arXiv subjects

Daniel Bossaller

Publications and source records attributed to Daniel Bossaller.

3 recordsLinked to original sources

arXiv2025

The trace dual of nonlinear skew cyclic codes

Codes which have a finite field $\mathbb{F}_{q^m}$ as their alphabet but which are only linear over a subfield $\mathbb{F}_q$ are a topic of much recent interest due to their utility in constructing quantum error correcting codes. In this article, we find generators for trace dual spaces of different families of $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^2}$. In particular, given the field extension $\mathbb{F}_q\leq \mathbb{F}_{q^2}$ with $q$ an odd prime power, we determine the trace Euclidean and trace Hermitian dual codes for the general $\mathbb{F}_q$-linear cyclic $\mathbb{F}_{q^2}$-code. In addition, we also determine the trace Euclidean and trace Hermitian duals for general $\mathbb{F}_q$-linear skew cyclic $\mathbb{F}_{q^2}$-codes, which are defined to be left $\mathbb{F}_q[X]$-submodules of $\mathbb{F}_{q^2}[X;\sigma]/(X^n-1)$, where $\sigma$ denotes the Frobenius automorphism and $\mathbb{F}_{q^2}[X;\sigma]$ the induced skew polynomial ring.

cs.IT↗

arXiv2025

Nonlinear Skew Quasi-Cyclic Codes

This article explores nonlinear analogues of skew quasi-cyclic codes of index~$\ell$, i.e., $\mathbb{F}_{q^m}[X;\sigma]$-submodules of $\left(\mathbb{F}_{q^m}[X;\sigma]/(X^n - 1)\right)^\ell$. After introducing nonlinear skew quasi-cyclic codes, we then determine the module structure of these codes by using a two-fold iteration of the Smith normal form of matrices over skew polynomial rings. We show that actually a single use of the Smith normal form will suffice to determine the elementary divisors of the code. Along the way, we also describe duals of our codes with respect to appropriately chosen inner products.

cs.IT↗

arXiv2022

Forcing a Basis into $\aleph_1$-Free Groups

In this paper, we address the question of when a non-free $\aleph_1$-free group $H$ can be be free in a transitive cardinality-preserving model extension. Using the $\Gamma$-invariant, denoted $\Gamma(H)$, we present a necessary and sufficient condition resolving this question for $\aleph_1$-free groups of cardinality $\aleph_1$. Specifically, if $\Gamma(H) = [\aleph_1]$, then $H$ will be free in a transitive model extension if and only if $\aleph_1$ collapses, while for $\Gamma(H) \ne [\aleph_1]$ there exist cardinality-preserving forcings that will add a basis to $H$. In particular, for $\Gamma(H) \neq [\aleph_1]$, we provide a poset $(\mathcal P_{\rm pb}, \leq)$ of partial bases for adding a basis to $H$ without collapsing $\aleph_1$.

math.GR↗