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 toO(n2)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.

In the realm of modern digital communication, cryptography, and signal processing, binary sequences with good correlation properties play a pivotal role. In the literature, considerable efforts have been dedicated to constructing good binary sequences of various lengths. As a consequence, numerous constructions of good binary sequences have been put forward. However, the majority of known constructions leverage the multiplicative cyclic group structure of finite fields Fpn, wherepis a prime andnis a positive integer. Recently, the authors made use of the cyclic group structure of all rational places of the rational function field over the finite field Fpn, and firstly constructed good binary sequences of lengthpn+ 1 via cyclotomic function fields over Fpnfor any primep[8], [10]. This approach has paved a new way for constructing good binary sequences. Motivated by the above constructions, we exploit the cyclic group structure of rational points of elliptic curves to design a family of binary sequences of length 2n+1+twith low correlation for many given integers |t| ⩽ 2(n+2)/2. Specifically, for any positive integerdwith gcd(d; 2n+1+t) = 1, we introduce a novel family of binary sequences of length 2n+1+t, sizeqd−1− 1, correlation bounded by (2d+ 1) · 2(n+2)/2+ |t|, and large linear complexity via elliptic curves.