Tristram Bogart, Anna-Lena Horlemann-Trautmann, David Karpuk, Alessandro Neri, Mauricio Velasco
We propose reducible algebraic curves as a mechanism to construct Partial MDS (PMDS) codes geometrically. We obtain new general existence results, new explicit constructions and improved estimates on the smallest field sizes over which such codes can exist. Our results are obtained by combining ideas from projective algebraic geometry, combinatorics and probability theory.
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Based on the fundamental results on MDS self-dual codes over finite fields constructed via generalized Reed-Solomon codes \cite{JX} and extended generalized Reed-Solomon codes \cite{Yan}, many series of MDS self-dual codes with different length have been obtained recently by a variety of constructions and individual computations. In this paper, we present an unified approach to get several previous results with concise statements and simplified proofs, and some new constructions on MDS self-dual codes. In the conclusion section we raise two open problems.
Entanglement-assisted quantum error-correcting (EAQEC) codes make use of preexisting entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQEC code from any classical linear code, unlike standard quantum error-correcting codes, which can only be constructed from dual-containing codes. However, the number $c$ of pre-shared maximally entangled states is usually calculated by computer search. In this paper, we first give a new formula for calculating the number $c$ of pre-shared maximally entangled states. Then, using this formula, we con…
A framework for constructing non-GRS MDS-NMDS codes from deep holes and its application
本文提出一种通过深孔构造非GRS MDS-NMDS码的统一框架，展示了如何从现有码族系统地生成更多码族，并应用该框架得到新的码族及研究码的单体等价性。
Recently, Roth and Skachek proposed two methods for constructing nearly maximum-distance separable (MDS) expander codes. We show that through the simple modification of using mixed-alphabet codes derived from MDS codes as constituent codes in their code designs, one can obtain nearly MDS codes of significantly smaller alphabet size, albeit at the expense of a (very slight) reduction in code rate.
The maximum independent sets in the Doob graphs D(m,n) are analogs of the distance-2 MDS codes in Hamming graphs and of the latin hypercubes. We prove the characterization of these sets stating that every such set is semilinear or reducible. As related objects, we study vertex sets with maximum cut (edge boundary) in D(m,n) and prove some facts on their structure. We show that the considered two classes (the maximum independent sets and the maximum-cut sets) can be defined as classes of completely regular sets with specified 2-by-2 quotient matrices. It is notable that for a set from the consi…
In this paper, we describe a procedure for constructing $q$--ary $[N,3,N-2]$--MDS codes, of length $N\leq q+1$ (for $q$ odd) or $N\leq q+2$ (for $q$ even), using a set of non--degenerate Hermitian forms in $PG(2,q^2)$.
The Galois hulls of linear codes are a generalization of the Euclidean and Hermitian hulls of linear codes. In this paper, we study the Galois hulls of (extended) GRS codes and present several new constructions of MDS codes with Galois hulls of arbitrary dimensions via (extended) GRS codes. Two general methods of constructing MDS codes with Galois hulls of arbitrary dimensions by Hermitian or general Galois self-orthogonal (extended) GRS codes are given. Using these methods, some MDS codes with larger dimensions and Galois hulls of arbitrary dimensions can be obtained and relatively strict con…
In distributed storage, erasure codes -- like Reed-Solomon Codes -- are often employed to provide reliability. In this setting, it is desirable to be able to repair one or more failed nodes while minimizing the repair bandwidth. In this work, motivated by Reed-Solomon codes, we study the problem of repairing multiple failed nodes in a scalar MDS code. We extend the framework of (Guruswami and Wootters, 2017) to give a framework for constructing repair schemes for multiple failures in general scalar MDS codes, in the centralized repair model. We then specialize our framework to Reed-Solomon cod…
Generalized GM-MDS: Polynomial Codes are Higher Order MDS
本文将 GM-MDS 定理推广到任意多项式码及其对偶码，并进一步推广到生成矩阵列取自不过原点的不可约簇上点的代数码及其对偶码，为构造达到容量的列表可译码提供了理论基础。
A Generic Construction of $q$-ary Near-MDS Codes Supporting 2-Designs with Lengths Beyond $q+1$
本文提出一种通用方法，构造出支持2设计的q-ary近MDS码，其长度超过q+1，结合椭圆曲线码、有限阿贝尔群、子集和与组合设计的新联系，得到无限族码及其重量分布。
Quantum maximal-distance-separable (MDS) codes form an important class of quantum codes. To get $q$-ary quantum MDS codes, it suffices to find linear MDS codes $C$ over $\mathbb{F}_{q^2}$ satisfying $C^{\perp_H}\subseteq C$ by the Hermitian construction and the quantum Singleton bound. If $C^{\perp_{H}}\subseteq C$, we say that $C$ is a dual-containing code. Many new quantum MDS codes with relatively large minimum distance have been produced by constructing dual-containing constacyclic MDS codes (see \cite{Guardia11}, \cite{Kai13}, \cite{Kai14}). These works motivate us to make a careful study…
Linear complementary-dual (LCD for short) codes are linear codes that intersect with their duals trivially. LCD codes have been used in certain communication systems. It is recently found that LCD codes can be applied in cryptography. This application of LCD codes renewed the interest in the construction of LCD codes having a large minimum distance. MDS codes are optimal in the sense that the minimum distance cannot be improved for given length and code size. Constructing LCD MDS codes is thus of significance in theory and practice. Recently, Jin (\cite{Jin}, IEEE Trans. Inf. Theory, 2016) con…
Two classes of reducible cyclic codes with large minimum symbol-pair distances
本文基于有限域分圆数和高斯周期理论，研究两类可约循环码在符号对度量下的参数，确定了其符号对重量分布，并证明在特定条件下最小符号对距离为最小汉明距离的两倍，建立了一类MDS符号对码。
Symbol-pair codes are block codes with symbol-pair metrics designed to protect against pair-errors that may occur in high-density data storage systems. MDS symbol-pair codes are optimal in the sense that it can attain the highest pair-error correctability within the same code length and code size. Constructing MDS symbol-pair codes is one of the main topics in symbol-pair codes. In this paper, we characterize the symbol-pair distances of some constacyclic codes of arbitrary lengths over finite fields and a class of finite chain rings. Using the characterization of symbol-pair distance, we pres…
New Constructions of Non-GRS MDS Codes, Recovery and Determination Algorithms for GRS Codes
本文提出了一种构造非GRS MDS码的新方法，并设计了两个高效算法用于GRS码的确定与恢复。
It has been a great challenge to construct new quantum MDS codes. In particular, it is very hard to construct quantum MDS codes with relatively large minimum distance. So far, except for some sparse lengths, all known $q$-ary quantum MDS codes have minimum distance less than or equal to $q/2+1$. In the present paper, we provide a construction of quantum MDS codes with minimum distance bigger than $q/2+1$. In particular, we show existence of $q$-ary quantum MDS codes with length $n=q^2+1$ and minimum distance $d$ for any $d\le q-1$ and $d= q+1$(this result extends those given in \cite{Gu11,Jin1…
The entanglement-assisted stabilizer formalism provides a useful framework for constructing quantum error-correcting codes (QECC), which can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this paper, we construct five classes of entanglement-assisted quantum MDS (EAQMDS) codes based on classical MDS codes by exploiting one or more pre-shared maximally entangled states. We show that these EAQMDS codes have much larger minimum distance than the standard quantum…
New families of non-Reed-Solomon MDS codes
本文提出一种基于求值多项式与求值点选择的通用框架来构造非Reed-Solomon型MDS码，并通过Schur平方证明其非RS性质，最终获得长度更灵活的新码族。
Entanglement-assisted quantum error correcting codes (EAQECCs) play a significant role in protecting quantum information from decoherence and quantum noise. Recently, constructing entanglement-assisted quantum maximum distance separable (EAQMDS) codes with flexible parameters has received much attention. In this work, four families of EAQMDS codes with a more general length are presented. And the method of selecting defining set is different from others. Compared with all the previously known results, the EAQMDS codes we constructed have larger minimum distance. All of these EAQMDS codes are n…
In ``Infinite families of near MDS codes holding $t$-designs, IEEE Trans. Inform. Theory, 2020, 66(9), pp. 5419-5428'', Ding and Tang made a breakthrough in constructing the first two infinite families of NMDS codes holding $2$-designs or $3$-designs. Up to now, there are only a few known infinite families of NMDS codes holding $t$-designs in the literature. The objective of this paper is to construct new infinite families of NMDS codes holding $t$-designs. We determine the weight enumerators of the NMDS codes and prove that the NMDS codes hold $2$-designs or $3$-designs. Compared with known $…
Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual is trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. On the other hand, being optimal codes, maximum distance separable codes (abbreviated MDS) have been of much interest from many researchers due to their theoretical significant and practical implications. However, little work has been done on LCD M…
Motivated by the application of high-density data storage technologies, symbol-pair codes are proposed to protect against pair-errors in symbol-pair channels, whose outputs are overlapping pairs of symbols. The research of symbol-pair codes with the largest minimum pair-distance is interesting since such codes have the best possible error-correcting capability. A symbol-pair code attaining the maximal minimum pair-distance is called a maximum distance separable (MDS) symbol-pair code. In this paper, we focus on constructing linear MDS symbol-pair codes over the finite field $\mathbb{F}_{q}$. W…
Construction of MDS Euclidean Self-Dual Codes via Multiple Subsets
本文致力于通过GRS码及其扩展码构造新的MDS欧氏自对偶码，提出从多个相交子集中选择求值集的方法，并基于范数函数和乘法子群及其陪集的并集构造了六类新码，使可能的码比例超过85%。
Column Twisted Reed-Solomon Codes as MDS Codes
本文研究列扭曲里德-所罗门码，建立其为MDS码的充分条件，证明其Schur平方码维数为2k，并展示其与里德-所罗门码不同。提出更灵活的参数设计，支持更长的码长，同时介绍列扭曲里德-所罗门码的对偶码。
In network communication, the source often transmits messages at several different information rates within a session. How to deal with information transmission and network error correction simultaneously under different rates is introduced in this paper as a variable-rate network error correction problem. Apparently, linear network error correction MDS codes are expected to be used for these different rates. For this purpose, designing a linear network error correction MDS code based on the existing results for each information rate is an efficient solution. In order to solve the problem more…
We consider constructing capacity-achieving linear codes with minimum message size for private information retrieval (PIR) from $N$ non-colluding databases, where each message is coded using maximum distance separable (MDS) codes, such that it can be recovered from accessing the contents of any $T$ databases. It is shown that the minimum message size (sometimes also referred to as the sub-packetization factor) is significantly, in fact exponentially, lower than previously believed. More precisely, when $K>T/\textbf{gcd}(N,T)$ where $K$ is the total number of messages in the system and $\textbf…
Constructions of block MDS LDPC codes from punctured circulant matrices
本文提出从穿孔循环矩阵构造块MDS LDPC码，具有无4-环的Tanner图和增强的随机错误校正能力。
This paper addresses the problem of constructing MDS codes that enable exact repair of each code block with small repair bandwidth, which refers to the total amount of information flow from the remaining code blocks during the repair process. This problem naturally arises in the context of distributed storage systems as the node repair problem [7]. The constructions of exact-repairable MDS codes with optimal repair-bandwidth require working with large sub-packetization levels, which restricts their employment in practice. This paper presents constructions for MDS codes that simultaneously prov…
Constructing $(h,d)$ cooperative MSR codes with sub-packetization $(d-k+h)(d-k+1)^{\lceil n/2 \rceil}$
针对MDS阵列码多节点故障修复的协作模型，本文提出了带$d$个辅助节点的最优MDS阵列码显式构造，其子分组化水平优于现有方案。
A new kind of Convolutional Codes generalizing Goppa Codes is proposed. This provides a systematic method for constructing convolutional codes with prefixed properties. In particular, examples of Maximum-Distance Separable (MDS) convolutional codes are obtained.
An $(n, M)$ vector code $\mathcal{C} \subseteq \mathbb{F}^n$ is a collection of $M$ codewords where $n$ elements (from the field $\mathbb{F}$) in each of the codewords are referred to as code blocks. Assuming that $\mathbb{F} \cong \mathbb{B}^{\ell}$, the code blocks are treated as $\ell$-length vectors over the base field $\mathbb{B}$. Equivalently, the code is said to have the sub-packetization level $\ell$. This paper addresses the problem of constructing MDS vector codes which enable exact reconstruction of each code block by downloading small amount of information from the remaining code …
Coding for distributed storage gives rise to a new set of problems in coding theory related to the need of reducing inter-node communication in the system. A large number of recent papers addressed the problem of optimizing the total amount of information downloaded for repair of a single failed node (the repair bandwidth) by accessing information on $d$ {\em helper nodes}, where $k\le d\le n-1.$ By the so-called cut-set bound (Dimakis et al., 2010), the repair bandwidth of an $(n,k=n-r)$ MDS code using $d$ helper nodes is at least $dl/(d+1-k),$ where $l$ is the size of the node. Also, a numbe…
In this paper, we shed light on how an adaptive, efficient error coding in the transport layer helps ensure the application requirements. We recap the use of MDS codes and show that binary coding can significantly reduce the complexity and hence increase the applicability also for embedded devices. We exploit the persymmetric structure of the generator matrix in polar codes to establish a duality of dispersion over channels (the polarization effect) and over packets (the generality required for multicast transmission), thereby constructing systematic polar codes for incremental redundancy whos…
Generic Construction of Optimal-Access Binary MDS Array Codes with Smaller Sub-packetization
针对二进制MDS阵列码单节点故障修复问题，提出两种通用构造方法，基于基础码构建出具有最优访问带宽的码C₁和C₂，其中C₂的子分组化程度为现有同类码中最小。
We present a new 'piggybacking' framework for designing distributed storage codes that are efficient in data-read and download required during node-repair. We illustrate the power of this framework by constructing classes of explicit codes that entail the smallest data-read and download for repair among all existing solutions for three important settings: (a) codes meeting the constraints of being Maximum-Distance-Separable (MDS), high-rate and having a small number of substripes, arising out of practical considerations for implementation in data centers, (b) binary MDS codes for all parameter…
Decoding Algorithms for Twisted GRS Codes
本文提出基于高斯消元法的扭曲广义RS码译码算法，可纠正MDS码最多$\floor{\frac{n-k}{2}}$个错误，复杂度$O(n^3)$，并适用于近MDS码。
We consider the problem of constructing linear Maximum Distance Separable (MDS) error-correcting codes with generator matrices that are sparsest and balanced. In this context, sparsest means that every row has the least possible number of non-zero entries, and balanced means that every column contains the same number of non-zero entries. Codes with this structure minimize the maximal computation time of computing any code symbol, a property that is appealing to systems where computational load-balancing is critical. The problem was studied before by Dau et al. where it was shown that there alw…
Explicit bases for Riemann-Roch spaces on elliptic curves and their application in constructing various elliptic code families
本文提出了一种构造任意椭圆曲线分歧Riemann-Roch基的方法，并探讨其在密码学中的应用，特别是用于改进McEliece公钥密码系统。
In a distributed storage systems (DSS), regenerating codes are used to optimize bandwidth in the repair process of a failed node. To optimize other DSS parameters such as computation and disk I/O, Distributed Replication-based Simple Storage (Dress) Codes consisting of an inner Fractional Repetition (FR) code and an outer MDS code are commonly used. Thus constructing FR codes is an important research problem, and several constructions using graphs and designs have been proposed. In this paper, we present an algorithm for constructing the node-packet distribution matrix of FR codes and thus enu…
Galois hulls of linear codes have important applications in quantum coding theory. In this paper, we construct some new classes of (extended) generalized Reed-Solomon (GRS) codes with Galois hulls of arbitrary dimensions. We also propose a general method on constructing GRS codes with Galois hulls of arbitrary dimensions from special Euclidean orthogonal GRS codes. Finally, we construct several new families of entanglement-assisted quantum error-correcting codes (EAQECCs) and MDS EAQECCs by utilizing the above results.
It is a well-known result that constructing codewords over $GF(2)$ to minimize the number of transmissions for a single-hop wireless multicasting is an NP-complete problem. Linearly independent codewords can be constructed in polynomial time for all the $n$ clients, known as maximum distance separable (MDS) code, when the finite field size $q$ is larger than or equal to the number of clients, $q\geq n$. In this paper we quantify the exact minimum number of transmissions for a multicast network using erasure code when $q=2$ and $n=3$, such that $q<n$. We first show that the use of Markov chain …
Linear Complementary Pairs of Algebraic Geometry Codes via Kummer Extensions
本文研究了线性互补对（LCPs）的构造问题，通过确定 Kummer 扩张上的非特殊除子显式构造，提出了多种基于 Kummer 扩张的代数几何码 LCPs 构建方法，并成功应用于椭圆及超椭圆等函数域生成 MDS AG Codes 和 LCD 码。
The repair problem in distributed storage addresses recovery of the data encoded using an erasure code, for instance, a Reed-Solomon (RS) code. We consider the problem of repairing a single node or multiple nodes in RS-coded storage systems using the smallest possible amount of inter-nodal communication. According to the cut-set bound, communication cost of repairing $h\ge 1$ failed nodes for an $(n,k=n-r)$ MDS code using $d$ helper nodes is at least $dhl/(d+h-k),$ where $l$ is the size of the node. Guruswami and Wootters (2016) initiated the study of efficient repair of RS codes, showing that…
In coding theory, constructing codes with good parameters is one of the most important and fundamental problems. Though a great many of good codes have been produced, most of them are defined over alphabets of sizes equal to prime powers. In this paper, we provide a new explicit construction of $(q+1)$-ary nonlinear codes via algebraic function fields, where $q$ is a prime power. Our codes are constructed by evaluations of rational functions at all rational places of the algebraic function field. Compared with algebraic geometry codes, the main difference is that we allow rational functions to…
Constructing locally repairable codes achieving Singleton-type bound (we call them optimal codes in this paper) is a challenging task and has attracted great attention in the last few years. Tamo and Barg \cite{TB14} first gave a breakthrough result in this topic by cleverly considering subcodes of Reed-Solomon codes. Thus, $q$-ary optimal locally repairable codes from subcodes of Reed-Solomon codes given in \cite{TB14} have length upper bounded by $q$. Recently, it was shown through extension of construction in \cite{TB14} that length of $q$-ary optimal locally repairable codes can be $q+1$ i…
On Linear Complementary Pairs of Algebraic Geometry Codes over Finite Fields
本文研究有限域上代数几何码的线性互补对，给出椭圆曲线显式构造、安全参数与对偶等价条件，并探讨MDS情形。
There has been a lot of effort to construct good quantum codes from the classical error correcting codes. Constructing new quantum codes, using Hermitian self-orthogonal codes, seems to be a difficult problem in general. In this paper, Hermitian self-orthogonal codes are studied from algebraic function fields. Sufficient conditions for the Hermitian self-orthogonality of an algebraic geometry code are presented. New Hermitian self-orthogonal codes are constructed from projective lines, elliptic curves, hyper-elliptic curves, Hermitian curves, and Artin-Schreier curves. In addition, over the pr…
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