Twisted GRS (TGRS) Codes

• Twisted GRS (TGRS) Codes are a family of MDS and NMDS linear codes constructed by twisting the message polynomial space of generalized Reed–Solomon codes to yield non-GRS structures.
• They feature explicit generator matrices and decoding algorithms using algebraic error-correcting pairs, Gaussian elimination, and Schur product analysis.
• Their rich duality, self-dual constructions, and complete weight distribution properties offer practical insights for error correction and cryptographic applications.

Twisted GRS (TGRS) Codes are a family of maximum distance separable (MDS) and near-MDS (NMDS) linear codes over finite fields, constructed as extensions of classical Generalized Reed–Solomon (GRS) codes by algebraically "twisting" the message polynomial space. These codes play a central role in algebraic coding theory, yielding large sets of non-GRS MDS codes with explicit algebraic structure, families of Hermitian self-dual codes, near-MDS codes, complementary dual (LCD) codes, and possessing rich duality, error-locator, decoding, and combinatorial properties (Li et al., 4 Aug 2025, Zhu et al., 2022).

1. Formalism, Construction, and Generator Matrix

A TGRS code is defined by selecting a collection of distinct field elements $\mathbf{a}=(a_1,\dots,a_n)\subset F_q$, a vector of nonzero multipliers $\mathbf{v}=(v_1,\dots,v_n)\in (F_q^*)^n$, and a "twisted" polynomial message space. For the $(+)$–TGRS (“plus-twisted”) case (the archetypal single-twist construction), the polynomial space is

$$\mathcal{V}_q[x]_k = \left\{f(x) = \sum_{i=0}^{k-1} f_i x^i + \eta f_{k-1} x^k : f_i \in F_q,\, \eta \in F_q^* \right\},$$

i.e. $f_k$ is algebraically tied to $f_{k-1}$ by the twist parameter $\eta$. The TGRS code is the evaluation code

$$\mathrm{TGRS}_k(\mathbf{a}, \mathbf{v}, \eta) = \{ (v_1f(a_1), ..., v_n f(a_n)) : f \in \mathcal{V}_q[x]_k \} \subset F_q^n,$$

with generator matrix $G$ in standard “twisted” form: the first $k-1$ rows are as in GRS, and the $k$-th is $$[v_1 (a_1^{k-1}+\eta a_1^{k}), ..., v_n (a_n^{k-1}+\eta a_n^{k})]$$ (Li et al., 4 Aug 2025, Zhu et al., 2022). The $(+)$–extended TGRS (ETGRS) code appends the coefficient $f_{k-1}$, yielding an $[n+1,k]$ code.

More general “multi-twist” constructions and arbitrary-twist variants (A-TGRS, $(\mathcal{L},\mathcal{P})$–TGRS) are parameterized by discrete twist sets, hook sets, and matrices, generalizing the single-twist case and subsuming all known TGRS families (Zhao et al., 2024, Hu et al., 7 Feb 2025).

2. MDS and NMDS Property, Weight Distribution

A $(+)$–TGRS $[n,k]$ code is MDS (i.e., minimum Hamming distance $d=n-k+1$) if and only if

$$-\eta^{-1} \not\in \mathcal{S}_k, \quad\text{where}\quad \mathcal{S}_k = \left\{ \sum_{a_i \in I} a_i : |I|=k\right\} \subset F_q.$$

Otherwise, the code is NMDS with $d=n-k$.

For the extended TGRS, the analogous condition is

$$\#\left\{A\subset\{\alpha_i\}:|A|=k,\;\sum_{a\in A}a=-\eta^{-1}\right\}=0$$

for MDS (Zhu et al., 2022, Li et al., 4 Aug 2025). The entire weight distribution is computed in closed form based on the $k$-subset sum structure (Li et al., 4 Aug 2025).

3. Non-GRS Structure and Schur Product

Almost all TGRS/ETGRS codes, when MDS, are not equivalent to any GRS code over $F_q$, as established by explicit Schur square (coordinatewise product) dimension arguments. For $k< \frac{n+1}{2}$,

$$\dim(\mathcal{C} * \mathcal{C}) = 2k-1 \quad\text{for GRS,}\qquad \dim(\mathcal{C}_{\text{TGRS}} * \mathcal{C}_{\text{TGRS}})\ge 2k.$$

Dually, the Schur square of the dual has distance 1 for high rates. This property holds for all $3\leq k < n/2$ or $n/2<k\leq n-3$ in TGRS, and for all $3\le k\le n-2$ in ETGRS (Li et al., 4 Aug 2025). The $(+)$–ETGRS code is not GRS or EGRS for $3 \leq k \leq n-2$ (Zhu et al., 2022).

4. Duality, Self-Dual, and Self-Orthogonal Codes

Euclidean and Hermitian duals of TGRS and ETGRS admit explicit matrix expressions. The dual of ETGRS has a parity-check matrix composed of weighted evaluations and twist-dependent terms (Li et al., 4 Aug 2025, Zhu et al., 2022). Hermitian self-dual TGRS and non-GRS MDS Hermitian self-dual TGRS codes are constructed by special choice of $\eta$ satisfying $\eta^q=-\eta$ over $F_{q^2}$, yielding two major explicit classes (Li et al., 4 Aug 2025). There are no Galois self-dual ETGRS codes of length $n+1$ and dimension $(n+1)/2$.

For $(+)$–TGRS with even $q$ and suitable parameters, self-dual and almost self-dual families exist by choice of evaluation set and twisting values. Explicit criteria (based on evaluation point sum) and construction algorithms guarantee self-duality and prescribe multipliers (Zhu et al., 2022).

5. Decoding Algorithms: Error-Correcting Pairs, Gaussian Elimination, List Decoding

Decoding TGRS and ETGRS codes employs algebraic error-correcting pairs (ECPs) and enables fast decoding. ETGRS codes always admit suitable ECPs: if $n-k$ is odd, an $(n-k-1)/2$–ECP exists; if even, an $(n-k)/2$–ECP. This allows polynomial-time (often $O(n^3)$) decoders correcting up to half the minimum distance. MDS $(+)$–TGRS codes with $n=4\ell,\,k=2\ell$ do not admit $\ell$–ECPs, distinguishing them from GRS (Li et al., 4 Aug 2025). Decoders based on Gaussian elimination solve certain structured polynomial systems associated with received words for both MDS and NMDS TGRS codes (Zhang et al., 5 Aug 2025). Complexity compares favorably to classical polynomial-time GRS decoders, and handles arbitrary twist positions.

There is no unique decoding algorithm based on ECPs for MDS $(+)$–TGRS codes when certain symmetry conditions are met, as any such would incorrectly imply GRS equivalence (Li et al., 4 Aug 2025).

6. Covering Radius, Deep Holes, and Extension Theory

For MDS TGRS codes, the covering radius of the dual code is exactly $k$ (the code dimension), and explicit $F_q$-affine families of deep holes are described. The $(+)$–ETGRS code can be interpreted as a second extension of $(+)$–TGRS and remains MDS if and only if a vector appended is a deep hole. The duals of TGRS codes constructed by Han and Zhang are shown to have covering radius $k$ and a complete family of deep holes via affine shifts (Li et al., 4 Aug 2025). This completes the analysis of maximal-likelihood decoding and geometric structure for non-GRS MDS codes in the TGRS/ETGRS family.

7. Concrete Examples and Practical Implications

Explicit small-field examples demonstrate all phenomena, including MDS/NMDS status, non-GRS nature, construction of ECPs and execution of decoding algorithms, and enumeration of deep holes. For instance, for $q=11$, $n=6$, $k=3$, $\mathbf{a} = \{2,5,7,9,10,12\}\subset F_{13}$, $\mathbf{v} = \mathbf{1}$, $\eta=1$, $-\eta^{-1}=12 \notin \mathcal{S}_3$, so $\mathrm{TGRS}_3$ is $[6,3,4]$–MDS and not GRS by Schur-square computation (Li et al., 4 Aug 2025).

Table: Key Theoretical Properties of $(+)$–TGRS and ETGRS Codes

Property	Conditions	Reference Section
MDS	$-\eta^{-1}\notin \mathcal{S}_k$	[(Li et al., 4 Aug 2025), §3]
Non-GRS MDS	$3\leq k<n/2$ or $n/2<k\leq n-3$	[(Li et al., 4 Aug 2025), §4]
Hermitian self-dual MDS	$\eta^q=-\eta$ (over $F_{q^2}$), explicit sets	[(Li et al., 4 Aug 2025), §5]
Dual code structure	Explicit parity-check matrix incl. twist terms	[(Li et al., 4 Aug 2025), §6]
ECP-based decoding	ECP exists for ETGRS for all $k$	[(Li et al., 4 Aug 2025), §7]
Covering radius	$\rho(\mathrm{TGRS}_k^*)=k$	[(Li et al., 4 Aug 2025), §8]

• “Properties and Decoding of Twisted GRS Codes and Their Extensions” (Li et al., 4 Aug 2025)
• “The $(+)$-extended twisted generalized Reed-Solomon code” (Zhu et al., 2022)