Non-GRS MDS Codes: Structure and Applications

• Non-GRS MDS codes are linear codes meeting the Singleton bound (d = n–k+1) by employing algebraic twists that prevent equivalence with generalized Reed-Solomon codes.
• Their construction involves methods like one-twist TGRS, multi-twist variants, and modified GRS designs that alter the Vandermonde structure to satisfy strict MDS criteria.
• These codes offer enhanced cryptographic security and practical benefits in communications and storage by diversifying classical algebraic constructions.

Non-GRS MDS Codes

A non-GRS MDS code is an $[n,k]$ linear code over a finite field $\mathbb{F}_q$ that achieves the Singleton bound ($d=n-k+1$) but is not equivalent (under scaling, permutation, or field automorphism) to any generalized Reed-Solomon (GRS) code. Non-GRS MDS codes provide new algebraic structures distinct from classical GRS codes, expanding the diversity of MDS constructions. Modern research, especially via “twisted” GRS paradigms and their generalizations, has produced large infinite families of such codes for applications in communication, storage, cryptography, and coding theory.

1. Structural Principles and Definitions

Generalized Reed-Solomon and Twisted Variants

Classically, a GRS code is defined as: $$\mathrm{GRS}_{n,k}(\bm{\alpha},\bm{v}) = \{ (v_1f(\alpha_1),\dots,v_nf(\alpha_n)) : f\in\mathbb{F}_q[x]_{<k} \}$$ where the $\alpha_i \in \mathbb{F}_q$ are distinct and $v_i \in \mathbb{F}_q^*$.

Non-GRS constructions often arise from twisting this paradigm by altering the monomial support, introducing non-Vandermonde structure, or by extending with parity coordinates that are nontrivial linear combinations of message coefficients. Principal constructions include:

• Twisted GRS (TGRS) Codes: Add extra terms (“twists”) to selected coefficients leading to codes of the form

$$f(x) = \sum_{i=0}^{k-1} a_i x^i + \sum_\mathrm{twists} \eta_j a_{h_j} x^{k-1+t_j}$$

where positions $h_j$ and degrees $t_j$ are chosen to break GRS equivalence (Zhang et al., 5 Aug 2025, Liu et al., 28 Jul 2025, Li et al., 4 Aug 2025, Hu et al., 7 Feb 2025, Zhao et al., 22 Aug 2024).

• Generalized Roth-Lempel (GRL) Codes: Appending $l$ parity coordinates that are linearly mixed combinations of the highest degree message coefficients (Liang et al., 4 Jun 2025).
• Extended/Punctured/Delete-Row/Monomial-twisted Families: Modifying the GRS generator matrix by removing rows, inserting higher-degree monomials, or extending codes by extra positions with specific combinatorial constraints (Abdukhalikov et al., 15 Dec 2025, Abdukhalikov et al., 4 Jun 2025, Wang et al., 2 Dec 2025, Zhi et al., 6 Jun 2024, Li et al., 9 Jan 2024).

These families are constructed so that the resulting codes are MDS but provably not equivalent to any GRS code for generic choices of parameters.

2. Necessary and Sufficient MDS and Non-GRS Criteria

Algebraic Characterizations

The canonical MDS property is checked via the nonvanishing of all $k\times k$ minors of a generator matrix. For non-GRS, two algebraic characterizations predominate:

• Cauchy/Minor Criterion: For a systematic generator matrix $[I_k|A]$, the code is GRS iff $A$ is a (possibly extended) Cauchy matrix (Liu et al., 28 Jul 2025). Any deviation (e.g., via rank–1 perturbation not absorbable by transformation) yields non-GRS.
• Schur-Square Criterion: A GRS code of dimension $k$ satisfies $\dim(C^{(2)}) = 2k-1$ for its componentwise Schur product. If this value is exceeded, the code is not GRS (Liu et al., 28 Jul 2025, Li et al., 4 Aug 2025, Hu et al., 7 Feb 2025, Abdukhalikov et al., 15 Dec 2025, Li et al., 9 Jan 2024, Zhi et al., 6 Jun 2024).

Specific non-GRS MDS criteria for parameterized families include:

• One-twist TGRS: MDS iff for all $k$-subsets $I$, $\eta\cdot\sum_{i\in I} \alpha_i \ne -1$ (Zhang et al., 5 Aug 2025, Li et al., 4 Aug 2025).
• GTRS with multi-twists: MDS iff all minors $\det(I_k + B F_T)\ne 0$, where $B$ encodes the twists and $F_T$ is a function of evaluation points (Hu et al., 7 Feb 2025).
• Roth-Lempel/GRL: See Theorem 4 in (Liang et al., 4 Jun 2025) for explicit conditions on parity-matrix entries and symmetric functions over evaluation sets.
• Delete-row (row-deletion twist) codes: No vanishing of certain symmetric polynomials or sum conditions over subsets of evaluation points (Abdukhalikov et al., 15 Dec 2025, Abdukhalikov et al., 4 Jun 2025).
• Modified-GRS family: MDS iff for all $(k-1)$-sets of evaluations, a prescribed polynomial in $\eta$ is nonzero (Wang et al., 2 Dec 2025).
• Extended codes: Additional minors (incorporating the extension columns) yield separate symmetric-sum conditions (Zhi et al., 6 Jun 2024, Li et al., 9 Jan 2024).

Proofs of non-GRS status are mainly via Schur-square dimension arguments or explicit exclusion under the Cauchy-matrix criterion.

3. Canonical Examples and Explicit Families

Non-GRS MDS codes appear in various explicit parametric constructions:

Construction Family	Generator Matrix Features	Reference(s)
One-twist TGRS	Vandermonde with one row “twisted” by $\eta x^k$	(Zhang et al., 5 Aug 2025, Li et al., 4 Aug 2025)
General multi-twist TGRS	Multiple twists at various positions (matrix $B$)	(Hu et al., 7 Feb 2025, Liu et al., 28 Jul 2025)
Modified GRS (MGRS)	Final coordinate is $f(0)+\eta f_t$	(Wang et al., 2 Dec 2025)
GRL codes	$l$ appended parity columns via $A\in GL_l$	(Liang et al., 4 Jun 2025)
Row/Column-deleted GRS	Remove or alter one/two rows in Vandermonde basis	(Abdukhalikov et al., 15 Dec 2025, Abdukhalikov et al., 4 Jun 2025)
Extended linear codes	GRS with appended columns: $x^k$ or related terms	(Zhi et al., 6 Jun 2024, Li et al., 9 Jan 2024)

For instance, the one-twist TGRS $C_{k,1,h}(\alpha,1,\eta)$ is explicitly not GRS for $0 < h < k-1$ and $\eta\ne 0$.

Extended or modified GRS families enable code lengths approaching $q/2$ for given $q$ and can achieve MDS property under systematic, explicitly checkable algebraic constraints (Wang et al., 2 Dec 2025, Abdukhalikov et al., 4 Jun 2025).

4. Decoding Algorithms and Complexity

Decoding for non-GRS MDS codes typically draws upon syndrome-based approaches analogous to those for GRS, with modifications to accommodate the structural twists.

• Gaussian Elimination Decoding: For one-twist (and related) TGRS codes, a key-equation approach via Gaussian elimination solves for error locator and evaluator polynomials directly from the linear system, enabling correction up to $\left\lfloor (d-1)/2\right\rfloor$ errors in $O(n^3)$ (Zhang et al., 5 Aug 2025).
• Error-Correcting Pairs (ECPs): Certain extended TGRS admit efficient ECP-based decoding. Performance can surpass standard polynomial or syndrome-based methods for appropriate parameter regimes (Li et al., 4 Aug 2025).
• Complexity: Linear-algebraic approaches are typically $O(n^3)$, versus $O(q n^2)$ or higher for some algebraic approaches reliant on GCD computations or interpolation (Zhang et al., 5 Aug 2025).

For code families sufficiently close in structure to GRS, traditional Berlekamp-Massey or Sugiyama-type algorithms can sometimes be adapted, provided the crucial twist-induced algebraic invariants are respected (Li et al., 4 Aug 2025).

5. Algebraic and Combinatorial Distinctions from GRS Codes

Non-GRS MDS codes are distinguished algebraically from GRS by their generator and parity-check matrices, Schur-square structures, and combinatoric properties.

• Generator Matrix: GRS is characterized by full monomial support; non-GRS codes exhibit gaps, twists, or appended columns involving linear forms in highest-degree coefficients (Zhang et al., 5 Aug 2025, Liu et al., 28 Jul 2025, Liang et al., 4 Jun 2025, Abdukhalikov et al., 15 Dec 2025).
• Parity-Check Matrix: Non-GRS codes often require parity-checks involving Lagrange interpolation constants, symmetric polynomials, or deep-hole vectors, rather than pure Vandermonde/Cauchy structure (Abdukhalikov et al., 4 Jun 2025, Zhi et al., 6 Jun 2024).
• Schur-Square Dimension: MDS property with $\dim(C\ast C) > 2k-1$ contradicts the GRS Cauchy characterization (Liu et al., 28 Jul 2025, Hu et al., 7 Feb 2025).
• Combinatorial Minor Tests: Some constructions employ nonvanishing symmetric polynomials or subset-sum obstructions to both the MDS and non-GRS properties (e.g., zero-sum-free criterion in Li–Zhu (Li et al., 9 Jan 2024)).

6. Applications and Cryptographic Implications

Non-GRS MDS codes possess several advantages in practical contexts:

• Enhanced Cryptographic Security: Classical GRS can be efficiently attacked (e.g., Sidelnikov-Shestakov), so non-GRS MDS codes enable more robust alternatives in code-based cryptosystems (e.g., McEliece) by masking algebraic structures (Zhang et al., 5 Aug 2025, Liu et al., 28 Jul 2025).
• Flexible Coding in Communication and Storage: The ability to construct MDS codes of diverse parameters, error-correction abilities, and structural properties enables their use in SSDs, RAID, satellite, and deep-space transmission (Zhang et al., 5 Aug 2025).
• Combinatorial Constructions and Projective Geometry: Certain classes relate to arc and o-polynomial constructions in finite projective spaces, enriching their connection to classical geometric and combinatorial designs (Abdukhalikov et al., 4 Jun 2025).
• Code Extension, Deep Holes, and Covering Radius: New non-GRS MDS codes enable the extension of codes by deep-hole vectors and explicit computation of covering radii for duals, with relevance to code extension theory and pseudorandomness (Li et al., 4 Aug 2025).

7. Infinite Families and Future Directions

The recent literature demonstrates that most parameter regimes admit infinite families of non-GRS MDS codes via combinatorial, algebraic, or field-extension techniques:

• Large Field Limit: As $q\to\infty$, the count of twist parameters or parity matrix choices for non-GRS MDS increases rapidly (see $B\in\mathcal{Q}$ in (Hu et al., 7 Feb 2025)).
• Generalized Multi-Twist Constructions: Arbitrary weight and position of “twists” in the polynomial or generator matrix enables further diversification, yielding codes beyond all prior single- and double-twist families (Zhao et al., 22 Aug 2024).
• Connections to Other Code Families: Many constructions unify and generalize Roth-Lempel, twisted GRS, extended codes, and their projective-geometric analogs (Liu et al., 28 Jul 2025, Abdukhalikov et al., 15 Dec 2025).

These advances position non-GRS MDS codes as a core area for future research in algebraic coding theory, both for their combinatorial structure and for their utility in post-quantum and high-integrity data systems.