Generalized Roth--Lempel Codes: NMDS Characterization, Hermitian Self-Orthogonality, and Quantum Constructions

Abstract: In their seminal 1989 work (IEEE Trans. Inf. Theory 35(3):655-657), Roth and Lempel constructed a well-known family of non-Reed-Solomon maximum distance separable (MDS) codes. For decades, this family of codes has attracted extensive research attention due to its algebraic structure, low-complexity decoding, and broad applications in cryptography and data storage. Most recently, in 2025, the generalized Roth-Lempel (GRL) framework unifies Roth-Lempel codes and its extensions under a flexible algebraic structure. However, explicit criteria for the near-MDS (NMDS) property of GRL codes have not been established, and no systematic construction of Hermitian self-orthogonal GRL codes has been reported, limiting their deployment in classical and quantum error correction. In this work, we make three contributions to address these gaps. First, we give explicit necessary and sufficient conditions for the NMDS property of the two most widely used subclasses of GRL codes. Second, we construct four new families of Hermitian self-orthogonal codes from GRL codes. Two of these families are NMDS, with parameters not covered by existing Hermitian self-orthogonal NMDS codes. Third, based on the proposed Hermitian self-orthogonal GRL codes, we construct four families of quantum GRL codes, including two infinite families of quantum NMDS codes that attain the quantum Singleton bound minus one. Compared to the known quantum error-correcting codes, we obtain many new or improved quantum error-correcting codes. This work bridges the gap between classical GRL code families and quantum error-correction applications.

• The paper introduces explicit algebraic and combinatorial conditions for achieving the NMDS property in Generalized Roth–Lempel codes.
• It constructs infinite families of Hermitian self-orthogonal codes for both s=2 and s=3 cases, enhancing quantum error correction potential.
• Utilizing the CSS framework, the authors develop new quantum NMDS codes with improved minimum distance and flexible parameters.

Generalized Roth–Lempel Codes: NMDS Characterization, Hermitian Self-Orthogonality, and Quantum Constructions

Introduction and Context

This paper advances the structural theory and quantum applications of Generalized Roth–Lempel (GRL) codes by providing complete algebraic and combinatorial criteria for their near-maximum distance separable (NMDS) property, constructing new infinite families of Hermitian self-orthogonal codes, and deploying these for quantum code construction. Building on the historical development from non-Reed-Solomon MDS codes by Roth and Lempel (1989), the GRL framework emerged recently as a flexible generalization encompassing further extensions. However, the literature lacked explicit NMDS criteria and systematic Hermitian self-orthogonal constructions, particularly for $s=2,3$ cases, limiting theoretical and practical utilization, especially in quantum error correction.

NMDS Characterization of GRL Codes

The authors derive necessary and sufficient algebraic conditions for the NMDS property of two central GRL subclasses ($s=2$ and $s=3$):

• For $s=2$, these codes generalize classic Roth-Lempel codes. The explicit NMDS criterion is formulated in terms of linear dependencies among the relevant generator matrix columns and subset-sum conditions over field elements. Notably, this also recovers and unifies prior special-case results for Roth-Lempel codes.
• For $s=3$, covering a class of generalized Roth-Lempel-type codes, the NMDS property is characterized with combinatorial invariants ($\Omega_k$ and $\Gamma_k$) that encode dependencies aligned with the generator matrix structure.

The technical analysis leverages a combinatorial lemma on the linear algebraic independence of columns (from Dodunekov et al.), projective geometry insights, and a careful combinatorial count via subset-sum problems. This explicit algebraic understanding enables both the classification of known NMDS instances and the construction of new ones.

Hermitian Self-Orthogonal GRL Codes

A critical step for quantum code applications is establishing Hermitian self-orthogonality. The authors provide:

• Two infinite families of Hermitian self-orthogonal NMDS GRL codes (for $s=2$) with parameters not previously realized. These families achieve flexible length and dimension, frequently exceeding the classical GRS or computer search-based results. The criterion for Hermitian self-orthogonality is given in terms of explicit trace and norm conditions over the evaluation points and generator matrix parameters, including closed-form conditions on certain field sums and matrix entries.
• Two further families for $s=3$, where the codes are Hermitian self-orthogonal and possess minimum distance bounds close to the best achievable for the structure, though a complete NMDS classification in this case remains technically challenging due to combinatorial complexity.

These constructions systematically apply selection of evaluation-point sets with prescribed algebraic invariants and design of generator matrices to satisfy the Hermitian orthogonality conditions. For code families where known computer search results limited length/dimension (e.g., over finite extensions such as $\mathbb{F}_{9^2}$), these constructions provide codes with dramatically broader parameters.

Quantum Code Constructions and QNMDS Attainment

Utilizing the CSS (Calderbank-Shor-Steane) framework with the constructed Hermitian self-orthogonal classical codes, the authors obtain four families of quantum GRL codes:

• For the $s=2$0 families, the resulting quantum codes are quantum NMDS (QNMDS), attaining the quantum Singleton bound minus one, and are infinite families with parameter sets not contained in previous Hermitian self-orthogonal or TGRS-derived families.
• For the $s=2$1 families, quantum codes with minimum distance at least $s=2$2 are constructed, supplying new or improved quantum codes with large length and dimension.

The parameter sets are analyzed relative to the quantum Singleton bound (using the defect formulation), and explicit examples demonstrate improved minimum distances over the previously best-known constructions in the literature (e.g., for $s=2$3 and $s=2$4, codes with higher minimum distance at the same length and dimension).

Implications and Future Directions

Theoretical implications include:

• A unified algebraic framework for NMDS characterization in rich families of algebraic-geometric codes, bridging classical and quantum code theory.
• Explicit combinatorial and algebraic conditions for Hermitian self-orthogonality, crucial for quantum error correction via CSS-type constructions, and uncoupled from GRS or TGRS constraints.

Practically, the work enables construction of quantum codes with greater flexibility in length and dimension, especially for $s=2$5-ary alphabets, relevant for advanced quantum communication and storage. The codes are competitive or superior in the quantum Singleton defect relative to existing tables.

The main open problem highlighted is whether Hermitian self-orthogonal GRL codes for $s=2$6 can always yield quantum NMDS codes, i.e., determining the precise extent of quantum optimality for higher $s=2$7. Further, extending explicit combinatorial NMDS criteria to broader classes, possibly incorporating automorphism group structures or higher genus curves, would generalize these results.

Conclusion

This paper closes several open theoretical gaps relating to the NMDS and Hermitian self-orthogonality properties of GRL codes and provides explicit, parameter-flexible constructions for quantum error correction, including new infinite families of quantum NMDS codes. The algebraic and combinatorial techniques employed should prove broadly applicable in classical and quantum coding theory and stimulate further research on explicit optimal code constructions in large parameter regimes.

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Reference: "Generalized Roth--Lempel Codes: NMDS Characterization, Hermitian Self-Orthogonality, and Quantum Constructions" (2604.11350)