Efficient Decoding of Gabidulin Codes over Galois Rings

TL;DR

This paper introduces a novel quadratic-complexity decoding algorithm for Gabidulin codes over Galois rings, addressing the challenge of decoding in ring structures with a two-step syndrome-based approach.

Contribution

It is the first to develop a decoding algorithm for Gabidulin codes over Galois rings with provable quadratic complexity.

Findings

Decoding algorithm with quadratic complexity over Galois rings.

Effective two-step syndrome-based decoding process.

Overcomes limitations of Euclidean algorithm over rings.

Abstract

This paper presents the first decoding algorithm for Gabidulin codes over Galois rings with provable quadratic complexity. The new method consists of two steps: (1) solving a syndrome-based key equation to obtain the annihilator polynomial of the error and therefore the column space of the error, (2) solving a key equation based on the received word in order to reconstruct the error vector. This two-step approach became necessary since standard solutions as the Euclidean algorithm do not properly work over rings.