Generalized Roth-Lempel Codes Overview

• Generalized Roth–Lempel (GRL) codes are a family of non-GRS linear codes defined by augmenting Reed–Solomon matrices with algebraically-structured columns.
• They enable construction of MDS, AMDS, NMDS, and self-dual codes, facilitating applications in quantum error correction, cryptography, and combinatorial design.
• GRL codes incorporate specialized decoding algorithms and clear algebraic criteria, ensuring efficient error correction and flexible control over code hull dimensions.

Generalized Roth–Lempel (GRL) codes are a family of non-GRS-type linear codes constructed via the augmentation of Reed–Solomon (RS) generator matrices with algebraically structured columns determined by invertible matrices acting on the polynomial coefficient space. These codes are parameterized by the choice of evaluation points, multipliers, and a tail matrix, enabling the construction of MDS, AMDS, NMDS, and self-dual codes with flexible hull dimensions. GRL codes have become central objects in the contemporary theory of code families not equivalent to generalized Reed–Solomon codes, with applications to quantum error correction (EAQECCs), LCD code constructions, and combinatorial design.

1. Algebraic Definition and Structural Properties

Let $\F_q$ be the finite field of order $q$. Fix integers $l+1 \le k+1 \le n \le q$. Select:

• $\boldsymbol{\alpha} = (\alpha_1, \dots, \alpha_n)$ with $\alpha_i \in \F_q$ all distinct,
• $\boldsymbol{v} = (v_1, \dots, v_n) \in (\F_q^*)^n$,
• $A \in GL_l(\F_q)$ (an invertible $l \times l$ matrix).

Define a message polynomial $f(x) = f_0 + f_1 x + \dots + f_{k-1} x^{k-1} \in \F_q[x]$. The appended $l$ coordinates are $$\boldsymbol{\beta} = (f_{k-l}, \dots, f_{k-1}) \cdot A$$. The generalized Roth–Lempel code is constructed as

$$\text{GRL}_k(\boldsymbol{\alpha}, \boldsymbol{v}, A) = \left\{ (v_1 f(\alpha_1), \dots, v_n f(\alpha_n), \beta_1, \dots, \beta_l) \mid f(x): \deg f < k \right\}$$

with parameters $[n+l, k]$ over $\F_q$ (Liang et al., 4 Jun 2025, Liang et al., 17 Aug 2025, Liang et al., 17 Mar 2026).

A prototypical generator matrix (for $l=3$) takes the block form

$$G = \begin{pmatrix} v_1 & \cdots & v_n & 0 & 0 & 0 \ \vdots & & \vdots & \vdots & \vdots & \vdots \ * & \cdots & * & a_{11} & a_{12} & a_{13} \ * & \cdots & * & a_{21} & a_{22} & a_{23} \ * & \cdots & * & a_{31} & a_{32} & a_{33} \end{pmatrix}$$

where the first $n$ columns form a "twisted" RS block and the last $l$ columns form the $A$ block. When $v_i=1$ and $l=3$, this recovers the original Roth–Lempel construction.

2. MDS, AMDS, NMDS, and Self-Dual Criteria

MDS Codes

The $[n+l, k]$ GRL code is MDS if and only if every $k$ columns of the generator matrix are linearly independent. For $l=3$ (classical RL codes), explicit combinatorial conditions relating minors of $A$ and structured sums over the set of evaluation points $\{\alpha_i\}$ are required to never vanish [(Liang et al., 4 Jun 2025), Theorem 3.3]. Specifically, MDS-ness is characterized by non-vanishing of determinant expressions partitioned over all subsets of $k-1$ and $k-2$ evaluation points.

AMDS and NMDS Codes

The dual of a GRL code is AMDS if every $(k-2)$ columns of the generator (or parity-check) matrix are independent, but some $k$ columns are dependent. Six distinct conditions (arising from types of minors) must be examined, mirroring the MDS criteria with suitable modifications [(Liang et al., 4 Jun 2025), Theorem 3.6]. Extended GRL codes (EGRL), with an additional appended coordinate, allow for precise rank and weight-distribution analysis, producing infinite families of explicit NMDS codes generalizing Han et al. using full and punctured evaluation sets (Liang et al., 17 Aug 2025, Liang et al., 24 Jun 2025).

Self-Duality

Self-dual GRL codes require $n+l=2k$ and two simultaneous algebraic constraints:

• Existence of a scalar $\lambda\in \F_q^*$ such that $v_i^2 u_i = \lambda$ for all $i$ (with $u_i = \prod_{j\neq i} (\alpha_i-\alpha_j)^{-1}$),
• $A A^T = \lambda M(\mathbf{\alpha})$, where $M(\mathbf{\alpha})$ is a structured symmetric matrix formed from the elementary symmetric sums of the $\{\alpha_i\}$ [(Liang et al., 4 Jun 2025), Theorem 4.3].

Non-RS self-dual MDS and AMDS GRL codes exist for specified field characteristics.

3. Decoding Algorithms and Complexity

List and unique decoding algorithms for GRL codes have been developed based on the Guruswami–Sudan interpolation paradigm. The decoding process consists of puncturing, interpolation, factorization, filtering, and final selection (potentially AMD-assisted):

1. Puncturing to a GRS code.
2. Interpolation: find $Q(X,Y) \in \F_q[X,Y]$ vanishing with multiplicity $s$ at each $(\alpha_i, r'_i)$, where $r'_i$ is the received symbol.
3. Factorization: enumerate all $f(X)$ of degree $<k$ such that $Q(X, f(X)) \equiv 0$.
4. Filtering and re-encoding; output codewords within radius $\tau$.
5. Use of systematic AMD codes over $\F_{q^b}$ enables high-probability identification of the correct codeword when list size exceeds one.

Under suitable parameters ($k+3 \leq n \leq q+1$), unique decoding can be achieved up to $\lfloor (n-k)/2 \rfloor$ errors, and list decoding exceeds this radius for $\sqrt{n-1} - \sqrt{k} > 1$, with near-linear complexity $O(n\log^2 n\log\log n)$ for fixed-rate codes (Zhu et al., 30 Dec 2025).

4. Linear Complementary Duality and Hull Structure

GRL codes support systematic construction of Euclidean and Hermitian LCD codes. A necessary and sufficient condition for a GRL code to be LCD is that $G G^T$ (Euclidean) or $G \overline{G}^T$ (Hermitian) is nonsingular.

Key LCD results:

• If $\ell < k/2$, a $[k+\ell,\,k]$ GRL code (with suitable structure) is Euclidean-LCD.
• For Hermitian-LCD codes, with evaluation points forming cosets in the multiplicative group and suitable $A$, upper bounds for the hull dimension are established and shown to be tight (Liang et al., 17 Mar 2026).

An upper bound $\dim\Hull_E(\mathrm{GRL})\leq \ell$ (similarly $\dim\Hull_H(\mathrm{GRL})\leq \ell$) holds in various parameter regimes, enabling controlled construction of small-hull codes.

5. Non-Equivalence to Generalized Reed–Solomon Codes

A fundamental property of generalized Roth–Lempel codes is that, for $k > \ell$, they are not monomially equivalent to any generalized Reed–Solomon code. The lower-right $\ell$ columns of any presumed systematic GRS generator matrix must satisfy Cauchy-matrix relations, but in GRL these columns arise from polynomials of degree $\ell < k$ and cannot satisfy the necessary vanishing conditions. This non-GRS property is provable via explicit row operations and analysis of the systematic form (Liang et al., 17 Mar 2026).

6. Applications: Quantum Codes, NMDS Codes, and Beyond

GRL codes are pivotal in constructing entanglement-assisted quantum error-correcting codes (EAQECCs), where the hull dimension determines the amount of required entanglement. For an $[n,k,d]_q$ code $\mathcal{C}$ with Euclidean-hull dimension $h$, there exists an MDS EAQECC $[[n, k-h, d, n-k-h]]_q$ (Liang et al., 17 Mar 2026). When working over $\F_{q^2}$, Hermitian-hull results yield similar quantum code parameters.

In the context of NMDS codes, GRL constructions have yielded explicit families with completely determined weight distributions:

• Class (A): $C_1 = \mathrm{GRL}_k(\F_q^*, 1, M)$ of parameters $[q+1, k, q+1-k]_q$
• Class (B): $C_2 = \mathrm{GRL}_k(\F_q, 1, M)$ of parameters $[q+2, k, q+2-k]_q$,

with duals of distance $k$ (AMDS). The MacWilliams identity relates the weight enumerators of $C$ and $C^\perp$, and all small-weight multiplicities admit closed-form expressions (Liang et al., 24 Jun 2025, Liang et al., 17 Aug 2025).

GRL codes (and their extensions EGRL) admit deterministic control over code parameters and hull size, making them significant not only for classical and quantum error correction but also in cryptographic, secret sharing, and combinatorial design applications.

7. Concrete Examples

Select explicit instances include:

Code Type	Field $\F_q$	Parameters	Matrix $A$	Special Properties
Non-RS MDS RL	$q=11$	$[8,4,5]$	$\begin{bmatrix}1&8&1\4&1&0\1&0&0\end{bmatrix}$	Non-RS, MDS
Non-RS AMDS Self-Dual GRL	$q=13$	$[8,4,4]$	$\begin{bmatrix}10&9&1\3&1&0\1&0&0\end{bmatrix}$, $\lambda=3$	Non-RS, self-dual, AMDS
NMDS GRL (EGRL, $q=3^m$, $k=5$)	$q=3^m$	$[q+2,5,q-3]$	Any invertible $2\times2$, nonzero	NMDS, explicit weight distribution

The examples illustrate the parameter flexibility and the feasibility of the conditions ensuring MDS, AMDS, NMDS, and self-duality (Liang et al., 4 Jun 2025, Liang et al., 17 Aug 2025, Liang et al., 24 Jun 2025).

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Generalized Roth–Lempel codes constitute a robust platform for constructing explicit, algebraically rich families of linear codes beyond the GRS paradigm, with broad impact on classical and quantum coding theory, combinatorial mathematics, and cryptography.