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Kathryn Haymaker

Publications and source records attributed to Kathryn Haymaker.

12 recordsLinked to original sources
arXiv2025

Hierarchical Quasi-cyclic Codes from Reed-Solomon and Polynomial Evaluation Codes

We introduce the first example of algebraically constructed hierarchical quasi-cyclic codes. These codes are built from Reed-Solomon codes using a 1964 construction of superimposed codes by Kautz and Singleton. We show both the number of levels in the hierarchy and the index of these Reed-Solomon derived codes are determined by the field size. We show that this property also holds for certain additional classes of polynomial evaluation codes. We provide explicit code parameters and properties as well as some additional bounds on parameters such as rank and distance. In particular, starting with Reed-Solomon codes of dimension $k=2$ yields hierarchical quasi-cyclic codes with Tanner graphs of girth 6. We present a table of small code parameters and note that some of these codes meet the best known minimum distance for binary codes, with the additional hierarchical quasi-cyclic structure. We draw connections to similar constructions in the literature, but importantly, while existing literature on related codes is largely simulation-based, we present a novel algebraic approach to determining new bounds on parameters of these codes.

cs.IT↗
arXiv2024

Metrics on permutations with the same descent set

Let $S_n$ be the symmetric group on the set $[n]:=\{1,2,\ldots,n\}$. Given a permutation $\sigma=\sigma_1\sigma_2 \cdots \sigma_n \in S_n$, we say it has a descent at index $i$ if $\sigma_i>\sigma_{i+1}$. Let $\mathcal{D}(\sigma)$ be the set of all descents of $\sigma$ and define $\mathcal{D}(S;n)=\{\sigma\in S_n\, | \,\mathcal{D}(\sigma)=S\}$. We study the Hamming metric and $\ell_\infty$-metric on the sets $\mathcal{D}(S;n)$ for all possible nonempty $S\subset[n-1]$ to determine the maximum possible value that these metrics can achieve when restricted to these subsets.

math.CO↗
arXiv2024

Hypergraphs of girth 5 and 6 and coding theory

In this paper, we study the maximum number of edges in an $N$-vertex $r$-uniform hypergraph with girth $g$ where $g \in \{5,6 \}$. Writing $\textrm{ex}_r ( N, \mathcal{C}_{<g} )$ for this maximum, it is shown that $\textrm{ex}_r ( N , \mathcal{C}_{ < 5} ) = \Omega_r ( N^{3/2 - o(1)} )$ for $r \in \{4,5,6 \}$. We address an unproved claim from [31] asserting a technique of Ruzsa can be used to show that this lower bound holds for all $r \geq 3$. We carefully explain one of the main obstacles that was overlooked at the time the claim from [31] was made, and show that this obstacle can be overcome when $r\in \{4,5,6\}$. We use constructions from coding theory to prove nontrivial lower bounds that hold for all $r \geq 3$. Finally, we use a recent result of Conlon, Fox, Sudakov, and Zhao to show that the sphere packing bound from coding theory may be improved when upper bounding the size of linear $q$-ary codes of distance $6$.

math.CO↗
arXiv2024

Metrics on permutations with the same peak set

Let $S_n$ be the symmetric group on the set $\{1,2,\ldots,n\}$. Given a permutation $\sigma=\sigma_1\sigma_2 \cdots \sigma_n \in S_n$, we say it has a peak at index $i$ if $\sigma_{i-1}<\sigma_i>\sigma_{i+1}$. Let $\text{Peak}(\sigma)$ be the set of all peaks of $\sigma$ and define $P(S;n)=\{\sigma\in S_n\, | \,\text{Peak}(\sigma)=S\}$. In this paper we study the Hamming metric, $\ell_\infty$-metric, and Kendall-Tau metric on the sets $P(S;n)$ for all possible $S$, and determine the minimum and maximum possible values that these metrics can attain in these subsets of $S_n$.

math.CO↗
arXiv2023

Parity-check Codes from Disjunct Matrices

The matrix representations of linear codes have been well-studied for use as disjunct matrices. However, no connection has previously been made between the properties of disjunct matrices and the parity-check codes obtained from them. This paper makes this connection for the first time. We provide some fundamental results on parity-check codes from general disjunct matrices (in particular, a minimum distance bound). We then consider three specific constructions of disjunct matrices and provide parameters of their corresponding parity-check codes including rate, distance, girth, and density. We show that, by choosing the correct parameters, the codes we construct have the best possible error-correction performance after one round of bit-flipping decoding with regard to a modified version of Gallager's bit-flipping decoding algorithm.

cs.IT↗
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.

cs.IT↗
arXiv2023

Spectral Radii of Arithmetical Structures on Cycle Graphs

Let $G$ be a finite, connected graph. An arithmetical structure on $G$ is a pair of positive integer-valued vectors $(\mathbf{d},\mathbf{r})$ such that $(\text{diag}(\mathbf{d})-A_G)\cdot \mathbf{r}=\textbf{0},$ where the entries of $\mathbf{r}$ have $\gcd$ 1 and $A_G$ is the adjacency matrix of $G$. In this article we find the arithmetical structures that maximize and minimize the spectral radius of $(\text{diag}(\mathbf{d})-A_G)$ among all arithmetical structures on the cycle graph $\mathcal{C}_n.$

math.CO↗
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.

math.NT↗
arXiv2017

Locally recoverable codes from algebraic curves and surfaces

A locally recoverable code is a code over a finite alphabet such that the value of any single coordinate of a codeword can be recovered from the values of a small subset of other coordinates. Building on work of Barg, Tamo, and Vlăduţ, we present several constructions of locally recoverable codes from algebraic curves and surfaces.

cs.IT↗
arXiv2016

The graph structure of graph groups that are subgroups of Thompson's group $V$

We determine exactly which graph products, also known as Right Angled Artin Groups, embed into Richard Thompson's group $V$. It was shown by Bleak and Salazar-Diaz that $\mathbb{Z}^2 * \mathbb{Z}$ was an obstruction. We show that this is the only obstruction. This is shown by proving a graph theory result giving an alternate description of simple graphs without an appropriate induced subgraph.

math.GR↗
arXiv2012

Geometric WOM codes and coding strategies for multilevel flash memories

This paper investigates the design and application of write-once memory (WOM) codes for flash memory storage. Using ideas from Merkx ('84), we present a construction of WOM codes based on finite Euclidean geometries over $\mathbb{F}_2$. This construction yields WOM codes with new parameters and provides insight into the criterion that incidence structures should satisfy to give rise to good codes. We also analyze methods of adapting binary WOM codes for use on multilevel flash cells. In particular, we give two strategies based on different rewrite objectives. A brief discussion of the average-write performance of these strategies, as well as concatenation methods for WOM codes is also provided.

cs.IT↗