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

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.

• The paper presents a universal LCD criterion for TLRS codes, stating that LCD holds if and only if η² ≠ -1, independent of evaluation set and code parameters.
• It leverages skew polynomial rings to explicitly construct MSRD codes in the sum-rank metric, unifying classical Reed–Solomon and Gabidulin frameworks.
• The paper also provides infinite families of additive ACD MDS codes over quadratic extensions, offering practical benefits for secure network coding and distributed storage.

Twisted and Twisted Linearized Reed--Solomon Codes: LCD and ACD MDS Constructions

Introduction and Background

This work provides a comprehensive treatment of a natural subfamily of twisted linearized Reed--Solomon (TLRS) codes in the sum-rank metric, focusing on twistings applied only to the constant term. TLRS codes are situated within the broader context of linearized Reed–Solomon (LRS) codes, which unify classical Reed–Solomon and Gabidulin codes under the sum-rank metric formalism. LRS and TLRS codes exhibit the maximum sum-rank distance (MSRD) property and are relevant for network coding, distributed storage, and related communication scenarios with both locality and global constraints.

Central to the development is the explicit construction and characterization of linear complementary dual (LCD) and additive complementary dual (ACD) codes with MDS or MSRD properties in the sum-rank metric and over quadratic extensions. The approach leverages the algebraic structure of skew polynomial rings and their evaluation homomorphisms, diverging from previous constructions which rely on classical commutative polynomial ring frameworks.

LCD Property in Twisted Linearized Reed--Solomon Codes

The main technical contribution is the derivation of a precise and universal criterion for when TLRS codes $\mathcal{L}_k^\theta(\eta,h)$ are LCD with respect to the standard trace inner product on the sum-rank module. Specifically, the paper presents a necessary and sufficient condition: the code is LCD if and only if $\eta^2 \ne -1$ in the ground (potentially extended) field. Crucially, this criterion is independent of the evaluation set $\Lambda \subseteq \mathbb{F}_q^*$, the twist exponent $h$, and the dimension $k$ (with the mild restriction $1 \le k \le \ell r - 1$ on the code length).

The analysis proceeds by explicit computation of the Gram matrix with respect to an explicit $\mathbb{F}_q$-basis, reducing its invertibility to the non-degeneracy of the trace bilinear form parametrized by $1+\eta^2$. This method demonstrates that the singularity of the Gram matrix is entirely controlled by the twisting parameter. Examples over $\mathbb{F}_{25}$ illustrate both LCD and non-LCD cases.

This result provides immediate explicit families of LCD MSRD codes in the sum-rank metric—an extension not previously available in the literature, where existing constructions are limited to the Hamming metric. The construction's simplicity, and its direct algebraic nature, stand in contrast to prior multistep approaches involving parity-check characterizations or self-orthogonality arguments in the commutative Reed–Solomon setting.

Additive Complementary Dual MDS Codes Over Quadratic Extensions

In the second part, the paper presents explicit infinite families of additive twisted RS codes over $\mathbb{F}_{q^2}$ possessing simultaneous ACD and MDS properties with respect to the trace-Hermitian inner product. These codes are defined via subspaces $\eta^2 \ne -1$0 of the polynomial algebra where the constant and highest-degree coefficients are $\eta^2 \ne -1$1-valued and the remaining coefficients are in $\eta^2 \ne -1$2.

For $\eta^2 \ne -1$3 and $\eta^2 \ne -1$4, and for all admissible lengths $\eta^2 \ne -1$5, the construction employs a twisting parameter $\eta^2 \ne -1$6 where $\eta^2 \ne -1$7. This choice guarantees that $\eta^2 \ne -1$8 is a nonsquare in $\eta^2 \ne -1$9 and $\Lambda \subseteq \mathbb{F}_q^*$0, which ensures the ACD property via an analysis of the trace of the Gram matrix of the code generator. An explicit matrix-based criterion is derived: the code is ACD if and only if two explicit block determinants—one corresponding to coefficients in $\Lambda \subseteq \mathbb{F}_q^*$1 and another involving the higher-dimensional subspace—are invertible.

A probabilistic and algebraic argument ensures the existence of evaluation sets $\Lambda \subseteq \mathbb{F}_q^*$2 such that all technical matrix conditions are satisfied for all permissible parameters. The resulting codes have parameters $\Lambda \subseteq \mathbb{F}_q^*$3 and generalize previous constructions by allowing flexible lengths and dimensions, again exceeding the reach of earlier approaches limited to classical or commutative settings.

Theoretical and Practical Implications

The established LCD criterion for TLRS codes significantly broadens the landscape of explicit LCD code constructions in the sum-rank metric, enabling their systematic deployment in applications requiring both maximum distance properties and security against various side-channel or fault injection attacks—a property of practical import in cryptography and secure storage systems.

Simultaneously, the explicit ACD MDS codes over quadratic extensions offer enhanced flexibility for code designers, particularly in settings such as space-time coding, quantum error correction, and multi-shot network coding protocols where additive duality or trace-Hermitian orthogonality are essential.

The algebraically transparent conditions on the twisting parameter and evaluation set facilitate practical code selection and parametrization. The universal nature of the LCD condition (independent of code or group parameters) represents a marked increase in construction simplicity over previous methods, and the trace-based block determinant criterion for ACD codes connects elegant algebraic structure with concrete code performance guarantees.

Future Directions

The framework suggests several avenues for continued investigation:

1. Field Extension: Extension of these constructions to finite fields where $\Lambda \subseteq \mathbb{F}_q^*$4 and to higher-degree extensions $\Lambda \subseteq \mathbb{F}_q^*$5, broadening the class of available ACD and LCD codes.
2. Quantum Codes: Application of these constructions to entanglement-assisted quantum error correction, utilizing the Hermitian or trace-Hermitian orthogonality for code-based quantum state encoding.
3. Network Coding and Secure Communication: Deployment of support-constrained TLRS and additive ACD MDS codes in networked systems where node or link behavior imposes zero patterns or local constraints on generator matrices.
4. Algorithmic Perspectives: Advanced decoding algorithms leveraging the highly structured basis and Gram matrix representations to support fast error correction in the sum-rank metric.

Conclusion

This work offers universal, explicit, and easily verifiable constructions of LCD and ACD MDS codes in the sum-rank metric and over field extensions. The simplicity of the LCD criterion for TLRS codes and the explicit matrix-based ACD condition, validated for all admissible parameters, enable robust and secure code design for a variety of applications, while opening new lines of inquiry in the intersection of algebraic coding theory and secure communication.