Twisted and Twisted Linearized Reed--Solomon Codes, LCD and ACD MDS constructions
2026-04-28 • Information Theory
Information Theory
AI summaryⓘ
The authors study a special kind of error-correcting codes called twisted linearized Reed–Solomon codes where the 'twist' affects only the constant term. They find a simple condition that tells when these codes have a property called linear complementary duality (LCD): the twist parameter must not square to -1 in the field. This condition does not depend on other code parameters. They also build large families of codes over certain fields that are both additive complementary dual (ACD) and maximum distance separable (MDS), meaning they are optimal in error correction under a special inner product.
twisted linearized Reed–Solomon codessum-rank metriclinear complementary dual (LCD)twisting parametercode dimensionadditive complementary dual (ACD)maximum distance separable (MDS)trace-Hermitian inner productfinite fieldserror-correcting codes
Authors
Sanjit Bhowmick, Kuntal Deka, Edgar Martínez-Moro
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.