Sub-Quadratic Decoding of One-Point Hermitian Codes

@article{Nielsen2015SubQuadraticDO,
  title={Sub-Quadratic Decoding of One-Point Hermitian Codes},
  author={Johan Sebastian Rosenkilde Nielsen and Peter Beelen},
  journal={IEEE Transactions on Information Theory},
  year={2015},
  volume={61},
  pages={3225-3240}
}
We present the first two sub-quadratic complexity decoding algorithms for one-point Hermitian codes. The first is based on a fast realization of the Guruswami-Sudan algorithm using state-of-the-art algorithms from computer algebra for polynomial-ring matrix minimization. The second is a power decoding algorithm: an extension of classical key equation decoding which gives a probabilistic decoding algorithm up to the Sudan radius. We show how the resulting key equations can be solved by the… 

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References

SHOWING 1-10 OF 48 REFERENCES
Algebraic Soft-Decision Decoding of Hermitian Codes
TLDR
An algebraic soft-decision decoder for Hermitian codes is presented and an interpolation algorithm is presented to find the Q-polynomial that plays a key role in the decoding.
Efficient list decoding of a class of algebraic-geometry codes
TLDR
A general class of one-point algebraic-geometry codes encompassing, among others, Reed-Solomon codes, Hermitian codes and norm-trace codes is defined, including Guruswami-Sudan list decoder codes.
List decoding of Hermitian codes using Gröbner bases
Simplified understanding and efficient decoding of a class of algebraic-geometric codes
TLDR
An efficient decoding algorithm for algebraic-geometric codes, including Hermitian curves, and a proof of d/sub min//spl ges/d* without directly using the Riemann-Roch theorem are presented.
A Fast Decoding Method of AG Codes from Miura-Kamiya Curves Cab up to Half the Feng-Rao Bound
We present a fast version of the Feng-Rao algorithm for decoding of one-point algebraic-geometric (AG) codes derived from the curves which Miura and Kamiya classified as Cab. Our algorithm performs
Generalized Berlekamp-Massey decoding of algebraic-geometric codes up to half the Feng-Rao bound
TLDR
A general class of algebraic geometry codes is treated and it is shown how to decode these up to half the Feng-Rao bound, using an extension and modification of the Sakata algorithm.
List Decoding of Algebraic Codes
TLDR
A fast maximum-likelihood list decoder based on the Guruswami– Sudan algorithm; a new variant of Power decoding, Power Gao, along with some new insights into Power decoding; and a new, module based method for performing rational interpolation for the Wu algorithm are given.
Ideal forms of Coppersmith's theorem and Guruswami-Sudan list decoding
TLDR
A framework for solving polynomial equations with size constraints on solutions is developed and powerful analogies from algebraic number theory allow us to identify the appropriate analogue of a lattice in each application and provide efficient algorithms to find a suitably short vector, thus allowing completely parallel proofs of the above theorems.
Bounds on collaborative decoding of interleaved Hermitian codes and virtual extension
TLDR
A decoding algorithm that achieves the maximum decoding radius for interleaved Hermitian (IH) codes if a collaborative decoding scheme is used is given and a bound on the code rate below which virtual extension of non-interleaved hermitian codes can improve the decoding capabilities is derived.
Performance analysis of a decoding algorithm for algebraic geometry codes
TLDR
It turns out that in the typical case, where the error points are "independent", one can prove that the algorithm always fails, that is gives a wrong or no answer, except for high rates where it does much better than expected.
...
...