MDS Codes Research Articles

Weight distribution of the dual code of the MDS code and an upper bound on the minimum distance of the code

Weight distribution of the dual code of the MDS code and an upper bound on the minimum distance of the code

Generating efficient circulant-like MDS matrices for implementation

The exploration of maximal distance separable codes (MDS codes) has been a longstanding focus in error-correcting code theory and holds significant relevance in cryptography. Numerous approaches have been investigated for constructing MDS matrices, including deriving them from MDS codes, utilizing Hadamard matrices, Cauchy matrices, Vandermonde matrices, circulant matrices, circulant-like matrices, among others. However, a major challenge for cryptography designers is finding MDS matrices with low implementation cost. In this paper, we propose algorithms for generating efficient circulant-like MDS matrices of size , and for implementation. Subsequently, we evaluate the fixed points, the number of XOR operations of the proposed MDS matrices, and compare them with MDS matrices of other well-known ciphers. These proposed MDS matrices can become promising candidates for many cryptographic algorithms in the future.

Improved Field Size Bounds for Higher Order MDS Codes

Improved Field Size Bounds for Higher Order MDS Codes

Bounds on the Minimum Field Size of Network MDS Codes

Bounds on the Minimum Field Size of Network MDS Codes

An Improvement for Error-Correcting Pairs of Some Special MDS Codes

The error-correcting pair is a general algebraic decoding method for linear codes. Since every linear code is contained in an MDS linear code with the same minimum distance over some finite field extensions, we focus on MDS linear codes. Recently, He and Liao showed that for an MDS linear code [Formula: see text] with minimum distance [Formula: see text], if it has an [Formula: see text]-error-correcting pair, then the parameters of the pair have three possibilities. Moreover, for the first case, they gave a necessary condition for an MDS linear code [Formula: see text] with minimum distance [Formula: see text] to have an [Formula: see text]-error-correcting pair, and for the other two cases, they only gave some counterexamples. For the second case, in this paper, we give a necessary condition for an MDS linear code [Formula: see text] with minimum distance [Formula: see text] to have an [Formula: see text]-error-correcting pair, and then basing on the Product Singleton Bound, we prove that there are two cases for such pairs, and then give some counterexamples basing on twisted generalized Reed–Solomon codes for these cases.

Many Non-Reed-Solomon Type MDS Codes From Arbitrary Genus Algebraic Curves

Many Non-Reed-Solomon Type MDS Codes From Arbitrary Genus Algebraic Curves

Private information retrieval from locally repairable databases with colluding servers

We consider information-theoretical private information retrieval (PIR) from a coded database with colluding servers. We target, for the first time, locally repairable storage codes (LRCs). We consider any number of local groups g, locality r, local distance δ and dimension k. Our main contribution is a PIR scheme for maximally recoverable (MR) LRCs based on linearized Reed–Solomon codes, which achieve the smallest field sizes among MR-LRCs for many parameter regimes. In our scheme, nodes are identified with codeword symbols and servers are identified with local groups of nodes. Only locally non-redundant information is downloaded from each server, that is, only r nodes (out of r+δ−1) are downloaded per server. The PIR scheme achieves the (download) rate R=(N−k−rt+1)/N, where N=gr is the length of the MDS code obtained after removing the local parities, and for any t colluding servers such that k+rt≤N. For an unbounded number of stored files, the obtained rate is strictly larger than those of known PIR schemes that work for any MDS code. Finally, the obtained PIR scheme can also be adapted when communication between the user and each server is performed via linear network coding, achieving the same rate as previous PIR schemes for this scenario but with polynomial finite field sizes, instead of exponential. Our rates are equal to those of PIR schemes for Reed–Solomon codes, but Reed–Solomon codes are incompatible with the MR-LRC property or linear network coding, thus our PIR scheme is less restrictive in its applications.

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.

An Investigation on Reed-Solomon Codes as Erasure Coding Technique on Its Properties and Utilizations

Erasure coding is an essential part of cloud computing, which is an important technology for effective data storage for recovering data that may be lost due to various reasons, there are various erasure coding techniques in the market. In this paper, linear MDS codes, which is a branch of erasure coding, will be investigated on their performance and usage. This paper will focus on the Reed-Solomon Code, which is the most implemented form of linear MDS codes, on three different aspects: 1) the methodologies of the encoding and decoding operations; 2) the pros and cons of different forms of Reed-Solomon Codes; 3) the different ways that different Reed-Solomon Codes are being employed. Moreover, the paper includes the definition of general Cloud Computing for the audience to understand its importance, how the erasure coding acts like a fault tolerance system of Cloud Computing, and how different kinds of Reed-Solomon code perform on tolerating erasures in cloud storage failures.

MDS and MHDR Cyclic Codes over Finite Chain Rings

This work establishes a unique set of generators for a cyclic code over a finite chain ring. Towards this, we first determine the minimal spanning set and rank of the code. Furthermore, sufficient as well as necessary conditions for a cyclic code to be an MDS code and for a cyclic code to be an MHDR code are obtained. Finally, to support our results, some examples of optimal cyclic codes are presented.

Exploring the Feasibility of Meeting the MDS Criterion in Random Matrices and Searching for Efficient Random Involutory Hadamard MDS Matrices

Maximum distance separable codes (MDS codes) have been studied for a long time in error correcting code theory and have important applications in cryptography. There are many different methods have been studied to build MDS matrices such as: from MDS codes, Cauchy matrices, Hadamard matrices, Vandermonde matrices, circulant and circulant-like matrices etc. In this paper, the construction of MDS matrix from random matrices is studied, and the possibility of satisfying the MDS condition of any random matrix is presented. Then, the method of random search of MDS matrices will be given. In addition, we propose algorithms to randomly search for efficient involutory Hadamard MDS matrices of size 4, and 8. These results will be meaningful to determine the possibility that MDS matrices can be obtained from random matrices and diversify the methods of searching for MDS matrices. The involutory Hadamard MDS matrices make the encryption and decryption processes identical, which makes them very efficient for both hardware and software implementation.

Вычисление пар, исправляющих ошибки для алгеброгеометрического кода

For an arbitrary algebrogeometric code and its dual, error-correcting pairs are explicitly calculated. Such a pair consists of codes that are necessary for an efficient decoding algorithm for a given code. The type of pairs depends on the powers of the divisors, with the help of which both the source code and one of the codes included in the pair are constructed. For an algebraic geometric code CL(D, G) of length and associated with a functional field F/Fq of genus g, pairs correcting t = ⌊(n - deg( G) - g - 1)/2⌋ errors, under certain restrictions on the degrees of divisors involved in their construction, are pairs of codes (Cl(D,F),Cl(D,G + F)⊥) or (CL(D,F⊥),Cl(D,F - G)). Restrictions on the degrees of divisors of codes (CL(D,F), CL(D,G - F)) constituting a pair correcting t = ⌊(deg( G) - 3g + 1)/2⌋ errors for the dual code CL(D, G)⊥) Cases where one of the codes involved in constructing a pair belongs to class of MDS codes and the parameters under which this situation is possible are derived. In addition, possible bounds for the divisors involved in the construction of error-correcting pairs for the subfield subcodes CL(D,G)|fp and CL< are calculated /sub>(D, G)⊥)|Fp of the original algebrogeometric code and its dual, with the expansion degree m = 2 (Fq = Fp2).

Asymmetric Entanglement-Assisted Quantum MDS Codes Constructed from Constacyclic Codes.

Due to the asymmetry of quantum errors, phase-shift errors are more likely to occur than qubit-flip errors. Consequently, there is a need to develop asymmetric quantum error-correcting (QEC) codes that can safeguard quantum information transmitted through asymmetric channels. Currently, a significant body of literature has investigated the construction of asymmetric QEC codes. However, the asymmetry of most QEC codes identified in the literature is limited by the dual-containing condition within the Calderbank-Shor-Steane (CSS) framework. This limitation restricts the exploration of their full potential in terms of asymmetry. In order to enhance the asymmetry of asymmetric QEC codes, we utilize entanglement-assisted technology and exploit the algebraic structure of cyclotomic cosets of constacyclic codes to achieve this goal. In this paper, we generalize the decomposition method of the defining set for constacyclic codes and apply it to count the number of pre-shared entangled states in order to construct four new classes of asymmetric entanglement-assisted quantum maximal-distance separable (EAQMDS) codes that satisfy the asymmetric entanglement-assisted quantum Singleton bound. Compared with the codes existing in the literature, the lengths of the constructed EAQMDS codes and the number of pre-shared entangled states are more general, and the codes constructed in this paper have greater asymmetry.

The Maximal Length of q-ary MDS Elliptic Codes Is Close to

Abstract Determining the maximal length of MDS codes with certain dimension is one of the central topics in coding theory and finite geometry. The MDS Main Conjecture states that the maximal length of a non-trivial $q$-ary MDS code of dimension $k$ is $q+1$ except when $q$ is even and $k=3$ or $k=q-1$. We prove that the maximal length of non-trivial $q$-ary MDS elliptic codes is close to $\frac{q}{2}$, which gives an affirmative answer to a conjecture of Li, Wan, and Zhang. Moreover, we apply our result to derive an answer to a question on subset sums in finite abelian groups from elliptic curves.

On constacyclic codes of length 9ps over 𝔽pm and their optimal codes

The problem of classifying constacyclic codes over a finite field, both the Hamming distance and the algebraic structure, is an interesting problem in algebraic coding theory. For the repeated-root constacyclic codes of length [Formula: see text] over [Formula: see text], where [Formula: see text] is a prime number and [Formula: see text] does not divide [Formula: see text], the problem has been solved completely for all [Formula: see text] and partially for [Formula: see text]. In this paper, we solve the problem for [Formula: see text] and all primes [Formula: see text] different from [Formula: see text] and [Formula: see text]. In particular, we characterize the Hamming distance of all repeated-root constacyclic codes of length [Formula: see text] over [Formula: see text]. As an application, we identify all optimal and near-optimal codes with respect to the Singleton bound of these types, namely, MDS, almost-MDS, and near-MDS codes.

A New Construction of Nonlinear Codes via Algebraic Function Fields

In coding theory, constructing codes with good parameters is one of the most important and fundamental problems. A great many good codes have been constructed over alphabets of sizes equal to prime powers, however, good block codes over other alphabet sizes are rare. In this paper, we provide a new explicit construction of ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> + 1)-ary nonlinear codes via algebraic function fields, where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> is a prime power. Our codes are constructed by evaluating rational functions at all rational places of an algebraic function field. Compared with algebraic geometry codes, the main difference is that we allow rational functions to be evaluated at pole places. After evaluating rational functions from a union of Riemann-Roch spaces, we obtain a family of nonlinear codes over the alphabet F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>q</i></sub> ⋃ {∞}. It turns out that our codes have better parameters than those obtained from MDS codes or good algebraic geometry codes via code alphabet extension and restriction.

Quantum MDS codes with new length and large minimum distance

According to the well-known CSS construction, constructing quantum MDS codes are extensively investigated via Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes. In this paper, given two Hermitian self-orthogonal GRS codes GRSk1(A,vA) and GRSk2(B,vB), we propose a sufficient condition to ensure that GRSk(A∪B,vA∪B) is still a Hermitian self-orthogonal code. Consequently, we first present a new general construction of infinitely families of quantum MDS codes from known ones. Moreover, applying the trace function and norm function over finite fields, we give another two new constructions of quantum MDS codes with flexible parameters. It turns out that the forms of the lengths of our quantum MDS codes are quite different from previous known results in the literature. Meanwhile, the minimum distances of all the q-ary quantum MDS codes are bigger than q/2+1.

Idempotents in simple skew group rings and applications to skew cyclic codes

We give the form of complete sets of pairwise orthogonal primitive idempotents in a simple skew group ring. Using these idempotents, we construct some classes of skew cyclic codes. We give also some examples of MDS codes and rows orthogonal and MDS matrices. Communicated by Eric Jespers

Two classes of quantum codes from almost MDS codes

Hermitian dual-containing codes are an important class of linear codes which have important applications in the construction of quantum codes. In this paper, two classes of Hermitian dual-containing almost MDS codes over finite fields are studied. By employing the Hermitian construction, a class of quantum codes with minimum distance [Formula: see text] and a class of quantum codes with minimum distance [Formula: see text] are constructed.

Generalized Weights of Convolutional Codes

In 1997 Rosenthal and York defined generalized Hamming weights for convolutional codes, by regarding a convolutional code as an infinite dimensional linear code endowed with the Hamming metric. In this paper, we propose a new definition of generalized weights of convolutional codes, that takes into account the underlying module structure of the code. We derive the basic properties of our generalized weights and discuss the relation with the previous definition. We establish upper bounds on the weight hierarchy of MDS and MDP codes and show that, depending on the code parameters, some or all of the generalized weights of MDS codes are determined by the length, rank, and internal degree of the code. We also prove an anticode bound for convolutional codes and define optimal anticodes as the codes which meet the anticode bound. Finally, we classify optimal anticodes and compute their weight hierarchy.