An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero

TL;DR

This paper extends Gabidulin codes to characteristic zero fields, introduces a new decoding method based on a Gao type key equation, and applies it to low-rank matrix recovery, improving decoding complexity.

Contribution

It presents a novel decoding approach for Gabidulin codes in characteristic zero fields, reducing complexity and linking decoding to established numerical stability problems.

Findings

Decoding complexity reduced to at least O(n^2)

Application to low-rank matrix recovery demonstrated

Connection established between decoding and coefficient growth issues

Abstract

Gabidulin codes, originally defined over finite fields, are an important class of rank metric codes with various applications. Recently, their definition was generalized to certain fields of characteristic zero and a Welch--Berlekamp like algorithm with complexity $O(n^3)$ was given. We propose a new application of Gabidulin codes over infinite fields: low-rank matrix recovery. Also, an alternative decoding approach is presented based on a Gao type key equation, reducing the complexity to at least $O(n^2)$. This method immediately connects the decoding problem to well-studied problems, which have been investigated in terms of coefficient growth and numerical stability.