[Paper Review] A Note on Self-Dual Generalized Reed-Solomon Codes

Weijun Fang, Jun Zhang+2 more|arXiv (Cornell University)|May 24, 2020

Coding theory and cryptography19 references4 citations

TL;DR

This paper presents four new families of MDS self-dual generalized Reed-Solomon (GRS) codes by constructing self-dual GRS codes using evaluation sets based on multiplicative subgroups and their cosets in finite fields. It introduces a systematic method using Möbius transformations (PGL₂ actions) to generate new self-dual GRS codes from known ones, proving that all self-dual extended GRS codes of length $ n < q+1 $ over $ \mathbb{F}_q $ can be derived from standard GRS codes without the infinity point.

ABSTRACT

A linear code is called an MDS self-dual code if it is both an MDS code and a self-dual code with respect to the Euclidean inner product. The parameters of such codes are completely determined by the code length. In this paper, we consider new constructions of MDS self-dual codes via generalized Reed-Solomon (GRS) codes and their extended codes. The critical idea of our constructions is to choose suitable evaluation points such that the corresponding (extended) GRS codes are self-dual. The evaluation set of our constructions is consists of a subgroup of finite fields and its cosets in a bigger subgroup. Four new families of MDS self-dual codes are obtained and they have better parameters than previous results in certain region. Moreover, by the Mobius action over finite fields, we give a systematic way to construct self-dual GRS codes with different evaluation points provided any known self-dual GRS codes. Specially, we prove that all the self-dual extended GRS codes over $\mathbb{F}_{q}$ with length $n&lt; q+1$ can be constructed from GRS codes with the same parameters.

Motivation & Objective

• To construct new families of MDS self-dual codes with improved parameters beyond existing literature.
• To develop a systematic method for generating self-dual GRS codes using automorphisms from the projective linear group PGL₂(𝔽_q).
• To prove that self-dual extended GRS codes of length $ n < q+1 $ over $ \mathbb{F}_q $ can be constructed from standard GRS codes without the infinity point.
• To explore the role of multiplicative subgroups and their cosets in finite fields as evaluation sets for self-dual GRS codes.

Proposed method

• Construct self-dual GRS codes by selecting evaluation sets as unions of multiplicative subgroups and their cosets in $ \mathbb{F}_q^* $.
• Utilize the action of PGL₂(𝔽_q) on evaluation points to generate new self-dual GRS codes from known ones via Möbius transformations.
• Leverage the automorphism property of GRS codes under PGL₂(𝔽_q) actions, using the transformation formula $ g_k G_k = G_k \Pi(g) \Delta_k(g) $.
• Apply the restriction of transformed generator matrices to subsets of columns corresponding to the new evaluation sets to derive new scaling vectors.
• Prove that any self-dual extended GRS code of length $ n < q+1 $ can be obtained from a standard GRS code by choosing an appropriate Möbius transformation that maps the infinity point out of the evaluation set.
• Use the fact that the Möbius transformation $ g = \begin{pmatrix} a & 1 \\ 1 & 0 \end{pmatrix} $ maps $ \infty $ to a finite element, enabling the removal of the infinity point from the evaluation set.

Experimental results

Research questions

• RQ1 Can new families of MDS self-dual GRS codes be constructed using structured evaluation sets based on multiplicative subgroups and their cosets in finite fields?
• RQ2 How can the PGL₂(𝔽_q) action be systematically used to generate new self-dual GRS codes from known ones?
• RQ3 Is it possible to eliminate the infinity point from the evaluation set of a self-dual extended GRS code while preserving self-duality?
• RQ4 What are the necessary and sufficient conditions on the evaluation set for a GRS code to be self-dual over $ \mathbb{F}_q $?
• RQ5 Can all self-dual extended GRS codes of length $ n < q+1 $ be constructed from standard GRS codes via Möbius transformations?

Key findings

• Four new families of MDS self-dual codes are constructed, with parameters not covered by previous results in certain regions.
• The Möbius action on GRS codes provides a systematic way to generate new self-dual GRS codes from any known self-dual GRS code.
• It is proven that all self-dual extended GRS codes of length $ n < q+1 $ over $ \mathbb{F}_q $ can be constructed from standard GRS codes without the infinity point.
• The evaluation set construction using subgroups and their cosets leads to MDS self-dual codes with improved parameter coverage.
• For any self-dual extended GRS code of length $ n < q+1 $, there exists a corresponding self-dual GRS code over $ \mathbb{F}_q $ alone (without infinity), via a suitable Möbius transformation.
• The method ensures that the new evaluation sets remain closed under the group action and preserve the self-dual property through appropriate scaling vector adjustment.