相关论文: On the generalization of $g$-circulant MDS matrice…
We show that the free distance, as a function on a space parameterizing a family of convolutional codes, is a lower-semicontinuous function and that, therefore, the property of being Maximum Distance Separable (MDS) is an open condition.…
We introduce the notion of Universally Decodable Matrices of Genus g (UDMG), which for g=0 reduces to the notion of Universally Decodable Matrices (UDM) introduced in [8]. A UDMG is a set of L matrices over a finite field, each with K rows,…
This work deals with partial MDS (PMDS) codes, a special class of locally repairable codes, used for distributed storage system. We first show that a known construction of these codes, using Gabidulin codes, can be extended to use any…
Self-dual maximum distance separable codes (self-dual MDS codes) and self-dual near MDS codes are very important in coding theory and practice. Thus, it is interesting to construct self-dual MDS or self-dual near MDS codes. In this paper,…
Euclidean distance matrices (EDM) are symmetric nonnegative matrices with several interesting properties. In this article, we introduce a wider class of matrices called generalized Euclidean distance matrices (GDMs) that include EDMs. Each…
This paper deals with the problem of constructing superregular matrices that lead to MDP convolutional codes. These matrices are a type of lower block triangular Toeplitz matrices with the property that all the square submatrices that can…
Maximum distance separable (MDS) codes are optimal where the minimum distance cannot be improved for a given length and code size. Twisted Reed-Solomon codes over finite fields were introduced in 2017, which are generalization of…
An MDS matrix is a matrix whose minors all have full rank. A question arising in coding theory is what zero patterns can MDS matrices have. There is a natural combinatorial characterization (called the MDS condition) which is necessary over…
A $q$-ary maximum distance separable (MDS) code $C$ with length $n$, dimension $k$ over an alphabet $\mathcal{A}$ of size $q$ is a set of $q^k$ codewords that are elements of $\mathcal{A}^n$, such that the Hamming distance between two…
The main conjecture on maximum distance separable (MDS) codes states that, execpt for some special cases, the maximum length of a q-ary linear MDS code is q+1. This conjecture does not hold true for near maximum distance separable codes…
Algebraic methods for the design of series of maximum distance separable (MDS) linear block and convolutional codes to required specifications and types are presented. Algorithms are given to design codes to required rate and required…
We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary…
The optimal branch number of MDS matrices makes them a preferred choice for designing diffusion layers in many block ciphers and hash functions. However, in lightweight cryptography, Near-MDS (NMDS) matrices with sub-optimal branch numbers…
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 code is said to be near maximum…
The determination of the maximal length of maximum distance separable (MDS) codes arising from elliptic curves is a central problem in coding theory. For an elliptic curve $E$ over $\mathbb{F}_q$, let $\operatorname{MEC}(k,q)$ denote the…
In this paper, we propose two algorithms for a hybrid construction of all $n\times n$ MDS and involutory MDS matrices over a finite field $\mathbb{F}_{p^m}$, respectively. The proposed algorithms effectively narrow down the search space to…
Multidimensional scaling (MDS) is a family of methods that embed a given set of points into a simple, usually flat, domain. The points are assumed to be sampled from some metric space, and the mapping attempts to preserve the distances…
A $q$-ary code of length $n$, size $M$, and minimum distance $d$ is called an $(n,M,d)_q$ code. An $(n,q^{k},n-k+1)_q$ code is called a maximum distance separable (MDS) code. In this work, some MDS codes over small alphabets are classified.…
Let $p$ be a prime and let $q$ be a power of $p$. In this paper, by using generalized Reed-Solomon (GRS for short) codes and extended GRS codes, we construct two new classes of quantum maximum-distance- separable (MDS) codes with parameters…
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…