The equivalent condition for GRL codes to be MDS, AMDS or self-dual

It’s well known that MDS, AMDS or self dual codes have good algebraic properties, and are applied in communication systems, data storage, quantum codes, and so on. In this paper, we focus on a class of generalized Roth-Lempel linear codes which are not not equivalent to linear codes in [21],[22] and give an equivalent condition for them or their dual to be non RS MDS, AMDS or non RS self-dual and some corresponding examples.

💡 Research Summary

The paper investigates a family of linear codes called generalized Roth‑Lempel (GRL) codes, which extend the classical Roth‑Lempel construction by appending three extra columns to the Vandermonde generator matrix of a Reed‑Solomon (RS) code and mixing these columns with an arbitrary invertible 3 × 3 matrix A₃×₃ over a finite field 𝔽_q. The authors focus on the case where the dimension k exceeds three and aim to characterize precisely when such a GRL code, or its dual, is a non‑RS maximum‑distance‑separable (MDS) code, an almost‑MDS (AMDS) code, or a non‑RS self‑dual code.

The paper begins with a concise review of basic coding‑theoretic notions: linear codes, the Singleton bound, MDS and AMDS definitions, self‑duality, and the classical RS construction. It then recalls the original Roth‑Lempel construction (adding two columns) and a recent extension that adds three columns with a fixed matrix A₂. The authors replace A₂ by a general invertible matrix A₃×₃, defining the GRL code GRL_k(α, v, A₃×₃) as the set of vectors (v₁ f(α₁),…,v_n f(α_n), β) where f runs over polynomials of degree < k and β is a linear combination of the highest k‑l coefficients of f with the rows of A₃×₃.

Non‑RS Property (Theorem 7).
For any k > 3, the GRL code is shown to be inequivalent to any RS code. The proof normalizes the multiplier vector v to all‑ones, constructs polynomials f_i(x)=∏_{j≠i}(x−α_j), and writes the generator matrix G₄ as a product C·V·F, where C and V are nonsingular Vandermonde‑type matrices. Assuming G₄ generated an RS code would force a quadratic polynomial to have more than two distinct roots (the α_i’s), a contradiction. Hence the code is non‑RS for all admissible parameters.

MDS Characterization (Theorem 8).
Using the well‑known criterion that a linear code is MDS iff every set of k columns of its generator matrix is linearly independent, the authors examine all possible k‑column submatrices of G₄. Four cases arise:

1. Submatrices containing none of the three special columns are ordinary Vandermonde matrices and are always nonsingular.
2. Submatrices containing exactly one special column u_s lead to a determinant equal to
   a₁s · Σ_{α∈J}∏{β∈J{α}}β − a₂s · Σ{α∈J}α + a₃s,
   where J is a (k‑1)-subset of evaluation points. The condition that this expression never vanishes for any J yields condition (1) of the theorem.
3. Submatrices containing exactly two special columns u_t and u_s (t < s) produce a more intricate determinant involving three symmetric sums S₁, S₂, S₃ over a (k‑2)-subset I. The vanishing of the determinant translates into condition (2) of the theorem.
4. Submatrices containing all three special columns have determinant det(A₃×₃) times a Vandermonde factor, which is nonzero because A₃×₃ is invertible.

Thus the GRL code is a non‑RS MDS code if and only if both (1) and (2) hold simultaneously.

Dual AMDS and Self‑Dual Conditions.
In Section 4 the authors derive an explicit parity‑check matrix for GRL_k(α, v, A₃×₃) and observe that its transpose is again a GRL matrix with complementary parameters. This symmetry allows them to formulate an equivalent condition for the dual code to be AMDS (i.e., to have minimum distance n − k). For self‑duality, they require k = n/2 and the parity‑check matrix to satisfy H Hᵗ = 0. This imposes algebraic relations among the entries of A₃×₃, which are expressed as a set of bilinear equations (essentially requiring the rows of A₃×₃ to be orthogonal under a specific inner product). When these equations are satisfied, the GRL code is a non‑RS self‑dual code.

Special Cases and Concrete Example.
Corollary 10 specializes A₃×₃ to the form
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