Solving Shift Register Problems over Skew Polynomial Rings using Module Minimisation

TL;DR

This paper introduces an algorithm for solving generalized shift register problems over skew polynomial rings, crucial for decoding certain algebraic codes, using module minimisation with efficient complexity.

Contribution

It presents a novel module minimisation approach for shift register problems over skew polynomial rings in the context of decoding interleaved Gabidulin codes.

Findings

Algorithm has time complexity $O(\ell \mu^2)$

Applicable to error and erasure decoding of $\ell$-Interleaved Gabidulin codes

Enhances decoding efficiency for algebraic codes

Abstract

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.