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.Comment: 27 pages; A previous lemma was incorrect and has been removed; Corrected computation has identified 18 new such curves (63 in total

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 $2$ Dieudonn\'{e} 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 $2$ Dieudonn\'{e} module. For $m=1$ and $2$, we determine the complete structure of the mod $2$ Dieudonn\'{e} module

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 $p$

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

A Graph Theoretic Approach for Generating Hypotheses About Phonetic Cues in Speech

Current models of speech perception suggest that combining acoustic cues and factoring out contextual variability allows listeners to recognize speech across different talkers. However, it remains unclear which specific cues are necessary and how their use varies between individual talkers. We use graph theoretic techniques to address these problems by constructing networks connecting talkers and possible cues. We identify subgraphs (Steiner trees) that connect talkers via cues consistently used to indicate specific phonemes. Classifiers trained on these cues match listeners\u27 data better than those trained on all cues, suggesting that Steiner trees can identify the cues necessary for speech recognition

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. Theses 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 construct codes with hierarchical locality from a geometric perspective, using fiber products of curves. We demonstrate how the constructed hierarchical codes can be viewed as punctured subcodes of Reed-Muller codes. This point of view provides natural structures for local recovery at each level in the hierarchy

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