by allowing $\rho$ to be any bijective linear map (not just automorphisms) and $i_0$ to be any index in the polynomial degree range.

2. MRD Code Construction:
   The paper extends this to construct linear MRD codes. By considering the minimal central left multiple $h$ of $f$ (where $h$ has $k$ irreducible factors) and an integer $l$ ($1 < l < k$), the author defines a set $P_{i_0, l}$ of polynomials of degree up to $lm$. The image of this set in the quotient ring $R/Rh$ defines a code $C_{i_0, n, m, l}$ in the matrix algebra $M_k(B)$, where $B$ is the right nucleus of the associated Petit algebra.

3. Analytical Tools:
   The analysis relies heavily on the theory of similar skew polynomials. Two polynomials are similar if the left $R$-modules $R/Rf$ and $R/Rg$ are isomorphic. The author utilizes the set of polynomials similar to $f$ and their coefficients to derive necessary and sufficient conditions for the algebraic and coding properties.

Key Contributions and Results

• Generalization of Constructions: The paper demonstrates that the existing frameworks for Sheekey's twisted cyclic semifields and the algebras in [LobSanSh2026] are special cases of the proposed framework where $i_0 = 0$ and $\rho$ is an automorphism. The new framework allows for arbitrary bijective linear maps $\rho$ and arbitrary indices $i_0$.
• Criteria for Division Algebras:
  • Theorem 3.2: Establishes that $S_f(K, i_0, \nu, \rho)$ is a division algebra over $F_0$ if and only if the set $P_{i_0}$ (polynomials of the form $g(t) + \nu\rho(g_{i_0})t^m$) contains no polynomial similar to $f$.
  • Theorem 3.5 & 3.8: Provide concrete criteria based on the coefficients of polynomials similar to $f$. Specifically, the algebra is a division algebra if for all monic polynomials $g$ similar to $f$, the condition $\nu\rho(g_{i_0}) \neq 1$ (when $g_{i_0} \neq 0$) holds, or equivalently, if a specific norm condition involving $g_{i_0}$ and $\nu$ is not met.
  • Corollaries 3.7 and 3.9: Recover Sheekey's previous results for twisted cyclic algebras as special cases and provide alternative conditions for them to be pre-semifields.
• Isotopy and Unitality:
  • Theorem 5.4: Shows that under certain conditions, the non-unital algebras constructed are isotopic to unital Petit division algebras. This implies that some constructions previously thought to be distinct (specifically those in [LobSanSh2026, Theorem 5.4]) are actually isotopic to known Petit algebras.
  • Theorem 5.7: Corrects a claim in [LobSanSh2026, Theorem 5.7], demonstrating that the algebra constructed there is not unital but possesses a left unit element, and is isotopic to the Petit algebra.
• MRD Code Criteria:
  • Theorem 4.1: Provides a necessary and sufficient condition for the constructed code $C_{i_0, n, m, l}$ to be an MRD code with minimum distance $k - l + 1$. The condition requires that the set $P_{i_0, l}$ contains no polynomial of degree $lm$ with $l$ irreducible factors all similar to $f$.
  • Theorem 4.2: Offers a sufficient condition based on norms for the code to be MRD, generalizing previous results to non-automorphic $\rho$.
• Invariants and Nuclei:
  • Section 6: Computes the left, middle, and right nuclei, as well as the centralizer and center, for these algebras and codes.
  • Theorem 6.2 & 6.7: For finite fields, the paper derives explicit descriptions of the nuclei in terms of fixed fields of compositions of $\rho$ and powers of $\sigma$. It highlights that choosing $\rho$ as a non-automorphic linear map can yield "new" codes where the idealizers differ from those in classical constructions.

Significance and Claims
The paper claims to provide a unifying framework that subsumes recent and classical constructions of division algebras and MRD codes. Its primary significance lies in:

1. Broadening the Scope: By relaxing the requirement that $\rho$ be an automorphism and allowing $i_0$ to vary, the framework encompasses a substantially larger class of potential constructions.
2. Clarifying Isotopy Classes: The work clarifies the relationship between new constructions and existing ones, showing that some "new" algebras are actually isotopic to known Petit algebras, while identifying specific conditions under which genuinely new isotopy classes (and thus new MRD codes) may arise.
3. Correcting Literature: The paper corrects assertions regarding the unitality of specific algebras found in prior literature.
4. Computational Feasibility: While the criteria for infinite fields are abstract, the paper notes that for finite fields, the conditions are tractable (involving finite checks of coefficients and norms), though explicit computation for large parameters requires computer algebra systems.

The author concludes that while the setup is more general, the computational effort to verify the division property or MRD property remains comparable to previous methods, and the framework successfully generalizes the theory of skew polynomial-based division algebras and codes.