Reed–Solomon Codes over Fields of Characteristic Zero

@article{Sippel2019ReedSolomonCO,
  title={Reed–Solomon Codes over Fields of Characteristic Zero},
  author={Carmen Sippel and Cornelia Ott and Sven Puchinger and Martin Bossert},
  journal={2019 IEEE International Symposium on Information Theory (ISIT)},
  year={2019},
  pages={1537-1541}
}
  • Carmen Sippel, C. Ott, +1 author M. Bossert
  • Published 2019
  • Computer Science, Mathematics
  • 2019 IEEE International Symposium on Information Theory (ISIT)
We study Reed–Solomon codes over arbitrary fields, inspired by several recent papers dealing with Gabidulin codes over fields of characteristic zero. Over the field of rational numbers, we derive bounds on the coefficient growth during encoding and the bit complexity of decoding, which is polynomial in the code length and in the bit width of error and codeword values. The results can be generalized to arbitrary number fields. 

References

SHOWING 1-10 OF 34 REFERENCES
An alternative decoding method for Gabidulin codes in characteristic zero
TLDR
A new application of Gabidulin codes over infinite fields: low-rank matrix recovery and an alternative decoding approach is presented based on a Gao type key equation, reducing the complexity to at least O(n2). Expand
Rank metric and Gabidulin codes in characteristic zero
TLDR
The theory of rank metric and Gabidulin codes are transposed to the case of fields of characteristic zero and the Frobenius automorphism is replaced by any element of the Galois group. Expand
Generalized Gabidulin codes over fields of any characteristic
TLDR
It is shown that the codes obtained by reduction are the classical Gabidulin codes over finite fields, which means that under some conditions, decoding generalized Gabidoxin codes over integer rings can be reduced to decoding Gabidul codes over a finite field. Expand
New List Decoding Algorithms for Reed–Solomon and BCH Codes
  • Yingquan Wu
  • Computer Science, Mathematics
  • IEEE Transactions on Information Theory
  • 2008
TLDR
A rational curve fitting algorithm is devised and applied to the list decoding of Reed-Solomon and Bose-Chaudhuri-Hocquenghen codes and the resulting list decoding algorithms exhibit the following significant properties. Expand
Syndrome Decoding of Reed–Solomon Codes Beyond Half the Minimum Distance Based on Shift-Register Synthesis
TLDR
To decode errors beyond half the minimum distance, the new decoder is allowed to fail for some high-weight error patterns with a very small probability, like classical algebraic decoding algorithms. Expand
A quadratic Welch-Berlekamp algorithm to decode generalized Gabidulin codes, and some variants
  • Gwezheneg Robert
  • Mathematics, Computer Science
  • 2016 IEEE International Symposium on Information Theory (ISIT)
  • 2016
TLDR
It is proved that the output of the algorithm is a solution of the reconstruction problem, and it is established that one of the variant of this algorithm is quadratic. Expand
Power Decoding Reed-Solomon Codes Up to the Johnson Radius
TLDR
This paper shows how to incorporate multiplicities, the parameter allowing the Guruswami-Sudan algorithm to decode up to the Johnson radius, and describes how to efficiently solve the resulting key equations. Expand
Improved decoding of Reed-Solomon and algebraic-geometric codes
  • V. Guruswami, M. Sudan
  • Computer Science, Mathematics
  • Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
  • 1998
TLDR
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. Expand
Encoding and error-correction procedures for the Bose-Chaudhuri codes
TLDR
A simple error-correction procedure for binary codes which for arbitrary m and t are t -error correcting and have length 2^m - 1 of which no more than mt digits are redundancy is described. Expand
Low-Rank Matrix Recovery using Gabidulin Codes in Characteristic Zero
TLDR
It is shown that LRMR can be reduced to decoding of Gabidulin codes and which field extensions can be used in the code construction. Expand
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