Erasure codes play an important role in storage systems to prevent data loss. In this work, we study a class of erasure codes called Multi-Erasure Locally Recoverable Codes (ME-LRCs) for storage arrays. Compared to previous related works, we focus on the construction of ME-LRCs over small fields. We first develop upper and lower bounds on the minimum distance of ME-LRCs. Our main contribution is to propose a general construction of ME-LRCs based on generalized tensor product codes, and study their erasure-correcting properties. A decoding algorithm tailored for erasure recovery is given, and correctable erasure patterns are identified. We then prove that our construction yields optimal ME-LRCs with a wide range of code parameters, and present some explicit ME-LRCs over small fields. Finally, we show that generalized integrated interleaving (GII) codes can be treated as a subclass of generalized tensor product codes, thus defining the exact relation between these codes.
Streaming codes are a class of packet-level erasure codes that are designed with the goal of ensuring recovery in low-latency fashion, of erased packets over a communication network. It is well-known in the streaming code literature, that diagonally embedding codewords of a $[τ+1,τ+1-a]$ Maximum Distance Separable (MDS) code within the packet stream, leads to rate-optimal streaming codes capable of recovering from $a$ arbitrary packet erasures, under a strict decoding delay constraint $τ$. Thus MDS codes are geared towards the efficient handling of the worst-case scenario corresponding to the …
Maximally recoverable codes with locality and availability
IMUVa-Mathematics Research Institute, University of Valladolid（瓦拉多利德大学数学研究所）；Signal Processing and Communications Research Center (SPCRC), International Institute of Information Technology, Hyderabad（海得拉巴国际信息学院信号处理与通信研究中心）
Locally recoverable codes are widely used in distributed and cloud storage systems. The objective of this paper is to present a construction of near MDS codes with oval polynomials and then determine the locality of the codes. It turns out that the near MDS codes and their duals are both distance-optimal and dimension-optimal locally recoverable codes. The lengths of the locally recoverable codes are different from known ones in the literature.
Maximally recoverable codes are a class of codes which recover from all potentially recoverable erasure patterns given the locality constraints of the code. In earlier works, these codes have been studied in the context of codes with locality. The notion of locality has been extended to hierarchical locality, which allows for locality to gradually increase in levels with the increase in the number of erasures. We consider the locality constraints imposed by codes with two-level hierarchical locality and define maximally recoverable codes with data-local and local hierarchical locality. We deri…
The traditional definition of Integrated Interleaved (II) codes generally assumes that the component nested codes are either Reed-Solomon (RS) or shortened Reed-Solomon codes. By taking general classes of codes, we present a recursive construction of Extended Integrated Interleaved (EII) codes into multiple layers, a problem that brought attention in literature for II codes. The multiple layer approach allows for a hierarchical scheme where each layer of the code provides for a different locality. In particular, we present the erasure-correcting capability of the new codes and we show that the…
An $(n,r,h,a,q)$-Local Reconstruction Code (LRC) is a linear code over $\mathbb{F}_q$ of length $n$, whose codeword symbols are partitioned into $n/r$ local groups each of size $r$. Each local group satisfies `$a$' local parity checks to recover from `$a$' erasures in that local group and there are further $h$ global parity checks to provide fault tolerance from more global erasure patterns. Such an LRC is Maximally Recoverable (MR), if it offers the best blend of locality and global erasure resilience -- namely it can correct all erasure patterns whose recovery is information-theoretically fe…
An $(m,n,a,b)$-tensor code consists of $m\times n$ matrices whose columns satisfy `$a$' parity checks and rows satisfy `$b$' parity checks (i.e., a tensor code is the tensor product of a column code and row code). Tensor codes are useful in distributed storage because a single erasure can be corrected quickly either by reading its row or column. Maximally Recoverable (MR) Tensor Codes, introduced by Gopalan et al., are tensor codes which can correct every erasure pattern that is information theoretically possible to correct. The main questions about MR Tensor Codes are characterizing which era…
A linear code with parameters $[n,k,n-k]$ is said to be almost maximum distance separable (AMDS for short). An AMDS code whose dual is also AMDS is referred to as an near maximum distance separable (NMDS for short) code. NMDS codes have nice applications in finite geometry, combinatorics, cryptography and data storage. In this paper, we first present several constructions of NMDS codes and determine their weight enumerators. In particular, some constructions produce NMDS codes with the same parameters but different weight enumerators. Then we determine the locality of the NMDS codes and obtain…
Locally Recoverable Codes with Availability $t\geq 2$ from Fiber Products of Curves
本文利用曲线的纤维积的结构，将Barg，Tamo和Vlaud Duffel给出的代数曲线上的局部可恢复码的构造推广到具有任意多个恢复集的代数曲线上.采用极大曲线，我们创建了几个新的家庭的本地可恢复的代码与多个恢复集，包括代码与两个恢复集的广义Giulietti和Kommarros（GK）曲线和铃木曲线，和新的本地可恢复的代码与许多恢复集的基础上的Hermitian曲线，使用一个纤维积建设的van der Geer和van der Vlugt。此外，我们考虑局部误差恢复和全局误差校正之间的关系，以及本地恢复任何固定数量的擦除模式所需的可用性。
It was shown in \cite{GXY18} that the length $n$ of a $q$-ary linear locally recoverable code with distance $d\ge 5$ is upper bounded by $O(dq^3)$. Thus, it is a challenging problem to construct $q$-ary locally recoverable codes with distance $d\ge 5$ and length approaching the upper bound. The paper \cite{GXY18} also gave an algorithmic construction of $q$-ary locally recoverable codes with locality $r$ and length $n=Ω_r(q^2)$ for $d=5$ and $6$, where $Ω_r$ means that the implicit constant depends on locality $r$. In the present paper, we present an explicit construction of $q$-ary locally re…
Maximally recoverable local reconstruction codes (MR LRCs for short) have received great attention in the last few years. Various constructions have been proposed in literatures. The main focus of this topic is to construct MR LRCs over small fields. An $(N=nr,r,h,\Gd)$-MR LRC is a linear code over finite field $\F_\ell$ of length $N$, whose codeword symbols are partitioned into $n$ local groups each of size $r$. Each local group can repair any $\Gd$ erasure errors and there are further $h$ global parity checks to provide fault tolerance from more global erasure patterns. MR LRCs deployed in p…
In this work we construct sequences of locally recoverable AG codes arising from a tower of function fields and give bound for the parameters of the obtained codes. In a particular case of a tower over $\mathbb{F}_{q^2}$ for any odd $q$, defined by Garcia and Stichtenoth in [GS2007], we show that the bound is sharp for the first code in the sequence, and we include a detailed analysis for the following codes in the sequence based on the distribution of rational places that split completely in the considered function field extension.
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.
Hierarchical Locally Recoverable Codes on surfaces
IMPA（巴西数学纯应用与计算研究所）；Colorado College Department of Mathematics and Computer Science（科罗拉多学院数学与计算机科学系）；Department of Mathematics and Statistics, Oakland University（奥克兰大学数学与统计系）；Faculty of Science and Engineering - Bernoulli Institute, University of Groningen（格罗宁根大学理学院伯努利研究所）
