Related papers: On subcodes of the generalized Reed-Solomon codes
We construct a new family of quantum MDS codes from classical generalized Reed-Solomon codes and derive the necessary and sufficient condition under which these quantum codes exist. We also give code bounds and show how to construct them…
In this paper, deep holes of Reed-Solomon (RS) codes are studied. A new class of deep holes for generalized affine RS codes is given if the evaluation set satisfies certain combinatorial structure. Three classes of deep holes for projective…
The dual of the Kasami code of length $q^2-1$, with $q$ a power of $2$, is constructed by concatenating a cyclic MDS code of length $q+1$ over $F_q$ with a Simplex code of length $q-1$. This yields a new derivation of the weight…
In this paper, we give a parity check matrix for the $(+)$-extended twisted generalized Reed Solomon (in short, ETGRS) code, and then not only prove that it is MDS or NMDS, but also determine the weight distribution. Especially, based on…
Lifted Reed-Solomon and multiplicity codes are classes of codes, constructed from specific sets of $m$-variate polynomials. These codes allow for the design of high-rate codes that can recover every codeword or information symbol from many…
MDS codes have diverse practical applications in communication systems, data storage, and quantum codes due to their algebraic properties and optimal error-correcting capability. In this paper, we focus on a class of linear codes and…
Regenerating codes provide an efficient way to recover data at failed nodes in distributed storage systems. It has been shown that regenerating codes can be designed to minimize the per-node storage (called MSR) or minimize the…
Reed-Solomon (RS) codes are constructed over a finite field that have been widely employed in storage and communication systems. Many fast encoding/decoding algorithms such as fast Fourier transform (FFT) and modular approach are designed…
The performance of Reed--Solomon codes (RS codes, for short) in the presence of insertion and deletion errors has attracted growing attention in recent literature. In this work, we further study this intriguing mathematical problem,…
The hull of a linear code is defined to be the intersection of the code and its dual. When the size of the hull is small, it has been proved that some algorithms for checking permutation equivalence of two linear codes and computing the…
By using the notion of $d$-embedding $\Gamma$ of a (canonical) subgeometry $\Sigma$ and of exterior set with respect to the $h$-secant variety $\Omega_{h}(\mathcal{A})$ of a subset $\mathcal{A}$, $ 0 \leq h \leq n-1$, in the finite…
We characterize the generator matrix in standard form of generalized Gabidulin codes. The parametrization we get for the non-systematic part of this matrix coincides with the $q$-analogue of generalized Cauchy matrices, leading to the…
In this work, doubly extended linearized Reed--Solomon codes and triply extended Reed--Solomon codes are generalized. We obtain a general result in which we characterize when a multiply extended code for a general metric attains the…
In this paper, we obtain some new results on the existence of MDS self-dual codes utilizing (extended) generalized Reed-Solomon codes over finite fields of odd characteristic. For some fixed $q$, our results can produce more classes of MDS…
One of the central tasks in quantum error-correction is to construct quantum codes that have good parameters. In this paper, we construct three new classes of quantum MDS codes from classical Hermitian self-orthogonal generalized…
Let $\mathrm{SLAut}(\mathbb{F}_{q}^{n})$ denote the group of all semilinear isometries on $\mathbb{F}_{q}^{n}$, where $q=p^{e}$ is a prime power. In this paper, we investigate general properties of linear codes associated with $\sigma$…
In this paper, we prove that the sub-field images of generalized Reed-Solomon (RS) codes can achieve the symmetric capacity of p-ary memoryless channels. Unlike the totally random linear code ensemble, as a class of maximum distance…
Maximum distance separable (MDS) and near maximum distance separable (NMDS) codes have been widely used in various fields such as communication systems, data storage, and quantum codes due to their algebraic properties and excellent…
We provide a geometric characterization of $k$-dimensional $\mathbb{F}_{q^m}$-linear sum-rank metric codes as tuples of $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^k$. We then use this characterization to study one-weight codes in the…
The classical family of Reed-Solomon codes consist of evaluations of polynomials over the finite field $\mathbb{F}_q$ of degree less than $k$, at $n$ distinct field elements. These are arguably the most widely used and studied codes, as…