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Susanne Pumpluen

Publications and source records attributed to Susanne Pumpluen.

At least 19 recordsLinked to original sources
arXiv2026

Menichetti's nonassociative $G$-crossed product algebras and Menichetti codes

We define the nonassociative Menichetti algebras which can be viewed as nonassociative crossed product algebras and then use them as ambient algebras for new linear error-correcting codes. More precisely, we take their left principal ideals to define linear codes which are in canonical one-to-one correspondence with these ideals, imitating the approach taken when defining right skew polycyclic codes as left principal ideals of Petit algebras. The approach is novel and will create large classes of new linear codes which are well behave because of their algebraic definition as ideals of an algebra. With the right choice of algebra the codes display symmetric and cyclic properties which promise efficient decoding algorithms.

math.RA↗
arXiv2026

Three-Dimensional Geometry in Exceptional Algebra

We review some topics in "exceptional mathematics'' from the perspective of 3-dimensional geometry: the octonions $\mathbb{O}$, the split octonions $\mathbb{O}'$, the bioctonions $\mathbb{O}_\mathbb{C} \cong \mathbb{C} \otimes_\mathbb{R} \mathbb{O}$, the complex Albert algebra $\mathfrak{h}_3(\mathbb{O}_\mathbb{C})$, and the complex form of the exceptional Lie algebra $\mathfrak{e}_6$. We show how to functorially build an algebra isomorphic to $\mathbb{O}$ from any 3d complex vector space equipped with an inner product and complex volume form. Similarly, we build one isomorphic to $\mathbb{O}'$ starting from a 3d real vector space equipped with a volume form, and one isomorphic to $\mathbb{O}_\mathbb{C}$ starting from a 3d complex vector space equipped with a complex volume form. We give applications to 3-dimensional real and complex manifolds. Finally, we describe how to build an Jordan algebra isomorphic to $\mathfrak{h}_3(\mathbb{O}_\mathbb{C})$ starting from three 3d complex vector spaces equipped with complex volume forms. This last construction gives a nice explicit description of the complex Lie algebra $\mathfrak{e}_6$ and its subalgebra $\mathfrak{sl}(3,\mathbb{C}) \oplus \mathfrak{sl}(3,\mathbb{C}) \oplus \mathfrak{sl}(3,\mathbb{C})$.

math.RA↗
arXiv2026

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

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.

cs.IT↗
arXiv2026

Explicit constructions of anti-automorphisms of cyclic and generalized cyclic algebras

We present norm criteria for the existence of anti-automorphisms, as well as explicit constructions of anti-automorphisms, both on cyclic and generalized cyclic algebras. Our approach describes anti-automorphisms as polynomial maps and unifies existing approaches. It recovers classical criteria for the existence of involutions as special cases. We obtain norm conditions for the existence of anti-automorphisms of the second kind on the ring of twisted Laurent series $K((t;σ))$ over a field $K$ and the ring of twisted Laurent series $D((t;σ))$ over a division algebra $D$ that is finite-dimensional over its center. Our constructions rely on the isomorphisms between an algebra and its opposite algebra. Along the way, we hence describe monomial isomorphisms between cyclic or generalized cyclic algebras, which ties in with studying the isomorphism problem for central simple algebras. Proper nonassociative cyclic and generalized cyclic algebras, which are canonical generalizations of associative cyclic and generalized cyclic central simple algebras, are included here.

math.RA↗
arXiv2026

When isometry and equivalence for skew constacyclic codes coincide

We work in the setting of linear skew constacyclic codes over a commutative base ring $S$. We show that the notions of $(n,σ)$-isometry and $(n,σ)$-equivalence introduced by Ou-azzou et al coincide for most skew $(σ,a)$-constacyclic codes of length $n$. To prove this, we show that all Hamming-weight preserving isomorphisms between their ambient rings which extend some automorphism $τ$ of $S$ that commutes with $σ$ must have degree one, when those rings are not associative. In the process we determine isomorphisms between their nonassociative ambient rings, the Petit rings $S[t;σ]/S[t;σ](t^n-a)$, which give rise to skew constacyclic codes. As a consequence, we propose new definitions of equivalence and isometry of skew constacyclic codes that exactly capture all Hamming-weight preserving isomorphisms between the ambient rings of skew constacyclic codes which extend $τ\in {\rm Aut}(S)$ that commute with $σ$, and lead to tighter classifications.

cs.IT↗
arXiv2026

Using nonassociative algebras to classify skew polycyclic codes up to isometry and equivalence

Employing isomorphisms between their ambient rings, we propose new definitions of equivalence and isometry for skew polycyclic codes that will lead to tighter classifications than existing ones. This reduces the number of previously known isometry and equivalence classes. In the process, we classify classes of skew $(f,σ,δ)$-polycyclic codes with the same performance parameters, to avoid duplicating already existing codes, and state precisely when different notions of equivalence coincide. The generator of a skew polycyclic code is in one-one correspondence with the generator of a principal left ideal in its nonassociative unital ambient ring. By allowing the ambient rings to be nonassociative, we eliminate the need on restrictions on the length of the codes. Ring isomorphisms that preserve the Hamming distance (called isometries) map generators of principal left ideals to generators of principal left ideals and preserve length, dimension, and Hamming distance of the corresponding isometric skew polycyclic codes.

cs.IT↗
arXiv2025

The isotopy classes of Petit division algebras

Let $R=K[t;σ]$ be a skew polynomial ring, where $K$ is a cyclic Galois field extension of degree $n$ with Galois group generated by $σ$. We show that two irreducible similar skew polynomials $f,g\in R$ are similar if and only if they have the same bound. We prove that for two irreducible similar skew polynomials $f,g\in R$ the nonassociative Petit division algebras $R/Rf$ and $R/Rg$ are isotopic. We then refine this result and demonstrate that $f$ and $g$ also yield two isotopic nonassociative Petit algebras $R/Rf$ and $R/Rg$, when the two irreducible polynomials in $F[x]$ that define the minimal central left multiples of $f$ and $g$ have identical degree and lie in the same orbit of some group $G$. For finite field we explicitly compute the upper bound for the number of non-isotopic algebras $R/Rf$ obtained by Lavrauw and Sheekey.

math.RA↗
arXiv2025

A closer look at some cyclic semifields

We show that different choices of generators $σ$ of the Galois group of $\mathbb{F}_{q^n}/\mathbb{F}_{q}$ produce non-isomorphic cyclic semifields $\mathbb{F}_{q^n}[t;σ]/\mathbb{F}_{q^n}[t;σ](t^m-a)$ when $n\geq m-1$: there are thus $φ(n)$ non-isomorphic classes of Sandler semifields $\mathbb{F}_{q^n}[t;σ]/\mathbb{F}_{q^n}[t;σ](t^m-a)$, one class for each generator $σ$ involved in their construction, where $φ$ is the Euler function. We prove that when $n=m$, two Sandler semifields constructed from different generators $σ_1$ and $σ_2$ of ${\rm Gal}(\mathbb{F}_{q^n}/\mathbb{F}_{q})$ are not isotopic. Hence when $n=m$ there are $φ(m)$ non-isotopic classes of these semifields, each class belonging to one choice of generator. We then present a full parametrization of the non-isomorphic Sandler semifields $\mathbb{F}_{q^m}[t;σ]/\mathbb{F}_{q^m}[t;σ](t^m-a)$ , when $m$ is prime and $\mathbb{F}_{q}$ contains a primitive $m$th root of unity. Since for $m=n$, two Sandler semifields constructed from the same generator are isotopic if and only if they are isomorphic, this parametrizes these Sandler semifields up to isotopy, and thus parametrizes both the corresponding non-Desarguesian projective planes, and maximum rank distance codes. Most of our results are proved in all generality for any cyclic Galois field extension.

math.RA↗
arXiv2025

Using cyclic $(f,σ)$-codes over finite chain rings to construct $\mathbb{Z}_p$- and $\mathbb{F}_q[\![t]\!]$-lattices

We construct $\mathbb{Z}_p$-lattices and $\mathbb{F}_q[\![t]\!]$-lattices from cyclic $(f,σ)$-codes over finite chain rings, employing quotients of natural nonassociative orders and principal left ideals in carefully chosen nonassociative algebras. This approach generalizes the classical Construction A that obtains $\mathbb{Z}$-lattices from linear codes over finite fields or commutative rings to the nonassociative setting. We mostly use proper nonassociative cyclic algebras that are defined over field extensions of $p$-adic fields. This means we focus on $σ$-constacyclic codes to obtain $\mathbb{Z}_p$-lattices, hence $\mathbb{Z}_p$-lattice codes. We construct linear maximum rank distance (MRD) codes that are $\mathbb{Z}_p$-lattice codes employing the left multiplication of a nonassociative algebra over a finite chain ring. Possible applications of our constructions include post-quantum cryptography involving $p$-adic lattices, e.g. learning with errors, building rank-metric codes like MRD-codes, or $p$-adic coset coding, in particular wire-tap coding.

math.RA↗
arXiv2024

Colour algebras over rings

Colour algebras over fields of odd characteristic are well-known noncommutative Jordan algebras. We define colour algebras more generally over a unital commutative associative ring with $\frac{1}{2}\in R$, and show that colour algebras can be constructed canonically by employing nondegenerate ternary hermitian forms with trivial determinant. We investigate their structure, automorphism group and derivations. As over fields, colour algebras over $R$ are closely related to octonion algebras over $R$.

math.RA↗
arXiv2024

A closer look at Witt rings for forms of higher degree

Witt rings for nondegenerate forms $φ$ of degree $d$ over a field of characteristic 0 or greater than $d$ were defined by Harrison and Pareigis. We revisit and discuss their definition as well as some special cases, classify the $H$-forms employed in their definition, and define Witt rings of diagonal forms of degree $d$. We also define two new Witt rings for nondegenerate forms $φ$ of degree $d$.

math.RA↗
arXiv2024

A parametrization of nonassociative cyclic algebras of prime degree

We determine and explicitly parametrize the isomorphism classes of nonassociative quaternion algebras over a field of characteristic different from two, as well as the isomorphism classes of nonassociative cyclic algebras of odd prime degree when the base field contains a primitive $m$th root of unity. In the course of doing so, we prove that any two such algebras can be isomorphic only if the cyclic field extension and the chosen generator of the Galois group are the same. As an application, we give a parametrization of nonassociative cyclic algebras of prime degree over a local nonarchimedean field $F$, which is entirely explicit under mild hypotheses on the residual characteristic. In particular, this gives a rich understanding of the important class of nonassociative quaternion algebras up to isomorphism over nonarchimedean local fields.

math.RA↗
arXiv2024

The automorphisms of differential extensions of characteristic $p$

Nonassociative differential extensions are generalizations of associative differential extensions, either of a purely inseparable field extension $K$ of exponent one of a field $F$, $F$ of characteristic $p$, or of a central division algebra over a purely inseparable field extension of $F$. Associative differential extensions are well known central simple algebras first defined by Amitsur and Jacobson. We explicitly compute the automorphisms of nonassociative differential extensions. These are canonically obtained by restricting automorphisms of the differential polynomial ring used in the construction of the algebra. In particular, we obtain descriptions for the automorphisms of associative differential extensions of $D$ and $K$, which are known to be inner.

math.RA↗
arXiv2024

A generalization of the first Tits construction

Let $F$ be a field of characteristic not 2 or 3. The first Tits construction is a well-known tripling process to construct separable cubic Jordan algebras, especially Albert algebras. We generalize the first Tits construction by choosing the scalar employed in the tripling process outside of the base field. This yields a new family of nonassociative unital algebras which carry a cubic map, and maps that can be viewed as generalized adjoint and generalized trace maps. These maps display properties often similar to the ones in the classical setup. In particular, the cubic norm map permits some kind of weak Jordan composition law.

math.RA↗
arXiv2023

Division algebras and MRD codes from skew polynomials

Let $D$ be a division algebra, finite-dimensional over its center, and $R=D[t;σ,δ]$ a skew polynomial ring. Using skew polynomials $f\in R$, we construct division algebras and a generalization of maximum rank distance codes consisting of matrices with entries in a noncommutative division algebra or field. These include a class of codes constructed by Sheekey (in particular, generalized Gabidulin codes), as well as Jha Johnson semifields.

math.RA↗
arXiv2022

Albert's twisted field construction using division algebras with a multiplicative norm

We generalize Albert's twisted field construction, applying it to unital division algebras with a multiplicative norm. We give conditions for the resulting algebras to be division algebras.Four- and eight-dimensional real unital and non-unital division algebras with large derivation algebras are constructed out of Hamilton's quaternion and Cayley's octonion algebra.

math.RA↗
arXiv2022

The eigenspaces of twisted polynomials over cyclic field extensions

Let $K$ be a field and $σ$ an automorphism of $K$ of order $n$.Employing a nonassociative algebra, we study the eigenspace of a bounded skew polynomial $f\in K[t;σ]$. We mainly treat the case that $K/F$ is a cyclic field extension of degree $n$ with Galois group generated by $σ$. We obtain lower bounds on the dimension of the eigenspace, and compute it in special cases as a quotient algebra. Conditions under which a monic polynomial $f\in F[t]\subset K[t;σ]$ is reducible are obtained in special cases.

math.RA↗