相关论文: Generalized Reed-Solomon Codes with Sparsest and B…

We consider the repair problem for Reed--Solomon (RS) codes, evaluated on an $\mathbb{F}_q$-linear subspace $U\subseteq\mathbb{F}_{q^m}$ of dimension $d$, where $q$ is a prime power, $m$ is a positive integer, and $\mathbb{F}_q$ is the…

A linear $ [n,k]_q $ code $ C $ is said to be a full weight spectrum (FWS) code if there exist codewords of each nonzero weight less than or equal to $ n $. In this brief communication we determine necessary and sufficient conditions for…

For a symmetric positive semidefinite linear system of equations $\mathcal{Q} {\bf x} = {\bf b}$, where ${\bf x} = (x_1,\ldots,x_s)$ is partitioned into $s$ blocks, with $s \geq 2$, we show that each cycle of the classical block symmetric…

In this paper, we consider Griesmer codes, namely those linear codes meeting the Griesmer bound. Let $C$ be an $[n,k,d]_q$ Griesmer code with $q=p^f$, where $p$ is a prime and $f\ge1$ is an integer. In 1998, Ward proved that for $q=p$, if…

We introduce a consistent and efficient method to construct self-dual codes over $GF(q)$ with symmetric generator matrices from a self-dual code over $GF(q)$ of smaller length where $q \equiv 1 \pmod 4$. Using this method, we improve the…

Maximum distance separable (MDS) and almost maximum distance separable (AMDS) codes have been widely used in various fields such as communication systems, data storage, and quantum codes because of their algebraic properties and excellent…

Many classical constructions, such as Plotkin's and Turyn's, were generalized by matrix product (MP) codes. Quasi-twisted (QT) codes, on the other hand, form an algebraically rich structure class that contains many codes with best-known…

The complexity of maximal likelihood decoding of the Reed-Solomon codes $[q-1, k]_q$ is a well known open problem. The only known result in this direction states that it is at least as hard as the discrete logarithm in some cases where the…

Recently, the construction of new MDS Euclidean self-dual codes has been widely investigated. In this paper, for square q, we utilize generalized Reed-Solomon (GRS) codes and their extended codes to provide four generic families of q-ary…

It is known that for every spatial complete graph on $n\ge 7$ vertices, the summation of the second coefficients of the Conway polynomials over the Hamiltonian knots is congruent to $r_{n}$ modulo $(n-5)!$, where $r_{n} = (n-5)!/2$ if…

In this paper, by calculating the dual code of the Schur square for the standard twisted Reed-Solomon code, we give a sufficient and necessary condition for the generalized twisted Reed-Solomon code with $h+t\le k-1$ to be self-orthogonal,…

A local tester for an error correcting code $C\subseteq \Sigma^{n}$ is a tester that makes $Q$ oracle queries to a given word $w\in \Sigma^n$ and decides to accept or reject the word $w$. An optimal local tester is a local tester that has…

Consider an ordinary generating function $\sum_{k=0}^{\infty}c_kx^k$, of an integer sequence of some combinatorial relevance, and assume that it admits a closed form $C(x)$. Various instances are known where the corresponding truncated sum…

In this paper, we introduce a novel explicit family of subcodes of Reed-Solomon (RS) codes that efficiently achieve list decoding capacity with a constant output list size. Our approach builds upon the idea of large linear subcodes of RS…

A new class of structured codes called Quasi Group Codes (QGC) is introduced. A QGC is a subset of a group code. In contrast with group codes, QGCs are not closed under group addition. The parameters of the QGC can be chosen such that the…

Self-dual maximum distance separable (MDS) codes over finite fields are linear codes with significant combinatorial and cryptographic applications. Twisted generalized Reed-Solomon (TGRS) codes can be both MDS and self-dual. In this paper,…

We investigate two classes of extended codes and provide necessary and sufficient conditions for these codes to be non-GRS MDS codes. We also determine the parity check matrices for these codes. Using the connection of MDS codes with arcs…

Many literatures consider the extended Reed-Solomon (RS) codes, including their dual codes and covering radii, but few focus on extended algebraic geometry (AG) codes of genus $g\ge1$. In this paper, we investigate extended AG codes and…

In 2005, Boman et al introduced the concept of factor width for a real symmetric positive semidefinite matrix. This is the smallest positive integer $k$ for which the matrix $A$ can be written as $A=VV^T$ with each column of $V$ containing…

Both MDS and Euclidean self-dual codes have theoretical and practical importance and the study of MDS self-dual codes has attracted lots of attention in recent years. In particular, determining existence of $q$-ary MDS self-dual codes for…