# On error distance of Reed-Solomon codes

@article{Li2008OnED, title={On error distance of Reed-Solomon codes}, author={Yujuan Li and Daqing Wan}, journal={Science in China Series A: Mathematics}, year={2008}, volume={51}, pages={1982-1988} }

The complexity of decoding the standard Reed-Solomon code is a well known open problem in coding theory. The main problem is to compute the error distance of a received word. Using the Weil bound for character sum estimate, we show that the error distance can be determined precisely when the degree of the received word is small. As an application of our method, we give a significant improvement of the recent bound of Cheng-Murray on non-existence of deep holes (words with maximal error distance…

## 32 Citations

### On the error distance of extended Reed-Solomon codes

- Computer ScienceAdv. Math. Commun.
- 2016

Using some algebraic constructions, this work is able to determine the error distance of words whose degrees are $k+1$ and $k-2$ to the extended Reed-Solomon codes.

### On Reed-Solomon codes

- Computer Science
- 2011

Li and Wan showed that the error distance can be determined when the degree of the received word as a polynomial is small, and a formula for the dimension of the generalized trace Reed-Solomon codes in some cases is obtained.

### Computing Error Distance of Reed-Solomon Codes

- Computer ScienceTAMC
- 2012

Using the Weil bound and a new sieve for distinct coordinates counting, this work is able to compute the error distance for a large class of received words and improves the existing results on the Cheng-Murray conjecture about the complete classification of deep holes for standard Reed-Solomon codes.

### On error distance of received words with fixed degrees to Reed-Solomon code

- Computer Science
- 2015

Using some elegant algebraic constructions, it is able to determine the error distance of received words whose degree is k+1 to the Standard Reed-Solomon code or Primitive Reed- Solomon code exactly.

### Improved error bounds for the distance distribution of Reed-Solomon codes

- Computer Science, MathematicsIEEE Transactions on Information Theory
- 2023

The generating function approach is used to derive simple expressions for the factorial moments of the distance distribution over Reed-Solomon codes, which leads to new results on the classification of deep holes of Reed Solomon codes.

### On deep holes of generalized Reed-Solomon codes

- Computer Science
- 2013

This work shows that the received word u is a deep hole of the standard Reed-Solomon codes [q-1, k] q if its Lagrange interpolation polynomial is the sum of monomial of degree q-2 and a polynometric of degree at most k-1.

### On deep holes of generalized Reed-Solomon codes

- Computer ScienceArXiv
- 2012

This work shows that the received word u is a deep hole of the standard Reed-Solomon codes [q− 1, k]q if its Lagrange interpolation polynomial is the sum of monomial of degree q− 2 and a polynometric of degree at most k− 1.

### Explicit Deep Holes of Reed-Solomon Codes

- Computer ScienceArXiv
- 2017

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.…

### On Determining Deep Holes of Generalized Reed–Solomon Codes

- Computer ScienceIEEE Transactions on Information Theory
- 2016

This paper classify deep holes completely for GRS codes RSp(D, k), where p is a prime, |D| > k ≥ (p - 1)/2, and is built on the idea of deep hole trees, and several results concerning the Erdös-Heilbronn conjecture.

### Determining deep hole trees of generalized Reed-Solomon codes and an application

- Computer Science
- 2017

This paper describes expected deep holes of generalized Reed-Solomon codes in an explicit way based on Newton interpolation and applies conclusions about expected deepholes to give a result on restricted sumset.

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