Han Cai, Ying Miao, Moshe Schwartz, Xiaohu Tang
We construct maximally recoverable codes (corresponding to partial MDS codes) which are based on linearized Reed-Solomon codes. The new codes have a smaller field size requirement compared with known constructions. For certain asymptotic regimes, the constructed codes have order-optimal alphabet size, asymptotically matching the known lower bound.
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Explicit Maximally Recoverable Codes with Locality
考虑系统线性码，其中一些（局部）奇偶校验符号取决于几个规定符号，而其他（重）奇偶校验符号可以取决于所有数据符号。本地奇偶校验允许在擦除任何单个符号时快速恢复该符号，而重奇偶校验则允许大量同时擦除。如果上面的代码校正了所有擦除模式，则该代码是最大可恢复的，这些擦除模式在给定代码拓扑的情况下是理论上可恢复的信息。在本文中，我们提出了明确的家庭的最大可恢复代码的地方。我们还开始研究最大可恢复性和字母表大小之间的权衡。
In this letter, locally recoverable codes with maximal recoverability are studied with a focus on identifying the MDS codes resulting from puncturing and shortening. By using matroid theory and the relation between MDS codes and uniform minors, the list of all the possible uniform minors is derived. This list is used to improve the known non-asymptotic lower bound on the required field size of a maximally recoverable code.
On Partial Maximally-Recoverable and Maximally-Recoverable Codes
一个[n，k]线性码C受到由奇偶检验矩阵H0施加的局部性约束，如果它可以从H0的零空间的某个k维子码可以从其恢复的任何擦除模式中恢复，则该线性码C被称为最大可恢复（MR）码。本文的重点是限制MR码具有全符号局部性r。鉴于它是具有挑战性的，以构建MR码具有小字段大小，我们目前的结果在两个方向。在第一个，我们放松MR约束，只需要除了要求是一个最佳的所有符号的局部代码，代码必须产生MDS码时，打孔在一个单一的，特定的模式，确保每个本地代码被打孔正好在一个坐标，没有两个本地代码共享相同的打孔坐标。我们将这些代码称为部分最大可恢复（PMR）代码。我们提供了一个简单的高速率PMR码的建设，然后提供了一个普遍的，有前途的方法，需要进一步调查。在第二个方向，我们提出了三个结构的MR码与改进的参数，主要是规模的有限域中采用的建设
We construct a family of linear maximally recoverable codes with locality $r$ and dimension $r+1.$ For codes of length $n$ with $r\approx n^α, 0\leα\le 1$ the code alphabet is of the order $n^{1+3α},$ which improves upon the previously known constructions of maximally recoverable codes.
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…
Weight Enumerators and Higher Support Weights of Maximally Recoverable Codes
本文建立了数据局部和局部最大可恢复码（MRC）对应的拟阵结构。这些码的拟阵结构可以用来确定相关的Tutte多项式。Greene证明了任何码的权枚举数都可以从其相关的Tutte多项式中确定。我们将使用这个结果来推导出显式表达式的权重枚举的数据本地和本地MRC。此外，Britz证明了任何代码的更高支持权重可以从其相关的Tutte多项式中确定。我们将使用这个结果来推导出表达式的数据本地和本地MRC与两个本地代码的更高的支持权重。
In recent years, the rapidly increasing amounts of data created and processed through the internet resulted in distributed storage systems employing erasure coding based schemes. Aiming to balance the tradeoff between data recovery for correlated failures and efficient encoding and decoding, distributed storage systems employing maximally recoverable codes came up. Unifying a number of topologies considered both in theory and practice, Gopalan et al. \cite{Gopalan2017} initiated the study of maximally recoverable codes for grid-like topologies. In this paper, we focus on the maximally recovera…
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…
Design of MDP Convolutional Codes and Maximally Recoverable Codes Through the Lens of Matrix Completion
本文通过矩阵补全框架研究具有最大可恢复性和最大距离谱的卷积码，提出通用构造方法，强调生成矩阵的稀疏性和子域元素特性。
Improved Constructions and Lower Bounds for Maximally Recoverable Grid Codes
本文提出改进的极大可恢复网格码构造，适用于大$n$和常数$m,h$的场景，提供多项式域大小的构造和下界。
Secure Storage using Maximally Recoverable Locally Repairable Codes
针对 MR-LRC 在全局修复中易被窃听的问题，提出基于 CPU 与局部多项式的安全修复框架，分析直接修复和转发修复并实现非零保密维度。
Maximally recoverable codes with locality and availability
本文提出具有局部性和可用性的最大可恢复码，通过局部可修复码的构造方法，减少存储开销并提升全局纠错能力，针对t=1时提供三种构造方法并扩展有限域大小下界。
Given a topology of local parity-check constraints, a maximally recoverable code (MRC) can correct all erasure patterns that are information-theoretically correctable. In a grid-like topology, there are $a$ local constraints in every column forming a column code, $b$ local constraints in every row forming a row code, and $h$ global constraints in an $(m \times n)$ grid of codeword. Recently, Gopalan et al. initiated the study of MRCs under grid-like topology, and derived a necessary and sufficient condition, termed as the regularity condition, for an erasure pattern to be recoverable when $a=1…
Partial-MDS (PMDS) codes are a family of locally repairable codes, mainly used for distributed storage. They are defined to be able to correct any pattern of $s$ additional erasures, after a given number of erasures per locality group have occurred. This makes them also maximally recoverable (MR) codes, another class of locally repairable codes. It is known that MR codes in general, and PMDS codes in particular, exist for any set of parameters, if the field size is large enough. Moreover, some explicit constructions of PMDS codes are known, mostly (but not always) with a strong restriction on …
The explosion in the volumes of data being stored online has resulted in distributed storage systems transitioning to erasure coding based schemes. Yet, the codes being deployed in practice are fairly short. In this work, we address what we view as the main coding theoretic barrier to deploying longer codes in storage: at large lengths, failures are not independent and correlated failures are inevitable. This motivates designing codes that allow quick data recovery even after large correlated failures, and which have efficient encoding and decoding. We propose that code design for distributed …
We consider a communication problem in which an update of the source message needs to be conveyed to one or more distant receivers that are interested in maintaining specific linear functions of the source message. The setting is one in which the updates are sparse in nature, and where neither the source nor the receiver(s) is aware of the exact {\em difference vector}, but only know the amount of sparsity that is present in the difference-vector. Under this setting, we are interested in devising linear encoding and decoding schemes that minimize the communication cost involved. We show that t…
A construction of Maximally Recoverable LRCs for small number of local groups
本文提出一种在较小有限域上构造最大可恢复局部重建码（MR LRCs）的新方法，推广了Hu和Yekhanin的构造，适用于局部组数量较少的实际场景，提高了容错与存储效率。
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…
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…
The explosion in the volumes of data being stored online has resulted in distributed storage systems transitioning to erasure coding based schemes. Local Reconstruction Codes (LRCs) have emerged as the codes of choice for these applications. These codes can correct a small number of erasures by accessing only a small number of remaining coordinates. An $(n,r,h,a,q)$-LRC is a linear code over $\mathbb{F}_q$ of length $n$, whose codeword symbols are partitioned into $g=n/r$ local groups each of size $r$. It has $h$ global parity checks and each local group has $a$ local parity checks. Such an LR…
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.
Local Reconstruction Codes (LRCs) allow for recovery from a small number of erasures in a local manner based on just a few other codeword symbols. A maximally recoverable (MR) LRC offers the best possible blend of such local and global fault tolerance, guaranteeing recovery from all erasure patterns which are information-theoretically correctable given the presence of local recovery groups. In an $(n,r,h,a)$-LRC, the $n$ codeword symbols are partitioned into $r$ disjoint groups each of which include $a$ local parity checks capable of locally correcting $a$ erasures. MR LRCs have received much …
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…
A Construction of Optimal Quasi-cyclic Locally Recoverable Codes using Constituent Codes
本文将准循环局部可恢复码分解为扩域上的组成码之和，并给出参数条件，用于构造维数和码长更大的近乎最优及最优码。
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…
Three Stories on a Two-sided Coin: Index Coding, Locally Recoverable Distributed Storage, and Guessing Games on Graphs
三个科学和工程问题最近的利益-索引编码，本地可恢复的分布式存储，和猜测游戏的图形-进行了讨论，并阐明其最佳解决方案之间的连接。通过推广Shanmugam和Dimakis以及Mazumdar关于有向图上的索引编码问题的最优广播速率与同一图上的局部可恢复分布式存储问题的归一化速率之间的互补性的最新结果，证明了这两个问题的容量区域和最优速率区域是互补的.建立这一结果的主要成分是Alon等人（2008）引入的混淆图的概念，混淆图的顶点传递性，通过混淆图的分数色数表征索引编码容量区域，以及通过混淆图的独立数表征局部可恢复分布式存储的最佳速率区域。作为互补性的第三个也是最后一个方面，Riis在图上的猜测游戏被讨论为局部可恢复分布式存储问题的特殊情况，并且表明猜测游戏的最优策略的获胜概率以及最优策略和随机猜测的获胜概率之间的比率可以分别被表征，通过索引编码的容量区域和分布式存储的最佳速率区域。
On a Duality Between Recoverable Distributed Storage and Index Coding
在本文中，我们介绍了一个模型的单故障本地可恢复的分布式存储系统。这个模型似乎引起了一个问题，似乎双重的研究良好的索引编码问题。本文建立了模型的最优索引码和最优分布存储码的维数之间的关系。我们还展示了向量码的一些扩展。
Recently, it was discovered by several authors that a $q$-ary optimal locally recoverable code, i.e., a locally recoverable code archiving the Singleton-type bound, can have length much bigger than $q+1$. This is quite different from the classical $q$-ary MDS codes where it is conjectured that the code length is upper bounded by $q+1$ (or $q+2$ for some special case). This discovery inspired some recent studies on length of an optimal locally recoverable code. It was shown in \cite{LXY} that a $q$-ary optimal locally recoverable code is unbounded for $d=3,4$. Soon after, it was proved that a $…
Improved bounds and optimal constructions of pure quantum locally recoverable codes
本文提出改进的纯量子局部可恢复码界限，并通过经典QECCs构造了最优qLRCs的无限族。
We present simple constructions of optimal erasure-correcting LRC codes by exhibiting their parity-check matrices. When the number of local parities in a parity group plus the number of global parities is smaller than the size of the parity group, the constructed codes are optimal with a field of size at least the length of the code. We can reduce the size of the field to at least the size of the parity groups when the number of global parities equals the number of local parities in a parity group plus one.
Aiming to recover the data from several concurrent node failures, linear $r$-LRC codes with locality $r$ were extended into $(r, δ)$-LRC codes with locality $(r, δ)$ which can enable the local recovery of a failed node in case of more than one node failure. Optimal LRC codes are those whose parameters achieve the generalized Singleton bound with equality. In the present paper, we are interested in studying optimal LRC codes over small fields and, more precisely, over $\mathbb{F}_4$. We shall adopt an approach by investigating optimal quaternary $(r,δ)$-LRC codes through their parity-check matr…
In this paper we construct new optimal hierarchical locally recoverable codes. Our construction is based on a combination of the ideas of \cite{ballentine2019codes,sasidharan2015codes} with an algebraic number theoretical approach that allows to give a finer tuning of the minimum distance of the intermediate code (allowing larger dimension of the final code), and to remove restrictions on the arithmetic properties of $q$ compared with the size of the locality sets in the hierarchy. In turn, we manage to obtain codes with a wide set of parameters both for the size $q$ of the base field, and for…
A family of optimal locally recoverable codes
如果编码中的每个符号都是少量（最多$r$）其他符号的函数，则有限字母表上的代码被称为局部可恢复（LRC）。我们提出了一个家庭的LRC码，达到最大可能的值的距离为一个给定的局部参数和代码基数。码字作为有限域上的特殊构造的多项式的评估而获得，并且如果局部参数$r$被设置为等于码维，则码字减少到Reed-Solomon码。大多数参数的代码字母表的大小仅略大于代码长度。恢复过程通过$r$点上的多项式插值来执行。我们还构造了几个不相交的恢复集的每一个符号的代码。这种结构使系统能够通过访问码字的不同部分来对特定符号进行多个独立且同时的恢复过程。此属性使频繁访问的数据（“热数据”）具有高可用性。
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…
Beyond Polynomials: Optimal Locally Recoverable Codes from Good Rational Functions
本文提出通过引入良好有理函数来扩展局部可恢复码的构造方法，利用代数函数域理论和伽罗瓦理论，构建了更优的LRC码家族。
Entanglement-Assisted Quantum Locally Recoverable Codes: Characterizations, Bounds, and Constructions
该研究针对量子局部可恢复码的约束限制，引入纠缠辅助量子局部可恢复码（EAQLRCs），建立其界与构造框架，得到首批明确的EAQLRCs最优码族。
Optimal pure quantum $(r,δ)$-locally recoverable codes from matrix-product construction
本文针对量子存储的删除错误修复需求，以矩阵积码为基础，构造出满足类Singleton界的最优纯量子$(r,δ)$-局部可修复码族，为量子存储系统提供了高效的纠错方案。
Bounds and Constructions of Quantum Locally Recoverable Codes from Quantum CSS Codes
本文借助经典 LRC 推导量子 LRC 的参数界并刻画最优纯量子 LRC，进而利用经典 LRC 构造量子 CSS 码，首次得到多族最优纯量子 LRC。
Let $q$ be a prime power and $\mathbb F_q$ be the finite field of size $q$. In this paper we provide a Galois theoretical framework that allows to produce good polynomials for the Tamo and Barg construction of optimal locally recoverable codes (LRC). Using our approach we construct new good polynomials and then optimal LRCs with new parameters. The existing theory of good polynomials fits entirely in our new framework. The key advantage of our method is that we do not need to rely on arithmetic properties of the pair $(q,r)$, where $r$ is the locality of the code.
A class of locally recoverable codes over finite chain rings
本文研究有限链环上的局部可恢复码，通过扩展Tamo和Barg在2014年提出的构造，提出了一类新的局部可恢复码并讨论其最优性，推广了域上码的已知结果和界。
Linear-time list recovery of high-rate expander codes
我们表明，扩展代码，当适当的实例化，是高速率的列表可恢复代码与线性时间列表恢复算法。最近，列表可恢复码在构造有效的列表可解码码以及用于压缩感知和组测试的矩阵的显式构造中是有用的。以前的列表可恢复代码与线性时间解码算法都有率最多为1/2，相比之下，我们的代码可以有率$1 - ε$的任何$ε> 0$。我们可以将我们的高速率代码插入Meir（2014）的构造中，以获得任意速率的线性时间列表可恢复代码，其接近所提供的非平凡列表的数量与代码的速率之间的最佳权衡。虽然列表恢复本身很有趣，但我们的主要动机是应用于列表解码。稍微加强我们的结果将意味着线性时间和最佳列表解码码的所有速率，我们的工作是解决这个重要问题的方向迈出了一步。
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 flash memory array. 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. These bounds explicitly take the field size into account. Our main contribution is to propose a general construction of ME-LRCs based on generalized tensor product codes, and study their erasure-correcting property. A deco…
Random Reed-Solomon Codes are List Recoverable with Optimal List Size
本文证明随机评估点的 Reed-Solomon 码在任意输入列表大小下可实现达到容量且输出列表大小最优的列表可恢复性，超越了列表恢复 Johnson 界，扩展了随机 RS 码列表译码的近期研究。
Locally recoverable algebro-geometric codes from projective bundles
本文提出了一种基于投影包构造的局部可恢复代数几何码，展示了其在不同维度下的最优性和高信息速率特性。
A code is said to be a Locally Recoverable Code (LRC) with availability if every coordinate can be recovered from multiple disjoint sets of other coordinates called recovering sets. The vector of sizes of recovering sets of a coordinate is called its recovery profile. In this work, we consider LRCs with availability under two different settings: (1) irregular recovery: non-constant recovery profile that remains fixed for all coordinates, (2) unequal locality: regular recovery profile that can vary with coordinates. For each setting, we derive bounds for the minimum distance that generalize pre…
Optimal locally repairable codes with information locality are considered. Optimal codes are constructed, whose length is also order-optimal with respect to a new bound on the code length derived in this paper. The length of the constructed codes is super-linear in the alphabet size, which improves upon the well known pyramid codes, whose length is only linear in the alphabet size. The recoverable erasure patterns are also analyzed for the new codes. Based on the recoverable erasure patterns, we construct generalized sector-disk (GSD) codes, which can recover from disk erasures mixed with sect…
By a locally recoverable code (LRC), we will in this paper, mean a linear code in which a given code symbol can be recovered by taking a linear combination of at most $r$ other code symbols with $r << k$. A natural extension is to the local recovery of a set of $t$ erased symbols. There have been several approaches proposed for the handling of multiple erasures. The approach considered here, is one of sequential recovery meaning that the $t$ erased symbols are recovered in succession, each time contacting at most $r$ other symbols for assistance in recovery. Under the constraint that each eras…
Entanglement-Assisted Quantum Locally Recoverable Codes: Bounds, Optimal Constructions, and Achievability
本文研究无需对偶包含的类CSS EA-qLRC，推导其各类参数界，给出等号达类Singleton界的条件，构造Tamo–Barg与循环码族的EA-qLRC，补充两个可达性界并统一所有界，明确循环码族可产生最优码。
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 …
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 c…
An Upper Bound On the Size of Locally Recoverable Codes
在{ \em局部可恢复}或{ \em可修复}码中，码字的任何符号都可以通过仅读取少量（恒定）的其它符号来恢复。本地可恢复性的概念在分布式存储领域非常重要，其中最常见的错误事件是单个存储节点故障（擦除）。一个共同的目标是通过从尽可能少的其他存储节点下载数据来修复节点。本文从码的长度、大小和局部性三个方面给出了码的最小距离的界。与以前的界限不同，我们的界限来自一个非常简单的分析，并取决于所使用的字母表的大小。事实证明，二进制单纯形码满足我们的界限与平等，因此单纯形码是第一个例子，一个最佳的二进制局部可修复的代码家庭。我们还提供了基于随机编码和级联码的可扩展性结果，这些结果在数值上被验证为接近我们的界限。
Locally recoverable $J$-affine variety codes
本文构造了$J$-仿射簇码子域子码的局部可恢复（LRC）码以纠正多个擦除，计算了其局部性$(r, δ)$，并证明了部分长度$n\gg q$的码为$(δ-1)$-最优。
Two families of linear codes with desirable properties from some functions over finite fields
本文基于增广技术从有限域函数构造了两族线性码，确定了其局部性、重量分布，并导出了最优或几乎最优可扩展的自正交局部可恢复码。
An $[n,k]$ code $\mathcal{C}$ is said to be locally recoverable in the presence of a single erasure, and with locality parameter $r$, if each of the $n$ code symbols of $\mathcal{C}$ can be recovered by accessing at most $r$ other code symbols. An $[n,k]$ code is said to be a locally recoverable code with sequential recovery from $t$ erasures, if for any set of $s \leq t$ erasures, there is an $s$-step sequential recovery process, in which at each step, a single erased symbol is recovered by accessing at most $r$ other code symbols. This is equivalent to the requirement that for any set of $s …
Locally recoverable codes were introduced by Gopalan et al. in 2012, and in the same year Prakash et al. introduced the concept of codes with locality, which are a type of locally recoverable codes. In this work we introduce a new family of codes with locality, which are subcodes of a certain family of evaluation codes. We determine the dimension of these codes, and also bounds for the minimum distance. We present the true values of the minimum distance in special cases, and also show that elements of this family are "optimal codes", as defined by Prakash et al.
This thesis makes several significant contributions to the theory of both Regenerating (RG) and Locally Recoverable (LR) codes. The two principal contributions are characterizing the optimal rate of an LR code designed to recover from $t$ erased symbols sequentially, for any $t$ and the development of a tight bound on the sub-packetization level (length of a vector code symbol) of a sub-class of RG codes called optimal-access RG codes. There are however, several other notable contributions as well such as deriving the tightest-known bounds on the performance metrics such as minimum distance an…
The (n, k, r)-Locally recoverable codes (LRC) studied in this work are (n, k) linear codes for which the value of each coordinate can be recovered by a linear combination of at most r other coordinates. In this paper, we are interested to find the largest possible minimum distance of (n,k,r)-LRCs, denoted D(n,k,r). We refer to the problem of finding the value of D(n,k,r) as the largest minimum distance (LMD) problem. LMD can be approximated within an additive term of one; it is known in the literature that D(n,k,r) is either equal to d* or d*-1, where d*=n-k-ceil(k/r) +2. Also, in the literatu…
Quantum Locally Recoverable Codes via Good Polynomials
本研究提出可使用任意优多项式的量子局部可恢复码构造方法，设计了基于仿射一般线性群子群的优多项式构建方式，并结合扩张器混合引理得到无r+1为素数限制的qLRC最小距离下界，推动量子存储编码研究。
Locally recoverable (LRC) codes have recently been a focus point of research in coding theory due to their theoretical appeal and applications in distributed storage systems. In an LRC code, any erased symbol of a codeword can be recovered by accessing only a small number of other symbols. For LRC codes over a small alphabet (such as binary), the optimal rate-distance trade-off is unknown. We present several new combinatorial bounds on LRC codes including the locality-aware sphere packing and Plotkin bounds. We also develop an approach to linear programming (LP) bounds on LRC codes. The result…
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 cod…
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