Decoding Algorithms for Twisted GRS Codes

TL;DR

This paper introduces new decoding algorithms for MDS twisted generalized Reed-Solomon (TGRS) codes using Gaussian elimination, capable of correcting errors efficiently and expanding decoding capabilities beyond existing methods.

Contribution

The paper presents Gaussian elimination-based decoding algorithms for MDS TGRS codes, improving error correction and addressing previously unconsidered decoding scenarios.

Findings

Algorithms correct up to loor((n-k)/2)rrors for odd n-k

Algorithms correct up to loor((n-k)/2)-1 errors for even n-k

Computational complexity is O(n^3) for the proposed decoding methods

Abstract

Twisted generalized Reed-Solomon (TGRS) codes were introduced to extend the algebraic capabilities of classical generalized Reed-Solomon (GRS) codes. This extension holds the potential for constructing new non-GRS maximum distance separable (MDS) codes and enhancing cryptographic security. It is known that TGRS codes with $1$ twist can either be MDS or near-MDS. In this paper, we employ the Gaussian elimination method to propose new decoding algorithms for MDS TGRS codes with parameters $[n,k,n-k+1]$. The algorithms can correct up to $\lfloor \frac{n-k}{2}\rfloor$ errors when $n-k$ is odd, and $\lfloor \frac{n-k}{2}\rfloor-1$ errors when $n-k$ is even. The computational complexity for both scenarios is $O(n^3)$. %, where $\omega\approx 2.37286$ is the matrix multiplication exponent. Our approach diverges from existing methods based on Euclidean algorithm and addresses situations that have not been considered in the existing literature \cite{SYJL}. Furthermore, this method is also applicable to decoding near-MDS TGRS codes with parameters $[n, k, n-k]$, enabling correction of up to $\lfloor \frac{n-k-1}{2} \rfloor$ errors, while maintaining polynomial time complexity in $n$.