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

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

Locally recoverable codes from automorphism groups of function fields of genus g≥1g \geq 1

A Locally Recoverable Code is a code such that the value of any single coordinate of a codeword can be recovered from the values of a small subset of other coordinates. When we have $\delta$ non overlapping subsets of cardinality $r_i$ that can be used to recover the missing coordinate we say that a linear code $\mathcal{C}$ with length $n$, dimension $k$, minimum distance $d$ has $(r_1,\ldots, r_\delta)$-locality and denote it by $[n, k, d; r_1, r_2,\dots, r_\delta].$ In this paper we provide a new upper bound for the minimum distance of these codes. Working with a finite number of subgroups of cardinality $r_i+1$ of the automorphism group of a function field $\mathcal{F}| \mathbb{F}_q$ of genus $g \geq 1$, we propose a construction of $[n, k, d; r_1, r_2,\dots, r_\delta]$-codes and apply the results to some well known families of function fields

Locally recoverable codes on surfaces

A linear error correcting code is a subspace of a finite-dimensional space over a finite field with a fixed coordinate system. Such a code is said to be locally recoverable with locality $r$ if, for every coordinate, its value at a codeword can be deduced from the value of (certain) $r$ other coordinates of the codeword. These codes have found many recent applications, e.g., to distributed cloud storage. We will discuss the problem of constructing good locally recoverable codes and present some constructions using algebraic surfaces that improve previous constructions and sometimes provide codes that are optimal in a precise sense. The main conceptual contribution of this paper is to consider surfaces fibered over a curve in such a way that each recovery set is constructed from points in a single fiber. This allows us to use the geometry of the fiber to guarantee the local recoverability and use the global geometry of the surface to get a hold on the standard parameters of our codes. We look in detail at situations where the fibers are rational or elliptic curves and provide many examples applying our methods.Comment: Revised version; incorporates suggestions by referee

Locally Recoverable Codes From Algebraic Curves

Locally recoverable (LRC) codes have the property that erased coordinates can be recovered by retrieving a small amount of the information contained in the entire codeword. An LRC code achieves this by making each coordinate a function of a small number of other coordinates. Since some algebraic constructions of LRC codes require that $n \leq q$, where $n$ is the length and $q$ is the size of the field, it is natural to ask whether we can generate codes over a small field from a code over an extension. Trace codes achieve this by taking the field trace of every coordinate of a code. In this thesis, we give necessary and sufficient conditions for when the local recoverability property is retained when taking the trace of certain LRC codes. This thesis also explores a subfamily of LRC codes with hierarchical locality (H-LRC) which have tiers of recoverability. We provide a general construction of codes with 2 levels of hierarchy from maps between algebraic curves and present several families from quotients of curves by a subgroup of automorphisms. We consider specific examples from rational, elliptic, Kummer, and Artin-Schrier curves and examples of asymptotically good families of H-LRC codes from curves related to the Garcia-Stichtenoth tower

Curve-lifted codes for local recovery using lines

In this paper, we introduce curve-lifted codes over fields of arbitrary characteristic, inspired by Hermitian-lifted codes over $\mathbb{F}_{2^r}$. These codes are designed for locality and availability, and their particular parameters depend on the choice of curve and its properties. Due to the construction, the numbers of rational points of intersection between curves and lines play a key role. To demonstrate that and generate new families of locally recoverable codes (LRCs) with high availabilty, we focus on norm-trace-lifted codes. In some cases, they are easier to define than their Hermitian counterparts and consequently have a better asymptotic bound on the code rate.Comment: 22 pages. Comments welcom