Maximum Distance Separable Codes Research Articles

Obtaining new classes of optimal linear codes by puncturing and shortening optimal cyclic codes

AbstractIn this paper we use the puncturing and shortening techniques on two already-known classes of optimal cyclic codes in order to obtain three new classes of optimal linear codes achieving the Griesmer bound. The weight distributions for these codes are settled. We also investigate their dual codes and show that they are either optimal or almost optimal with respect to the sphere-packing bound. Moreover, these duals contain classes of almost maximum distance separable codes which are shown to be proper for error detection. Further, some of the obtained optimal linear codes are suitable for constructing secret sharing schemes with nice access structures.

Near MDS and near quantum MDS codes via orthogonal arrays

Near maximum distance separable (NMDS) codes are closely related to interesting objects in finite geometry and have nice applications in combinatorics and cryptography. But there are many unsolved problems about construction of NMDS codes. In this paper, by using symmetrical orthogonal arrays (OAs), we construct a lot of NMDS, m-MDS and almost extremal NMDS codes. Quantum error-correcting codes (QECCs) play a central role in quantum information processing and can protect quantum information from various quantum noises. We present a general method for constructing QECCs over mixed alphabets through asymmetrical OAs. Since quantum maximum distance separable (QMDS) codes over mixed alphabets with the dimension equal to one have not been found in all the literature so far, the definition of a near QMDS code over mixed alphabets is proposed. By using asymmetrical OAs, we obtain many such codes.

Three New Classes of Subsystem Codes

In this paper, we construct three classes of Clifford subsystem maximum distance separable (MDS) codes based on Reed-Solomon codes and extended generalized Reed-Solomon codes over finite fields [see formula in PDF] for specific code lengths. Moreover, our Clifford subsystem MDS codes are new because their parameters differ from the previously known ones.

Near-MDS codes from maximal arcs in PG(2,q)

The singleton defect of an [n,k,d] linear code C is defined as s(C)=n−k+1−d. Codes with S(C)=0 are called maximum distance separable (MDS) codes, and codes with S(C)=S(C⊥)=1 are called near maximum distance separable (NMDS) codes. Both MDS codes and NMDS codes have good representations in finite projective geometry. MDS codes over Fq with length n and n-arcs in PG(k−1,q) are equivalent objects. When k=3, NMDS codes of length n are equivalent to (n,3)-arcs in PG(2,q). In this paper, we deal with the NMDS codes with dimension 3. By adding some suitable projective points in maximal arcs of PG(2,q), we can obtain two classes of (q+5,3)-arcs (or equivalently [q+5,3,q+2] NMDS codes) for any prime power q. We also determine the exact weight distribution and the locality of such NMDS codes and their duals. It turns out that the resultant NMDS codes and their duals are both distance-optimal and dimension-optimal locally recoverable codes.

Classification of a class of A3MDS code and applications in secret-sharing schemes

A [Formula: see text]-ary linear [Formula: see text] code [Formula: see text] is known as [Formula: see text] code. If [Formula: see text] the code is called a maximum distance separable code. [Formula: see text] codes are known to be useful in the secret-sharing schemes and coding theory. In this paper, we classify all binary self-orthogonal [Formula: see text] codes. We have introduced a subclass of [Formula: see text] codes, called [Formula: see text] codes, to form a special type of secret-sharing scheme where for any participant, there exist [Formula: see text] other participants who can collectively reveal the secret by combining their shares and there exists another set of [Formula: see text] participants including the given participant, who cannot reveal the secret.

Design of Degree Distribution with Maximum Distance Separable Codes for User Priority

Design of Degree Distribution with Maximum Distance Separable Codes for User Priority

Several classes of Galois self-orthogonal MDS codes and related applications

Let q=ph be a prime power and e be an integer with 0≤e≤h−1. e-Galois self-orthogonal codes are generalizations of Euclidean self-orthogonal codes (e=0) and Hermitian self-orthogonal codes (e=h2 and h is even). In this paper, we propose two general methods to construct e-Galois self-orthogonal (extended) generalized Reed-Solomon (GRS) codes. As a consequence, eight new classes of e-Galois self-orthogonal (extended) GRS codes with odd q and 2e|h are obtained. Based on the Galois dual of a code, we also study its punctured and shortened codes. As applications, new e′-Galois self-orthogonal maximum distance separable (MDS) codes for all possible e′ satisfying 0≤e′≤h−1, new e-Galois self-orthogonal MDS codes via the shortened codes, and new MDS codes with prescribed dimensional e-Galois hull via the punctured codes are derived. Moreover, some new q-ary quantum MDS codes with length greater than q+1 and minimum distance greater than q2+1 are obtained.

MDS coding enabled proxy‐based Internet censorship circumvention system

AbstractIn this letter, a novel maximum distance separable coding enabled proxy‐based Internet censorship circumvention system is proposed to handle severe security threats. By introducing redundancy encoding and multi‐proxy transmission, its anti‐censorship capability is significantly enhanced in terms of transmission reliability and data security. A generalized characterization method is constructed for the proposed system utilizing a six‐parameter tuple. Defining successful transmission probability and secure transmission probability as the evaluation metrics of system performances, their mathematical formulations are further derived theoretically. Extensive simulation results are provided to demonstrate the effectiveness and advantages of the proposed system.

Block codes on pomset metric

Given a regular multiset [Formula: see text] on [Formula: see text], a partial order [Formula: see text] on [Formula: see text], and a label map [Formula: see text] defined by [Formula: see text] with [Formula: see text], we define a pomset block metric [Formula: see text] on the direct sum [Formula: see text] of [Formula: see text] based on the pomset [Formula: see text]. The pomset block metric extends the classical pomset metric introduced by Sudha and Selvaraj and generalizes the poset block metric introduced by Alves et al. over [Formula: see text]. The space [Formula: see text] is called the pomset block space and we determine the complete weight distribution of it. Further, [Formula: see text]-perfect pomset block codes for ideals with partial and full counts are described. Then, for block codes with chain pomset, the packing radius and Singleton bound are established. The relation between maximum distance separable (MDS) codes and [Formula: see text]-perfect codes for any ideal [Formula: see text] is investigated. Moreover, the duality theorem for an MDS pomset block code is established when all the blocks have the same size.

Near MDS codes of non-elliptic-curve type from Reed-Solomon codes

An [n,k,d] linear code is said to be maximum distance separable (MDS) or almost maximum distance separable (AMDS) if d=n−k+1 or d=n−k, respectively. If a code and its dual code are both AMDS, then the linear code is called a near maximum distance separable (NMDS) code. NMDS codes correspond to some interesting objects in finite geometry and have nice applications in cryptography. The NMDS codes constructed from elliptic curves are referred as elliptic curve NMDS codes. In this paper, by adding two suitable columns to the generator matrices of famous Reed-Solomon codes, we can obtain two families of NMDS codes with parameters [q+3,4,q−1]q (q is a prime power with gcd⁡(q−1,3)=1) and [2m+3,5,2m−2]2m respectively. These NMDS codes are shown to be linearly inequivalent to elliptic curve NMDS codes. Besides, the weight enumerators and locality of these NMDS codes are completely determined. It turns out that the resultant NMDS codes and their dual codes are mostly distance-optimal and dimension-optimal locally recoverable codes.

A Novel Maximum Distance Separable Coded OFDM-RIS for 6G Wireless Communications

The vision of the sixth generation mobile communication (6G) calls for extremely reliable data transmission over complex and diverse wireless channels. By combining the maximum distance separable (MDS) codes with the classic orthogonal frequency division multiplexing (OFDM), we present a new paradigm with the assistance of the reconfigurable intelligent surfaces (RIS). In our design, the RIS with limited phase shifts are adopted and the pairwise error probability (PEP) of the proposed system is first derived. Then, in order to further minimize the bit error rate (BER), we provide a novel method based on phase alignment to obtain the optimal solution for the discrete phase shifts. Our simulation results show that the proposed scheme provides considerable BER performance improvements compared to conventional OFDM systems. Moreover, the proposed discrete phase optimization algorithm is capable of achieving performance similar to that of the system with continuous phase shifts.

Enhancing data protection with a distributed storage system based on the redundant residue number system

AbstractBig data becomes the key for ubiquitous computing and intelligence, and Distributed Storage Systems (DSS) are widely used in large-scale data centers or in the cloud for efficient data management. However, the data on stored are likely to be unavailable due to hardware failures and cyberattacks, e.g. DDoS. Maximum Distance Separable (MDS) codes are commonly used for the recovery of faulty storage nodes or unavailable data. However, the recovery of data nodes usually involves access to multiple nodes, which introduces significant communication overheads to the DSS. In this paper, a new DSS based on the Redundant Residue Number System (RRNS) is proposed, where efficient recovery is enabled by applying the second version of Chinese Remainder Theorem (CRT-II). The complexity and network traffic of the proposed data protection scheme is analyzed theoretically and compared with that of traditional MDS based DSSs. Experimental results show that the proposed DSS achieves lower encoding complexity, lower recovery complexity and lower network traffic than the MDS based schemes. Although the proposed data protection scheme introduces computation overheads for the case on which there are no failing nodes, its complexity is still lower for scenarios with frequent data updates. In addition, the proposed scheme introduces additional advantages in terms of security and storage flexibility.

Estimation for the number of MDS Matrices, Recursive MDS Matrices and Symmetric Recursive MDS Matrices from the Reed-Solomon Codes

The diffusion layer of the SPN block ciphers is usually built on the basis of the MDS (Maximum Distance Separable) matrices which is the matrix of the maximum distance separable code (MDS code). MDS codes have long been studied in error correcting code theory and have applications not only in coding theory but also in the design of block ciphers and hash functions. Thanks to that important role, there have been many studies on methods of building MDS matrices. In particular, the recursive MDS matrices and the symmetric recursive MDS matrices have particularly important applications because they are very efficient for execution. In this paper, we will give an estimate of the number of MDS matrices, recursive MDS matrices and symmetric recursive MDS matrices built from Reed-Solomon codes. This result is meaningful in determining the efficiency from this method of building matrices based on the Reed-Solomon codes. From there, this method can be applied to find out many MDS matrices, secure and efficient symmetric recursive MDS matrices for execution to apply in current block ciphers. Furthermore, recursive MDS matrices can be efficiently implemented using Linear Feedback Shift Registers (LFSR), making them well suited for lightweight cryptographic algorithms, so suitable for limited resources application.

New Families of MDS Symbol-Pair Codes From Matrix-Product Codes

In emerging storage technologies, the outputs of the channels consist of overlapping pairs of symbols. The errors are no longer individual symbols. Controlling them calls for a different approach. Symbol-pair codes have been proposed as a solution. The error-correcting capability of such a code depends on its minimum pair distance instead of the usual minimum Hamming distance. Longer codes can be conveniently constructed from known shorter ones by a matrix-product approach. The parameters of a matrix-product code can be determined from the parameters of the ingredient codes. We construct a new family of maximum distance separable (MDS) symbol-pair matrix-product codes. Codes which are permutation equivalent to matrix-product codes may have improved minimum pair distances. We present four new families of MDS symbol-pair codes and a new family of almost MDS symbol-pair codes. The codes in these five new families are permutation equivalent to matrix-product codes. Each of our five constructions identifies permutations that can increase the minimum pair distances. We situate the new families among previously known families of MDS symbol-pair codes to highlight the versatility of our matrix-product construction route.

Constructions of near MDS codes which are optimal locally recoverable codes

A linear code with length n, dimension k and minimum distance 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 a 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 many families of distance-optimal and dimension-optimal locally recoverable codes.

PMDS Array Codes With Small Sub-Packetization, Small Repair Bandwidth/Rebuilding Access

Partial maximum distance separable (PMDS) codes are a kind of erasure codes where the nodes are divided into multiple groups with each forming an MDS code with a smaller code length, thus they allow repairing a failed node with only a few helper nodes and can correct all erasure patterns that are information-theoretically correctable. However, the repair of a failed node of PMDS codes still requires a large amount of communication if the group size is large. Recently, PMDS array codes with each local code being an MSR code were introduced to reduce the repair bandwidth further. However, they require extensive rebuilding access and unavoidably a significant sub packetization level. In this paper, we first propose two constructions of PMDS array codes with two global parities that have smaller sub-packetization levels and much smaller finite fields than the existing one. One construction can support an arbitrary number of local parities and has $(1+\epsilon)$-optimal repair bandwidth (i.e., $(1+\epsilon)$ times the optimal repair bandwidth), while the other one is limited to two local parities but has significantly smaller rebuilding access and its sub packetization level is only $2$. In addition, we present a construction of PMDS array code with three global parities, which has a smaller sub-packetization level as well as $(1+\epsilon)$-optimal repair bandwidth, the required finite field is significantly smaller than existing ones.

Some quantum MDS codes from GRS codes

The construction of quantum maximum-distance-separable (MDS) codes has become one of the major goals in quantum coding theory. In this paper, we construct several classes of Hermitian self-orthogonal generalized Reed–Solomon codes. Based on these classical MDS codes, we obtain several new classes of quantum MDS codes with large minimum distance. It turns out that these constructed quantum MDS codes have fewer constraints on the selection of code length, and in some cases, have larger minimum distance than previous literature. Notably, some of the distance parameters of our codes are greater than .

AMDS symbol-pair codes from repeated-root cyclic codes

Symbol-pair codes are proposed to guard against pair-errors in symbol-pair read channels. The minimum symbol-pair distance is of significance in determining the error-correcting capability of a symbol-pair code. One of the central themes in symbol-pair coding theory is the constructions of symbol-pair codes with the largest possible minimum symbol-pair distance. Maximum distance separable (MDS) and almost maximum distance separable (AMDS) symbol-pair codes are optimal and sub-optimal regarding the Singleton bound, respectively. In this paper, six new classes of AMDS symbol-pair codes are explicitly constructed through repeated-root cyclic codes. Remarkably, one class of such codes has unbounded lengths and the minimum symbol-pair distance of another class can reach 13.

Quantum error-correcting codes and their geometries

This is an expository article aiming to introduce the reader to the underlying mathematics and geometry of quantum error correction. Information stored on quantum particles is subject to noise and interference from the environment. Quantum error-correcting codes allow the negation of these effects in order to successfully restore the original quantum information. We briefly describe the necessary quantum-mechanical background to be able to understand how quantum error correction works. We go on to construct quantum codes: firstly qubit stabilizer codes, then qubit non-stabilizer codes, and finally codes with a higher local dimension. We will delve into the geometry of these codes. This allows one to deduce the parameters of the code efficiently, deduce the inequivalence between codes that have the same parameters, and presents a useful tool in deducing the feasibility of certain parameters. We also include sections on quantum maximum distance separable codes and the quantum MacWilliams identities.

MDS Code Based Ultralightweight Authentication Protocol for RFID System

Privacy and security are central issues in the deployment of an RFID system. It is vulnerable to several attacks, such as replay attacks, location tracking, man-in-middle attack, de-synchronization attack, etc., due to the inherent weaknesses of underlying wireless communications. In order to tackle these privacy and security concerns, this paper presents an ultralightweight authentication protocol based on group homomorphism and maximum distance separable (MDS) code. We use group homomorphism properties to make a server lookup table that reduces searching complexity and overcomes scalability issues. We develop an ultralightweight protocol using MDS code properties that employs only bit-wise operators. Formal and informal security analysis of the proposed protocol shows that our proposed protocol resists various attacks. In addition, we use automated security protocol verification tools, AVISPA and Scyther, to validate the security features of the proposed protocol. We demonstrate the correctness proof of the proposed protocol using BAN logic. To measure the level of privacy of the proposed protocol, we use two benchmark metrics to simulate the proposed protocol. The performance analysis indicates that the proposed protocol is efficient for a low-cost environment.