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

PIR Array Codes with Optimal Virtual Server Rate

There has been much recent interest in Private information Retrieval (PIR) in models where a database is stored across several servers using coding techniques from distributed storage, rather than being simply replicated. In particular, a recent breakthrough result of Fazelli, Vardy and Yaakobi introduces the notion of a PIR code and a PIR array code, and uses this notion to produce efficient PIR protocols. In this paper we are interested in designing PIR array codes. We consider the case when we have $m$ servers, with each server storing a fraction $(1/s)$ of the bits of the database; here $s$ is a fixed rational number with $s > 1$. A PIR array code with the $k$-PIR property enables a $k$-server PIR protocol (with $k\leq m$) to be emulated on $m$ servers, with the overall storage requirements of the protocol being reduced. The communication complexity of a PIR protocol reduces as $k$ grows, so the virtual server rate, defined to be $k/m$, is an important parameter. We study the maximum virtual server rate of a PIR array code with the $k$-PIR property. We present upper bounds on the achievable virtual server rate, some constructions, and ideas how to obtain PIR array codes with the highest possible virtual server rate. In particular, we present constructions that asymptotically meet our upper bounds, and the exact largest virtual server rate is obtained when $1 < s \leq 2$. A $k$-PIR code (and similarly a $k$-PIR array code) is also a locally repairable code with symbol availability $k-1$. Such a code ensures $k$ parallel reads for each information symbol. So the virtual server rate is very closely related to the symbol availability of the code when used as a locally repairable code. The results of this paper are discussed also in this context, where subspace codes also have an important role

Locally repairable convertible codes with optimal access costs

Modern large-scale distributed storage systems use erasure codes to protect against node failures with low storage overhead. In practice, the failure rate and other factors of storage devices in the system may vary significantly over time, and leads to changes of the ideal code parameters. To maintain the storage efficiency, this requires the system to adjust parameters of the currently used codes. The changing process of code parameters on encoded data is called code conversion. As an important class of storage codes, locally repairable codes (LRCs) can repair any codeword symbol using a small number of other symbols. This feature makes LRCs highly efficient for addressing single node failures in the storage systems. In this paper, we investigate the code conversions for locally repairable codes in the merge regime. We establish a lower bound on the access cost of code conversion for general LRCs and propose a general construction of LRCs that can perform code conversions with access cost matching this bound. This construction provides a family of LRCs together with optimal conversion process over the field of size linear in the code length.Comment: 25 page

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