General Linearized Polynomial Interpolation and Its Applications

TL;DR

This paper introduces a general interpolation algorithm in linearized polynomial rings, enabling efficient decoding of various codes like Gabidulin, KK, and MV codes, with improved complexity over existing methods.

Contribution

It presents a novel general interpolation algorithm applicable to multiple code families, offering a unified decoding approach with lower computational complexity.

Findings

Decoding algorithm for Gabidulin codes differs from Loidreau's polynomial reconstruction.

Interpolation algorithm matches Sudan-style list decoding for KK codes.

Capable of decoding MV codes with reduced complexity.

Abstract

In this paper, we first propose a general interpolation algorithm in a free module of a linearized polynomial ring, and then apply this algorithm to decode several important families of codes, Gabidulin codes, KK codes and MV codes. Our decoding algorithm for Gabidulin codes is different from the polynomial reconstruction algorithm by Loidreau. When applied to decode KK codes, our interpolation algorithm is equivalent to the Sudan-style list-1 decoding algorithm proposed by K/"otter and Kschischang for KK codes. The general interpolation approach is also capable of solving the interpolation problem for the list decoding of MV codes proposed by Mahdavifar and Vardy, and has a lower complexity than solving linear equations.