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

On the Hardness of the Lee Syndrome Decoding Problem

In this paper we study the hardness of the syndrome decoding problem over finite rings endowed with the Lee metric. We first prove that the decisional version of the problem is NP-complete, by a reduction from the 3-dimensional matching problem. Then, we study the actual complexity of solving the problem, by translating the best known solvers in the Hamming metric over finite fields to the Lee metric over finite rings, as well as proposing some novel solutions. For the analyzed algorithms, we assess the computational complexity in both the finite and asymptotic regimes.Comment: Part of this work appeared as preliminary results in arXiv:2001.0842