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Beth Malmskog

Publications and source records attributed to Beth Malmskog.

At least 19 recordsLinked to original sources
arXiv2026

Simulating Single Transferable Voting for the Colorado House of Representatives

Social choice theory research demonstrates that single transferable voting (STV) results in more proportionally representative legislative bodies. We aim to understand how using multi-member districts and ranked ballots with STV would affect the representation of political parties in the Colorado House of Representatives. We investigated this objective by producing 10,000 multi-member districting plans of Colorado, generating ranked ballots for each of these plans using returns from the 2022 Colorado attorney general race, and simulating STV using these ballots. Our simulated STV elections for the Colorado House of Representatives gave more proportional representation for Democrats and Republicans than the current first-past-the-post system. Future research should explore how the implementation of STV would influence the representation of racial and language minority groups in the Colorado General Assembly to provide guidance on electoral reform in the state.

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arXiv2026

Computing Invariants of Artin-Schreier Curves

We present an algorithmic framework for computing generators for the ring of invariants of an Artin-Schreier curve. We give explicit invariants for almost all Artin-Schreier curves of genus up to~8 in standard form, and for a handful of curves of higher genus.

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arXiv2026

Hierarchical Locally Recoverable Codes on surfaces

We construct locally recoverable codes with hierarchy from surfaces in $\mathbb{A}^3$ admitting a fibration by curves of Artin-Schreier or Kummer type. We derive the parameters of our codes by leveraging the geometry and arithmetic of the fibration, which is obtained by projection onto one of the coordinates. As a byproduct, we obtain estimates for (and in one case an explicit count of) the number of rational points in certain families of surfaces.

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arXiv2024

Codes with Hierarchical Locality on Artin-Schreier Surfaces

In this article, we construct codes with hierarchical locality using natural geometric structures in Artin-Schreier surfaces of the form $y^p-y=f(x,z)$. Our main theorem describes the codes, their hierarchical structure and recovery algorithms, and gives parameters. We also develop a family of examples using codes defined over $\mathbb{F}_{p^2}$ on the surface $y^p-y=x^{p+1}z^2+x^2z^{p+1}$. We use elementary methods to count the $\mathbb{F}_{p^2}$-rational points on the surface, enabling us to provide explicit hierarchical parameters and a better bound on minimum distance for these codes. An additional example and some generalizations are also considered.

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arXiv2024

On invariants of Artin-Schreier curves

The main goal of this article is to expand the theory of invariants of Artin-Schreier curves by giving a complete classification in genus 3 and 4. To achieve this goal, we first establish standard forms of Artin-Schreier curves and determine all isomorphisms between curves in this form. We then compute reconstructing systems of invariants for curves in each connected component of the strata of the moduli spaces for Artin-Schreier curves of genus 3 and 4 for $p>2$.

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arXiv2023

Algebraic hierarchical locally recoverable codes with nested affine subspace recovery

Codes with locality, also known as locally recoverable codes, allow for recovery of erasures using proper subsets of other coordinates. These subsets are typically of small cardinality to promote recovery using limited network traffic and other resources. Hierarchical locally recoverable codes allow for recovery of erasures using sets of other symbols whose sizes increase as needed to allow for recovery of more symbols. In this paper, we describe a hierarchical recovery structure arising from geometry in Reed-Muller codes and codes with availability from fiber products of curves. We demonstrate how the fiber product hierarchical codes can be viewed as punctured subcodes of Reed-Muller codes, uniting the two constructions. This point of view provides natural structures for local recovery with availability at each level in the hierarchy.

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arXiv2023

Lower Rate Bounds for Hermitian-Lifted Codes for Odd Prime Characteristic

Locally recoverable codes are error correcting codes with the additional property that every symbol of any codeword can be recovered from a small set of other symbols. This property is particularly desirable in cloud storage applications. A locally recoverable code is said to have availability $t$ if each position has $t$ disjoint recovery sets. Hermitian-lifted codes are locally recoverable codes with high availability first described by Lopez, Malmskog, Matthews, Pi\~nero-Gonzales, and Wootters. The codes are based on the well-known Hermitian curve and incorporate the novel technique of lifting to increase the rate of the code. Lopez et al. lower bounded the rate of the codes defined over fields with characteristic 2. This paper generalizes their work to show that the rate of Hermitian-lifted codes is bounded below by a positive constant depending on $p$ when $q=p^l$ for any odd prime $p$.

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arXiv2022

Minimum Distance and Parameter Ranges of Locally Recoverable Codes with Availability from Fiber Products of Curves

We construct families of locally recoverable codes with availability $t\geq 2$ using fiber products of curves, determine the exact minimum distance of many families, and prove a general theorem for minimum distance of such codes. The paper concludes with an exploration of parameters of codes from these families and the fiber product construction more generally. We show that fiber product codes can achieve arbitrarily large rate and arbitrarily small relative defect, and compare to known bounds and important constructions from the literature.

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arXiv2021

Colorado in Context: Congressional Redistricting and Competing Fairness Criteria in Colorado

In this paper, we apply techniques of ensemble analysis to understand the political baseline for Congressional representation in Colorado. We generate a large random sample of reasonable redistricting plans and determine the partisan balance of each district using returns from state-wide elections in 2018, and analyze the 2011/2012 enacted districts in this context. Colorado recently adopted a new framework for redistricting, creating an independent commission to draw district boundaries, prohibiting partisan bias and incumbency considerations, requiring that political boundaries (such as counties) be preserved as much as possible, and also requiring that mapmakers maximize the number of competitive districts. We investigate the relationships between partisan outcomes, number of counties which are split, and number of competitive districts in a plan. This paper also features two novel improvements in methodology--a more rigorous statistical framework for understanding necessary sample size, and a weighted-graph method for generating random plans which split approximately as few counties as acceptable human-drawn maps.

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arXiv2020

A robust implementation for solving the $S$-unit equation and several applications

Let $K$ be a number field, and $S$ a finite set of places in $K$ containing all infinite places. We present an implementation for solving the $S$-unit equation $x + y = 1$, $x,y \in\mathscr{O}_{K,S}^\times$ in the computer algebra package SageMath. This paper outlines the mathematical basis for the implementation. We discuss and reference the results of extensive computations, including exponent bounds for solutions in many fields of small degree for small sets $S$. As an application, we prove an asymptotic version of Fermat's Last Theorem for totally real cubic number fields with bounded discriminant where 2 is totally ramified. In addition, we use the implementation to find all solutions to some cubic Ramanujan-Nagell equations.

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arXiv2020

Hermitian-Lifted Codes

In this paper, we construct codes for local recovery of erasures with high availability and constant-bounded rate from the Hermitian curve. These new codes, called Hermitian-lifted codes, are evaluation codes with evaluation set being the set of $\mathbb{F}_{q^2}$-rational points on the affine curve. The novelty is in terms of the functions to be evaluated; they are a special set of monomials which restrict to low degree polynomials on lines intersected with the Hermitian curve. As a result, the positions corresponding to points on any line through a given point act as a recovery set for the position corresponding to that point.

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arXiv2018

Locally Recoverable Codes with Availability $t\geq 2$ from Fiber Products of Curves

We generalize the construction of locally recoverable codes on algebraic curves given by Barg, Tamo and Vlăduţ to those with arbitrarily many recovery sets by exploiting the structure of fiber products of curves. Employing maximal curves, we create several new families of locally recoverable codes with multiple recovery sets, including codes with two recovery sets from the generalized Giulietti and Korchmáros (GK) curves and the Suzuki curves, and new locally recoverable codes with many recovery sets based on the Hermitian curve, using a fiber product construction of van der Geer and van der Vlugt. In addition, we consider the relationship between local error recovery and global error correction as well as the availability required to locally recover any pattern of a fixed number of erasures.

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arXiv2018

The de Rham cohomology of the Suzuki curves

For a natural number $m$, let $\mathcal{S}_m/\mathbb{F}_2$ be the $m$th Suzuki curve. We study the mod $2$ Dieudonné module of $\mathcal{S}_m$, which gives the equivalent information as the Ekedahl-Oort type or the structure of the $2$-torsion group scheme of its Jacobian. We accomplish this by studying the de Rham cohomology of $\mathcal{S}_m$. For all $m$, we determine the structure of the de Rham cohomology as a $2$-modular representation of the $m$th Suzuki group and the structure of a submodule of the mod $2$ Dieudonné module. For $m=1$ and $2$, we determine the complete structure of the mod $2$ Dieudonné module.

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arXiv2017

Variations of the McEliece Cryptosystem

Two variations of the McEliece cryptosystem are presented. The first one is based on a relaxation of the column permutation in the classical McEliece scrambling process. This is done in such a way that the Hamming weight of the error, added in the encryption process, can be controlled so that efficient decryption remains possible. The second variation is based on the use of spatially coupled moderate-density parity-check codes as secret codes. These codes are known for their excellent error-correction performance and allow for a relatively low key size in the cryptosystem. For both variants the security with respect to known attacks is discussed.

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arXiv2016

Picard curves over Q with good reduction away from 3

Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over Q with good reduction away from 3, up to Q-isomorphism. A correspondence between the isomorphism classes of such curves and certain quintic binary forms possessing a rational linear factor is established. An exhaustive list of integral models is determined, and an application to a question of Ihara is discussed.

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arXiv2015

Zeta functions of a class of Artin-Schreier curves with many automorphisms

This paper describes a class of Artin-Schreier curves, generalizing results of Van der Geer and Van der Vlugt to odd characteristic. The automorphism group of these curves contains a large extraspecial group as a subgroup. Precise knowledge of this subgroup makes it possible to compute the zeta functions of the curves in the class over the field of definition of all automorphisms in the subgroup. As a consequence, we obtain new examples of maximal curves.

math.AG↗
arXiv2012

The Automorphism Groups of a Family of Maximal Curves

The Hasse Weil bound restricts the number of points of a curve which are defined over a finite field; if the number of points meets this bound, the curve is called maximal. Giulietti and Korchmaros introduced a curve C_3 which is maximal over F_{q^6} and determined its automorphism group. Garcia, Guneri, and Stichtenoth generalized this construction to a family of curves C_n, indexed by an odd integer n greater than or equal to 3, such that C_n is maximal over F_{q^{2n}}. In this paper, we determine the automorphism group Aut(C_n) when n > 3; in contrast with the case n=3, it fixes the point at infinity on C_n. The proof requires a new structural result about automorphism groups of curves in characteristic p such that each Sylow p-subgroup has exactly one fixed point. MSC:11G20, 14H37.

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