Twisted and Twisted Linearized Reed--Solomon Codes, LCD and ACD MDS constructions

2026-04-28

AI Summary

This work investigates the construction of linear codes that are simultaneously maximum distance separable (MDS) and complementary dual under the sum-rank metric. Focusing on a class of twisted linearized Reed–Solomon codes in which only the constant term is twisted, the authors establish a linear complementary dual (LCD) criterion independent of the underlying subgroup, code dimension, and twist indices, and provide necessary and sufficient conditions for such codes to be LCD. Furthermore, they explicitly construct—for the first time—an infinite family of additive complementary dual (ACD) MDS codes under the trace-Hermitian inner product, valid for all admissible lengths, thereby unifying ACD and MDS properties in the sum-rank metric.

Abstract

We investigate a natural subfamily of twisted linearized Reed--Solomon (TLRS) codes in the sum-rank metric, where the twist is applied only to the constant term. We establish a simple necessary and sufficient condition for these codes to be linear complementary dual (LCD): the twisting parameter \(η\) must satisfy \(η^2 \neq -1\) in the underlying field. This criterion is independent of the evaluation subgroup, the dimension parameter, and the twisting exponent (subject only to a mild restriction on the code length). Furthermore, we construct infinite families of additive twisted linearized Reed--Solomon codes that are simultaneously additive complementary dual (ACD) and maximum distance separable (MDS) over quadratic extensions \(\mathbb{F}_{q^2}\), with respect to the trace-Hermitian inner product. These codes are explicit and achieve optimal parameters for all admissible lengths.

Problem

• Twisted Linearized Reed--Solomon Codes
• Linear Complementary Dual
• Additive Complementary Dual
• Maximum Distance Separable
• Sum-Rank Metric

Innovation

• Twisted Linearized Reed–Solomon Codes
• Linear Complementary Dual (LCD)
• Additive Complementary Dual (ACD)
• Maximum Distance Separable (MDS)
• Sum-Rank Metric