## One Citation

### 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.

## References

SHOWING 1-10 OF 28 REFERENCES

### 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 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 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 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.

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

- Computer ScienceFinite Fields Their Appl.
- 2016

### On the covering radius of binary codes (Corresp.)

- Computer ScienceIEEE Trans. Inf. Theory
- 1978

It is shown that the covering radius r_{m} of the first-order Reed-Muller code of lenglh 2^{m} satisfies 2-2 m-l-2 lceil m/2 r-rceil -1 r-m, which is equivalent to 2.m-1-2.m/2-1.

### Covering radius - Survey and recent results

- Computer ScienceIEEE Trans. Inf. Theory
- 1985

A number of upper and lower bounds, including asymptotic results, a few exact determinations of covering radius, and some extensive relations with other aspects of coding theory through the Reed-Muller codes are presented.

### Gabidulin Decoding via Minimal Bases of Linearized Polynomial Modules

- Computer Science
- 2014

This work shows how Gabidulin codes can be decoded via parametrization by using interpolation modules over the ring of linearized polynomials with composition, and strengthens the link betweenGabidulin decoding and Reed-Solomon decoding.

### Packing and Covering Properties of Rank Metric Codes

- Computer Science, MathematicsIEEE Transactions on Information Theory
- 2008

First, the packing properties of rank metric codes are investigated, and sphere covering properties are studied, to derive bounds on their parameters, and investigate their asymptotic covering properties.