Efficient Decoding of Twisted GRS Codes and Roth--Lempel Codes

TL;DR

This paper introduces efficient list and unique decoding algorithms for twisted GRS and Roth-Lempel codes, significantly improving decoding speed and capabilities, and extending the range of decodable parameters compared to previous methods.

Contribution

It presents the first efficient decoders for Roth-Lempel codes and extends decoding algorithms for TGRS codes to multiple twists, achieving near-linear time complexity.

Findings

Decoders operate in near-linear time for suitable parameters.

Support for up to O(n^2) twists in TGRS codes.

List decoders surpass classical decoding radius.

Abstract

MDS codes play a central role in practice due to their broad applications. To date, most known MDS codes are generalized Reed-Solomon (GRS) codes, leaving codes that are not equivalent to GRS codes comparatively less understood. Studying this non-GRS regime is therefore of intrinsic theoretical interest, and is also practically relevant since the strong algebraic structure of GRS codes can be undesirable in cryptographic settings. Among the known non-GRS codes, twisted generalized Reed-Solomon (TGRS) codes and Roth-Lempel codes are two representative families of non-GRS codes that have attracted significant attention. Though substantial work has been devoted to the construction and structural analysis of TGRS and Roth-Lempel codes, comparatively little attention has been paid to their decoding, and many problems remain open. In this paper, we propose list and unique decoding algorithms for TGRS codes and Roth-Lempel codes based on the Guruswami-Sudan algorithm. Under suitable parameter conditions, our algorithms achieve near-linear running time in the code length, improving upon the previously best-known quadratic-time complexity. Our TGRS decoder supports fixed-rate TGRS codes with up to O(n^2) twists, substantially extending prior work that only handled the single-twist case. For Roth-Lempel codes, we provide what appears to be the first efficient decoder. Moreover, our list decoders surpass the classical unique-decoding radius for a broad range of parameters. Finally, we incorporate algebraic manipulation detection (AMD) codes into the list-decoding framework, enabling recovery of the correct message from the output list with high probability.