On deep holes of projective Reed-Solomon codes

  title={On deep holes of projective Reed-Solomon codes},
  author={Jun Zhang and Daqing Wan},
  journal={2016 IEEE International Symposium on Information Theory (ISIT)},
  • Jun ZhangD. Wan
  • Published 9 May 2016
  • Computer Science
  • 2016 IEEE International Symposium on Information Theory (ISIT)
In this paper, we obtain new results on the covering radius and deep holes for projective Reed-Solomon (PRS) codes. 

On deep-holes of Gabidulin codes

Explicit Deep Holes of Reed-Solomon Codes

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 Deep Holes of Elliptic Curve Codes

A method to construct deep holes for elliptic curve codes and it is conjecture that this construction is complete in the sense that it gives all deep holes.

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

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.

On deep holes of primitive projective Reed-Solomon codes

A class of deepholes of primitive projective Reed-Solomon codes is obtained by using the generating matrix of maximaldistance separable codes over the finite field $\mathbb{F}_q$ and Vandermonde determinant.

On deep holes of generalized projective Reed-Solomon codes

D\"ur's theorem on the relation between the covering radius and minimum distance of the GPRS_q(D, k) is shown to be nonzero for any subset $I\subseteq D$ with $\#(I)=k$.

MDS, Near-MDS or 2-MDS Self-Dual Codes via Twisted Generalized Reed-Solomon Codes

A sufficient and necessary condition that a twisted Reed-Solomon (TRS) code with two twists is MDS is characterized and some constructions of self-dual TGRS codes are presented, which are MDS, NMDS or 2-MDS.

On integral weight spectra of the MDS codes cosets of weight 1, 2, and 3

The weight of a coset of a code is the smallest Hamming weight of any vector in the coset, and integral weight spectrum the overall numbers of weight w vectors in all the cosets of a fixed weight is called.

On Cosets Weight Distribution of Doubly-Extended Reed-Solomon Codes of Codimension 4

It is proved that the difference between the w-th components of the distributions is uniquely determined by the Difference between the 3-rd components, which implies an interesting (and in some sense unexpected) symmetry of the obtained distributions.

On the weight distribution of the cosets of MDS codes

The Bonneau formula is transformed into a more structured and convenient form and it is proved that all the cosets of weight of the MDS code have the same weight distribution.



On deep holes of generalized Reed-Solomon codes

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.

Counting generalized Reed-Solomon codes

The number of generalized Reed-Solomon (GRS) codes is counted, including the codes coming from a non-degenerate conic plus nucleus and known formulae for the number of MDS codes are compared.

On the main conjecture on geometric MDS codes

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  • Computer Science, Mathematics
    IEEE Trans. Inf. Theory
  • 1992
The conjecture for codes arising from curves of genus one or two when the cardinal of the ground field is large enough is proven and the main conjecture on MDS codes for geometric codes is attacked.

Deep Holes and MDS Extensions of Reed–Solomon Codes

  • K. Kaipa
  • Computer Science
    IEEE Transactions on Information Theory
  • 2017
This paper gives a complete classification of deep holes of Reed–Solomon codes with redundancy three in all dimensions and shows that this problem is equivalent to the problem of classifying maximum distance separable (MDS) extensions of Reed- Solomon codes by one digit.

On the error distance of extended Reed-Solomon codes

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 Determining Deep Holes of Generalized Reed–Solomon Codes

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.

On error distance of Reed-Solomon codes

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.

The Newton radius of MDS codes

The Newton radius of a code is the maximal Hamming weight of a correctable error and is studied in the context of MDS codes.

On deep holes of standard Reed-Solomon codes

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 the covering radius of Reed - Solomon codes

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    Discret. Math.
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