[Paper Review] Gabidulin Decoding via Minimal Bases of Linearized Polynomial Modules

Anna-Lena Trautmann, Margreta Kuijper|arXiv (Cornell University)|Aug 11, 2014

Coding theory and cryptography34 references3 citations

TL;DR

This paper presents a module-theoretic decoding algorithm for Gabidulin codes that computes all closest codewords to a received word using minimal bases of linearized polynomial modules. By leveraging the Predictable Leading Monomial property and parametrization, it achieves polynomial-time decoding within the unique decoding radius and outperforms chase list decoding beyond it when the radius is less than $ n-k $, strengthening the link to Reed-Solomon decoding.

ABSTRACT

We show how Gabidulin codes can be decoded via parametrization by using interpolation modules over the ring of linearized polynomials with composition. Our decoding algorithm computes a list of message words that correspond to all closest codewords to a given received word. This involves the computation of a minimal basis for the interpolation module that corresponds to the received word, followed by a search through the parametrization for valid message words. Our module-theoretic approach strengthens the link between Gabidulin decoding and Reed-Solomon decoding. Two subalgorithms are presented to compute the minimal basis, one iterative, the other an extended Euclidean algorithm. Both of these subalgorithms have polynomial time complexity. The complexity order of the overall algorithm, using the parametrization, is then compared to straightforward exhaustive search as well as to chase list decoding.

Motivation & Objective

• To develop a polynomial-time list decoding algorithm for Gabidulin codes that finds all closest codewords to a given received word.
• To strengthen the conceptual and algorithmic parallels between Gabidulin and Reed-Solomon decoding via module-theoretic parametrization.
• To analyze the computational complexity of the proposed decoding algorithm and compare it with existing methods like chase list decoding and exhaustive search.
• To investigate whether polynomial-size lists exist for Gabidulin codes in the region between the unique decoding radius and the Johnson radius.
• To provide two efficient subalgorithms—iterative and extended Euclidean—for computing minimal bases of interpolation modules.

Proposed method

• The method formulates Gabidulin list decoding as an interpolation problem over the ring of linearized polynomials, using four $ q $-linearized polynomials to define an interpolation module.
• It employs the Predictable Leading Monomial (PLM) property of minimal bases to enable systematic parametrization of solutions to the interpolation conditions.
• The algorithm computes a minimal basis of the interpolation module using two subalgorithms: one iterative and one based on an extended Euclidean algorithm for linearized polynomials.
• A parametrization step then searches through the basis to extract all valid message polynomials corresponding to codewords within the minimum rank distance.
• The decoding process is structured to avoid exhaustive search by exploiting algebraic structure and module properties.
• Complexity analysis is performed by comparing the algorithm’s time complexity to that of chase list decoding and exhaustive search.

Experimental results

Research questions

• RQ1 Can Gabidulin codes be list-decoded efficiently beyond the unique decoding radius using a structured algebraic approach?
• RQ2 What is the computational complexity of list decoding Gabidulin codes when the decoding radius is less than $ n-k $?
• RQ3 How does the proposed module-theoretic method compare in complexity to chase list decoding and exhaustive search?
• RQ4 To what extent can the parametrization framework used in Reed-Solomon decoding be adapted to Gabidulin codes?
• RQ5 Is there a polynomial-size list of codewords within the Johnson radius for Gabidulin codes, and can such a list be computed in polynomial time?

Key findings

• The proposed algorithm achieves polynomial-time complexity for decoding within the unique decoding radius, matching the complexity of known unique decoders.
• For decoding radii less than $ n-k $, the algorithm's complexity is strictly better than that of the chase list decoding algorithm.
• When the decoding radius is $ n-k $ or larger, the algorithm's complexity order matches that of the chase algorithm.
• The algorithm outperforms exhaustive search list decoding for all decoding radii less than $ n/2 $.
• The minimal basis computation via the iterative and extended Euclidean subalgorithms is shown to be efficient and equivalent to known algorithms, but now repurposed for list decoding.
• The parametrization approach successfully generalizes the decoding framework to find all closest codewords, not just one, thereby enabling full list decoding.