Decoding Reed-Solomon codes up to the Sudan radius with the Euclidean algorithm

International audienceWe modify the Euclidean algorithm of Feng and Tzeng to decode Reed-Solomon (RS) codes up to the Sudan radius. The basic steps are the virtual extension to an Interleaved RS code and the reformulation of the multi-sequence shift-register problem of varying length to a multi-sequence problem of equal length. We prove the reformulation and analyze the complexity of our new decoding approach. Furthermore, the extended key equation, that describes the multi-sequence problem, is derived in an alternative polynomial way

Solving Shift Register Problems over Skew Polynomial Rings using Module Minimisation

For many algebraic codes the main part of decoding can be reduced to a shift register synthesis problem. In this paper we present an approach for solving generalised shift register problems over skew polynomial rings which occur in error and erasure decoding of $\ell$-Interleaved Gabidulin codes. The algorithm is based on module minimisation and has time complexity $O(\ell \mu^2)$ where $\mu$ measures the size of the input problem.Comment: 10 pages, submitted to WCC 201

Decoding of Interleaved Reed-Solomon Codes Using Improved Power Decoding

We propose a new partial decoding algorithm for $m$-interleaved Reed--Solomon (IRS) codes that can decode, with high probability, a random error of relative weight $1-R^{\frac{m}{m+1}}$ at all code rates $R$, in time polynomial in the code length $n$. For $m>2$, this is an asymptotic improvement over the previous state-of-the-art for all rates, and the first improvement for $R>1/3$ in the last $20$ years. The method combines collaborative decoding of IRS codes with power decoding up to the Johnson radius.Comment: 5 pages, accepted at IEEE International Symposium on Information Theory 201