On the autotopism groups and the equivalence of finite cyclic semifields

Technical Summary: On the Autotopism Groups and the Equivalence of Finite Cyclic Semifields

Problem Statement
Finite cyclic semifields, originally constructed by Hughes, Kleinfeld, Sandler, and Knuth, and later generalized by Petit and rediscovered by Jha and Johnson, represent a significant class of nonassociative division algebras. Despite foundational work by Sandler (1962) and subsequent advances by Dempwolff (2011), two critical problems regarding this family remained unresolved:

1. The complete classification of cyclic semifields up to isotopy.
2. The full determination of their autotopism groups, specifically in cases where the field extension degree strictly divides the degree of the defining polynomial (a case left open by Dempwolff).

Furthermore, since cyclic semifields correspond to specific instances of Maximum Rank Distance (MRD) codes constructed via skew-polynomial rings, the equivalence and automorphism problems for these codes were also incompletely understood.

Methodology
The authors employ a dual-framework approach, leveraging two complementary descriptions of the same algebraic family:

1. The Skew-Polynomial Model: Utilizing the ring $R = \mathbb{F}_{q^n}[X; \sigma]$ and its quotient $R_F = R/R_F(X^n)$, where $F(Y)$ is an irreducible polynomial over $\mathbb{F}_q$. The codes $S_k(F)$ are defined as subspaces of $R_F$ spanned by powers of $X$ up to $k-1$.
2. The Semilinear-Transformation Model: Identifying the quotient $R_F$ with the endomorphism ring of a vector space $V$ over a field $E \cong \mathbb{F}_{q^s}$. In this setting, the codes correspond to spans of powers of an irreducible $\sigma$-semilinear operator $T_F$.

The core technical strategy involves:

• Establishing an explicit isomorphism between the skew-polynomial quotient and the algebra of semilinear operators (Theorem 3.6).
• Analyzing the structural properties of the centralizer of $\mathbb{F}_{q^n}$ within the quotient ring to characterize elements that preserve the code structure (Section 4).
• Reducing the equivalence problem to the comparison of the defining irreducible polynomials $F$ and $G$ under specific group actions involving field automorphisms and reciprocal involution (Section 5).
• Determining the full automorphism groups by combining linear automorphisms (derived from the centralizer structure) with semilinear symmetries induced by field automorphisms (Section 6).

Key Contributions and Results

1. Complete Classification up to Isotopy and Equivalence:
   The paper provides a necessary and sufficient condition for two cyclic semifields (or their associated MRD codes $S_k(F)$ and $S_k(G)$) to be isotopic or equivalent.

   • Linear Equivalence: $S_k(F)$ and $S_k(G)$ are linearly equivalent if and only if there exists $\lambda \in \mathbb{F}_q^\times$ such that $F(Y) = \lambda^s G(\lambda^{-1}Y)$, or, in the specific case $n=2$, $F(Y) = \lambda^s \hat{G}(\lambda^{-1}Y)$, where $\hat{G}$ is the reciprocal polynomial.
   • Semilinear Equivalence: Over the prime field $\mathbb{F}_p$, equivalence holds if and only if the polynomials are related by a field automorphism $\tau \in \text{Gal}(\mathbb{F}_q/\mathbb{F}_p)$ and the scalar $\lambda$ as described above.
   • These results yield explicit lower bounds on the number of pairwise non-isotopic cyclic semifields and inequivalent MRD codes, demonstrating a "genuinely large supply" of such objects, particularly for odd non-square orders.

2. Full Determination of Autotopism and Automorphism Groups:
   The authors resolve the open cases left by Dempwolff by fully characterizing the autotopism groups of cyclic semifields and the automorphism groups of the associated MRD codes.

   • Linear Automorphisms: The linear automorphism group $\text{Aut}_E(S_k(F))$ is explicitly described as a set of pairs $(\bar{g}, \bar{g}^{-1}\lambda)$ where $\bar{g}$ belongs to a specific subgroup of the centralizer. The order is given by the formula:
     $$|\text{Aut}_E(S_k(F))| = \frac{n(q^s - 1)(q^n - 1)^2}{q - 1}$$
     (excluding the special case $(n, s, F) = (2, 2, Y^2 - \nu)$ which corresponds to Hughes-Kleinfeld semifields).
   • Semilinear Automorphisms: The full automorphism group fits into a short exact sequence:
     $$1 \longrightarrow \text{Aut}_E(C_k(F)) \longrightarrow \text{Aut}(C_k(F)) \longrightarrow H_F \longrightarrow 1$$
     where $H_F$ is a cyclic subgroup of $\text{Aut}_{\mathbb{F}_p}(E)$ determined by the roots of $F$.
   • Autotopism Groups: For the semifield case ($k=1$), the semilinear automorphism group of the spread set is identified with the full autotopism group of the cyclic semifield, providing a complete description previously unavailable for the general case.

3. Enumeration:
   The classification theorems allow for the exact counting of isotopy classes of cyclic semifields and equivalence classes of MRD codes in terms of the orbits of the defining polynomials under the action of a specific group $G$ (or $\hat{G}$ for semilinear equivalence).

Significance
The paper claims to close the remaining open cases regarding the autotopism groups of cyclic semifields, a problem that has persisted since Sandler's foundational work in 1962. By providing a complete classification up to isotopy and an explicit description of the full autotopism groups, the authors resolve the isotopy problem for this family.

Furthermore, the results extend to coding theory, offering a complete classification of a broad family of MRD codes constructed via skew polynomials up to linear and semilinear equivalence over the prime field. This resolves open questions regarding the equivalence of these codes raised in previous literature (e.g., Sheekey [40]). The work establishes that cyclic semifields provide the largest explicit family of odd-order semifields with completely determined isotopy classes and explicit autotopism groups, surpassing previous bounds known for square orders.