The equivalent condition for GRL codes to be MDS, AMDS or self-dual
Zhonghao Liang, Yongkang Wan, Qunying Liao
TL;DR
This work advances the theory of generalized Roth–Lempel codes by analyzing GRL_k(α, v, A_{3×3}) with a non-singular 3×3 matrix, establishing explicit equivalent conditions for the code and its dual to be non-RS MDS/AMDS/self-dual. The authors prove GRL_k(k>3) is non-RS and derive two key criteria for non-RS MDS and AMDS (for the dual) that depend on combinatorial sums over α and the entries of A_{3×3}. They further provide a concrete parity-check construction and derive a non-RS self-dual condition linking a scalar λ, the v_i weights, and A_{3×3} A_{3×3}^T, enabling explicit self-duality tests. The results generalize prior Roth–Lempel constructions (A22) by replacing A_2 with a GL_3(F_q) matrix, yielding a broader set of NMDS/MDS-like codes and self-dual examples with concrete parameter choices and finite-field realizations.
Abstract
It's well known that MDS, AMDS or self dual codes have good algebraic properties, and are applied in communication systems, data storage, quantum codes, and so on. In this paper, we focus on a class of generalized Roth-Lempel linear codes which are not not equivalent to linear codes in [21],[22] and give an equivalent condition for them or their dual to be non RS MDS, AMDS or non RS self-dual and some corresponding examples.