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

TL;DR

This paper studies twisted linearized Reed--Solomon codes in the sum-rank metric, providing conditions for LCD property and constructing optimal additive ACD and MDS codes over quadratic fields.

Contribution

It establishes a simple criterion for LCD property in twisted linearized Reed--Solomon codes and constructs explicit infinite families of additive ACD and MDS codes over quadratic extensions.

Findings

A necessary and sufficient condition for LCD codes: ta^2 neq -1.

Construction of infinite families of additive ACD and MDS codes.

Codes achieve optimal parameters over quadratic extensions.

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 \(\eta\) must satisfy \(\eta^2 \neq -1\) 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.