On deep holes of projective Reed-Solomon codes

  title={On deep holes of projective Reed-Solomon codes},
  author={Jiandi Zhang and Daqing Wan},
  journal={2016 IEEE International Symposium on Information Theory (ISIT)},
  • J. Zhang, D. Wan
  • Published 9 May 2016
  • Computer Science, Mathematics
  • 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
In this paper, we determine the covering radius and a class of deep holes for Gabidulin codes with both rank metric and Hamming metric. Moreover, we give a necessary and sufficient condition for
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.
Deep Holes of Projective Reed-Solomon Codes
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.
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 $\#(I)=k$.
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
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
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
  • Carlos Munuera
  • Mathematics, Computer Science
    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, Mathematics
    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
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
The Newton radius of MDS codes
The Newton radius of a code is the maximal Hamming weight of a correctable error. The Newton radius of MDS codes is studied.
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
  • A. Dür
  • Mathematics, Computer Science
    Discret. Math.
  • 1994
For doubly-extended Reed-Solomon codes over GF( q ) with minimum distance d the covering radius ϱ is either d − 1 or d − 2, and a characterization of the deep holes of the sphere packing in Hamming space defined by the code is given in terms of their syndromes.