Sub-quadratic Decoding of One-point Hermitian Codes

Johan Sebastian Rosenkilde Nielsen (1) , Peter Beelen (2)

1 GRACE - Geometry, arithmetic, algorithms, codes and encryption
2 DTU - Danmarks Tekniske Universitet = Technical University of Denmark

We present the first two sub-quadratic complexity decoding algorithms for one-point Hermitian codes. The first is based on a fast realisation of the Guruswami–Sudan algorithm by using state-of-the-art algorithms from computer algebra for polynomial-ring matrix minimisation. 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 matrix minimisation algorithms from computer algebra, yielding similar asymptotic complexities.

Mots clés: Hermitian codes; AG codes; list decoding; Guruswami-Sudan; Power decoding

Domaines: Théorie de l'information [cs.IT]

Johan Sebastian Rosenkilde Nielsen, Peter Beelen. Sub-quadratic Decoding of One-point Hermitian Codes. IEEE Transactions on Information Theory, 2015, 61 (6), pp.3225-3240 ⟨10.1109/TIT.2015.2424415⟩. ⟨hal-01245062⟩