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

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 ${\eta }^2\mathrm{\ne }-1$ 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.