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 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 V…
A locally recoverable code is an error-correcting code such that any erasure in a coordinate of a codeword can be recovered from a set of other few coordinates. In this article we introduce a model of local recoverable codes that also includes local error detection. The cases of the Reed-Solomon and Locally Recoverable Reed-Solomon codes are treated in some detail.
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.
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.
A Locally Recoverable code is an error-correcting code such that any erasure in a single coordinate of a codeword can be recovered from a small subset of other coordinates. We study Locally Recoverable Algebraic Geometry codes arising from certain curves defined by equations with separated variables. The recovery of erasures is obtained by means of Lagrangian interpolation in general, and simply by one addition in some particular cases.
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 locally recoverable code is an error-correcting code such that any erasure in a single coordinate of a codeword can be recovered from a small subset of other coordinates. In this article we develop an algorithm that computes a recovery structure as concise posible for an arbitrary linear code $\mathcal{C}$ and a recovery method that realizes it. This algorithm also provides the locality and the dual distance of $\mathcal{C}$. Complexity issues are studied as well. Several examples are included.
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 $…
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…
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.
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）
A monomial-Cartesian code is an evaluation code defined by evaluating a set of monomials over a Cartesian product. It is a generalization of some families of codes in the literature, for instance toric codes, affine Cartesian codes and $J$-affine variety codes. In this work we use the vanishing ideal of the Cartesian product to give a description of the dual of a monomial-Cartesian code. Then we use such description of the dual to prove the existence of quantum error correcting codes and MDS quantum error correcting codes. Finally we show that the direct product of monomial-Cartesian codes is …
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…
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…
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…
Integrated Interleaved (II) and Extended Integrated Interleaved (EII) codes are a versatile alternative for Locally Recoverable (LRC) codes, since they require fields of relatively small size. II and EII codes are generally defined over Reed-Solomon type of codes. A new comprehensive definition of EII codes is presented, allowing for EII codes over any field, and in particular, over the binary field $GF(2)$. The traditional definition of II and EII codes is shown to be a special case of the new definition. Improvements over previous constructions of LRC codes, in particular, for binary codes, …
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…
Entanglement-Assisted Quantum Locally Recoverable Codes: Characterizations, Bounds, and Constructions
Nanyang Technological University（南洋理工大学）；VinUniversity（越南大学）；City University of Hong Kong（香港城市大学）；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；Hefei University of Technology（合肥工业大学）
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.
Private Information Retrieval (PIR) problem has recently attracted a significant interest in the information-theory community. In this problem, a user wants to privately download one or more messages belonging to a database with copies stored on a single or multiple remote servers. In the single server scenario, the user must have prior side information, i.e., a subset of messages unknown to the server, to be able to privately retrieve the required messages in an efficient way. In the last decade, there has also been a significant interest in Locally Recoverable Codes (LRC), a class of storage…
Beyond Polynomials: Optimal Locally Recoverable Codes from Good Rational Functions
Southwest Jiaotong University（西南交通大学）；University of Paris VIII（巴黎第八大学）；CNRS, UMR 7539 LAGA, University of Paris XIII（法国国家科学研究中心，UMR 7539 LAGA，巴黎第十三大学）；China West Normal University（西华师范大学）
Local List Recovery of High-rate Tensor Codes and Applications
在这项工作中，我们给出了第一个高速率的本地列表可恢复代码的建设。列表恢复一直是编码理论中非常有用的构建块，我们的动机是使用这些代码作为这样的构建块。特别是，我们的建设给出了第一个容量实现本地列表可解码的代码（在恒定大小的字母表）;第一个容量实现全局列表可解码的代码与近线性时间列表解码算法（再一次，在恒定大小的字母表）;和Gilbert-Varshamov界上的二进制码的随机化结构，可以在近线性时间内唯一解码，具有比以前已知的更高的速率。我们的技术实际上非常简单，并且受到Gopalan，Guruswami和Raghavendra（Siam Journal on Computing，2011）用于列表解码张量代码的方法的启发。我们表明，张量的权力（全球）列表可恢复的代码是“近似”本地列表可恢复的，和“近似”的修改器可以通过预编码的消息与合适的本地可解码的代码被删除。用高速率全局列表可恢复代码和高速率局部可解码代码的已知构造来实例化该构造完成该构造。
