Twisted and Twisted Linearized Reed--Solomon Codes, LCD and ACD MDS constructions
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917444258365440 |
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| author | Bhowmick, Sanjit Deka, Kuntal Martínez-Moro, Edgar |
| author_facet | Bhowmick, Sanjit Deka, Kuntal Martínez-Moro, Edgar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25870 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Twisted and Twisted Linearized Reed--Solomon Codes, LCD and ACD MDS constructions Bhowmick, Sanjit Deka, Kuntal Martínez-Moro, Edgar Information Theory 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. |
| title | Twisted and Twisted Linearized Reed--Solomon Codes, LCD and ACD MDS constructions |
| topic | Information Theory |
| url | https://arxiv.org/abs/2604.25870 |