相关论文: New Deep Holes of Generalized Reed-Solomon Codes
Determining deep holes is an important topic in decoding Reed-Solomon codes. In a previous paper [8], we showed that the received word $u$ is a deep hole of the standard Reed-Solomon codes $[q-1, k]_q$ if its Lagrange interpolation…
For a linear code, deep holes are defined to be vectors that are further away from codewords than all other vectors. The problem of deciding whether a received word is a deep hole for generalized Reed-Solomon codes is proved to be…
Determining deep holes is an important open problem in decoding Reed-Solomon codes. It is well known that the received word is trivially a deep hole if the degree of its Lagrange interpolation polynomial equals the dimension of the…
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 deep hole problem is a fundamental problem in coding theory, and it has many important applications in code constructions and cryptography. The deep hole problem of Reed-Solomon codes has gained a lot of attention. As a generalization…
The deep holes of a linear code are the vectors that achieve the maximum error distance (covering radius) to the code. {Determining the covering radius and deep holes of linear codes is a fundamental problem in coding theory. In this paper,…
We study the problem of classifying deep holes of Reed-Solomon codes. We show that this problem is equivalent to the problem of classifying MDS extensions of Reed-Solomon codes by one digit. This equivalence allows us to improve recent…
For generalized Reed-Solomon codes, it has been proved \cite{GuruswamiVa05} that the problem of determining if a received word is a deep hole is co-NP-complete. The reduction relies on the fact that the evaluation set of the code can be…
Determining deep holes is an important topic in decoding Reed-Solomon codes. Let $l\ge 1$ be an integer and $a_1,\ldots,a_l$ be arbitrarily given $l$ distinct elements of the finite field ${\bf F}_q$ of $q$ elements with the odd prime…
For an $[n,k]$ Reed-Solomon code $\mathcal{C}$, it can be shown that any received word $r$ lies a distance at most $n-k$ from $\mathcal{C}$, denoted $d(r,\mathcal{C})\leq n-k$. Any word $r$ meeting the equality is called a deep hole.…
Projective Reed-Solomon (PRS) codes are Reed-Solomon codes of the maximum possible length q+1. The classification of deep holes --received words with maximum possible error distance-- for PRS codes is an important and difficult problem. In…
Under polynomial time reduction, the maximum likelihood decoding of a linear code is equivalent to computing the error distance of a received word. It is known that the decoding complexity of standard Reed-Solomon codes at certain radius is…
In this paper, we obtain new results on the covering radius and deep holes for projective Reed-Solomon (PRS) codes.
We use the generating function approach to derive simple expressions for the factorial moments of the distance distribution over Reed-Solomon codes. We obtain better upper bounds for the error term of a counting formula given by Li and Wan,…
Reed-Solomon (RS) codes are among the most ubiquitous codes due to their good parameters as well as efficient encoding and decoding procedures. However, RS codes suffer from having a fixed length. In many applications where the length is…
Because of their importance in applications and their quite simple definition, Reed-Solomon codes can be explained in any introductory course on coding theory. However, decoding algorithms for Reed-Solomon codes are far from being simple…
Polynomial remainder codes are a large class of codes derived from the Chinese remainder theorem that includes Reed-Solomon codes as a special case. In this paper, we revisit these codes and study them more carefully than in previous work.…
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…
A general class of polynomial remainder codes is considered. Such codes are very flexible in rate and length and include Reed-Solomon codes as a special case. As an extension of previous work, two joint error-and-erasure decoding approaches…
Extended Han-Zhang codes are a class of linear codes where each code is either a non-generalized Reed-Solomon (non-GRS) maximum distance separable (MDS) code or a near MDS (NMDS) code. They have important applications in communication,…