# New Deep Holes of Generalized Reed-Solomon Codes

@article{Zhang2012NewDH, title={New Deep Holes of Generalized Reed-Solomon Codes}, author={Jun Zhang and Fang-Wei Fu and Qunying Liao}, journal={ArXiv}, year={2012}, volume={abs/1205.6593} }

Deep holes play an important role in the decoding of generalized Reed-Solomon codes. Recently, Wu and Hong [11] found a new class of deep holes for standard Reed-Solomon codes. In the present paper, we give a concise method to obtain a new class of deep holes for generalized Reed-Solomon codes. In particular, for standard Reed-Solomon codes, we get the new class of deep holes given in [11]. Li and Wan [6] studied deep holes of generalized Reed-Solomon codes GRSk(Fq,D) and characterized deep…

## 10 Citations

### 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.

### 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.…

### Finite Geometry and Deep Holes of Reed-Solomon Codes over Finite Local Rings

- Mathematics, Computer ScienceCommunications in Mathematical Research
- 2022

This paper proposes the maximum arc problem, normal rational curve conjecture, and extensions of normal rational curves over finite local rings, analogously to the finite geometry over finite fields, and studies the deep hole problem of generalized Reed-Solomon codes over finiteLocal rings.

### 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 projective Reed-Solomon codes

- Computer Science2016 IEEE International Symposium on Information Theory (ISIT)
- 2016

New results on the covering radius and deep holes for projective Reed-Solomon (PRS) codes are obtained.

### On Deep Holes of Elliptic Curve Codes

- Computer Science, MathematicsIEEE Transactions on Information Theory
- 2023

It is conjecture that the construction of deep holes for long elliptic curve codes is complete in the sense that it gives all deep holes.

### Rational points on complete symmetric hypersurfaces over finite fields

- MathematicsDiscret. Math.
- 2020

### Deep holes in Reed-Solomon codes based on Dickson polynomials

- Computer ScienceFinite Fields Their Appl.
- 2016

### Deep Holes of Projective Reed-Solomon Codes

- Computer ScienceIEEE Transactions on Information Theory
- 2020

This paper uses algebraic methods to explicitly construct three classes of deep holes for PRS codes with redundancy four, and shows that these three classes completely classify all deep holes of PRS code with redundancyFour.

## References

SHOWING 1-10 OF 12 REFERENCES

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

- Computer ScienceArXiv
- 2011

This paper finds a new class of deep holes for standard Reed-Solomon codes [q − 1, k]q with q a power of the prime p and shows that the received word u is a deep hole if its Lagrange interpolation polynomial is the sum of monomial of degree q − 2 and a polynometric of degree at most k − 1.

### On Deciding Deep Holes of Reed-Solomon Codes

- Computer ScienceTAMC
- 2007

By applying Cafure-Matera estimation of rational points on algebraic varieties, it is proved that the received vector (f(α))α∈Fp for the Reed-Solomon, cannot be a deep hole, whenever f(x) is a polynomial of degree k + d for 1 ≤ d ≤ p3/13-Ɛ.

### On the List and Bounded Distance Decodibility of the Reed-Solomon Codes (Extended Abstract)

- Computer ScienceFOCS
- 2004

These results show that the decoding problems for the Reed-Solomon code are at least as hard as the discrete logarithm problem over finite fields.

### Improved decoding of Reed-Solomon and algebraic-geometric codes

- Computer ScienceProceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
- 1998

An improved list decoding algorithm for decoding Reed-Solomon codes and alternant codes and algebraic-geometric codes is presented, including a solution to a weighted curve fitting problem, which is of use in soft-decision decoding algorithms for Reed- Solomon codes.

### Reed–Solomon Codes

- Computer Science
- 2006

A Reed-Solomon (RS) code is an error-correcting code first described in a paper by Reed and Solomon in 1960 [9]. Since that time they’ve been applied in CD-ROMs, wireless communications, space…

### On error distance of Reed-Solomon codes

- Computer Science
- 2008

A significant improvement is given of the recent bound of Cheng-Murray on non-existence of deep holes (words with maximal error distance) using the Weil bound for character sum estimate.

### On the list and bounded distance decodibility of Reed-Solomon codes

- Computer Science45th Annual IEEE Symposium on Foundations of Computer Science
- 2004

This paper proves that list decoding can not be done for radius n - g(n, k: q) or larger, otherwise the discrete logarithm over F/sub qg(m, k, q)-k/ is easy, and shows that the discretelogarithM problem over F-sub qh/ can be efficiently reduced by a randomized algorithm to the bounded distance decoding problem of the Reed-Solomon code.

### Decoding of Reed Solomon Codes beyond the Error-Correction Bound

- Computer ScienceJ. Complex.
- 1997

To the best of the knowledge, this is the first efficient (i.e., polynomial time bounded) algorithm which provides error recovery capability beyond the error-correction bound of a code for any efficient code.

### Maximum-likelihood decoding of Reed-Solomon codes is NP-hard

- Computer ScienceIEEE Transactions on Information Theory
- 2005

It is proved that maximum-likelihood decoding is NP-hard for the family of Reed-Solomon codes, thereby strengthening a result of Bruck and Naor.

### On the subset sum problem over finite fields

- Computer Science, MathematicsFinite Fields Their Appl.
- 2008