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 $\mathbf{v}=(v_1,\dots,v_n)\in (F_q^*)^n$0-th is $\mathbf{v}=(v_1,\dots,v_n)\in (F_q^*)^n$1 (Li et al., 4 Aug 2025, Zhu et al., 2022). The $\mathbf{v}=(v_1,\dots,v_n)\in (F_q^*)^n$2–extended TGRS (ETGRS) code appends the coefficient $\mathbf{v}=(v_1,\dots,v_n)\in (F_q^*)^n$3, yielding an $\mathbf{v}=(v_1,\dots,v_n)\in (F_q^*)^n$4 code.

More general “multi-twist” constructions and arbitrary-twist variants (A-TGRS, $\mathbf{v}=(v_1,\dots,v_n)\in (F_q^*)^n$5–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 $\mathbf{v}=(v_1,\dots,v_n)\in (F_q^*)^n$6–TGRS $\mathbf{v}=(v_1,\dots,v_n)\in (F_q^*)^n$7 code is MDS (i.e., minimum Hamming distance $\mathbf{v}=(v_1,\dots,v_n)\in (F_q^*)^n$8) if and only if

$\mathbf{v}=(v_1,\dots,v_n)\in (F_q^*)^n$9

Otherwise, the code is NMDS with $(+)$0.

For the extended TGRS, the analogous condition is

$(+)$1

for MDS (Zhu et al., 2022, Li et al., 4 Aug 2025). The entire weight distribution is computed in closed form based on the $(+)$2-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 $(+)$3, as established by explicit Schur square (coordinatewise product) dimension arguments. For $(+)$4,

$(+)$5

Dually, the Schur square of the dual has distance 1 for high rates. This property holds for all $(+)$6 or $(+)$7 in TGRS, and for all $(+)$8 in ETGRS (Li et al., 4 Aug 2025). The $(+)$9–ETGRS code is not GRS or EGRS for $$\mathcal{V}_q[x]_k = \left\{f(x) = \sum_{i=0}^{k-1} f_i x^i + \eta f_{k-1} x^k : f_i \in F_q,\, \eta \in F_q^* \right\},$$0 (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 $$\mathcal{V}_q[x]_k = \left\{f(x) = \sum_{i=0}^{k-1} f_i x^i + \eta f_{k-1} x^k : f_i \in F_q,\, \eta \in F_q^* \right\},$$1 satisfying $$\mathcal{V}_q[x]_k = \left\{f(x) = \sum_{i=0}^{k-1} f_i x^i + \eta f_{k-1} x^k : f_i \in F_q,\, \eta \in F_q^* \right\},$$2 over $$\mathcal{V}_q[x]_k = \left\{f(x) = \sum_{i=0}^{k-1} f_i x^i + \eta f_{k-1} x^k : f_i \in F_q,\, \eta \in F_q^* \right\},$$3, yielding two major explicit classes (Li et al., 4 Aug 2025). There are no Galois self-dual ETGRS codes of length $$\mathcal{V}_q[x]_k = \left\{f(x) = \sum_{i=0}^{k-1} f_i x^i + \eta f_{k-1} x^k : f_i \in F_q,\, \eta \in F_q^* \right\},$$4 and dimension $$\mathcal{V}_q[x]_k = \left\{f(x) = \sum_{i=0}^{k-1} f_i x^i + \eta f_{k-1} x^k : f_i \in F_q,\, \eta \in F_q^* \right\},$$5.

For $$\mathcal{V}_q[x]_k = \left\{f(x) = \sum_{i=0}^{k-1} f_i x^i + \eta f_{k-1} x^k : f_i \in F_q,\, \eta \in F_q^* \right\},$$6–TGRS with even $$\mathcal{V}_q[x]_k = \left\{f(x) = \sum_{i=0}^{k-1} f_i x^i + \eta f_{k-1} x^k : f_i \in F_q,\, \eta \in F_q^* \right\},$$7 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 $$\mathcal{V}_q[x]_k = \left\{f(x) = \sum_{i=0}^{k-1} f_i x^i + \eta f_{k-1} x^k : f_i \in F_q,\, \eta \in F_q^* \right\},$$8 is odd, an $$\mathcal{V}_q[x]_k = \left\{f(x) = \sum_{i=0}^{k-1} f_i x^i + \eta f_{k-1} x^k : f_i \in F_q,\, \eta \in F_q^* \right\},$$9–ECP exists; if even, an $f_k$0–ECP. This allows polynomial-time (often $f_k$1) decoders correcting up to half the minimum distance. MDS $f_k$2–TGRS codes with $f_k$3 do not admit $f_k$4–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 $f_k$5–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 $f_k$6 (the code dimension), and explicit $f_k$7-affine families of deep holes are described. The $f_k$8–ETGRS code can be interpreted as a second extension of $f_k$9–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 $f_{k-1}$0 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 $f_{k-1}$1, $f_{k-1}$2, $f_{k-1}$3, $f_{k-1}$4, $f_{k-1}$5, $f_{k-1}$6, $f_{k-1}$7, so $f_{k-1}$8 is $f_{k-1}$9–MDS and not GRS by Schur-square computation (Li et al., 4 Aug 2025).

Table: Key Theoretical Properties of $\eta$0–TGRS and ETGRS Codes

Property	Conditions	Reference Section
MDS	$\eta$1	[(Li et al., 4 Aug 2025), §3]
Non-GRS MDS	$\eta$2 or $\eta$3	[(Li et al., 4 Aug 2025), §4]
Hermitian self-dual MDS	$\eta$4 (over $\eta$5), 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 $\eta$6	[(Li et al., 4 Aug 2025), §7]
Covering radius	$\eta$7	[(Li et al., 4 Aug 2025), §8]

• “Properties and Decoding of Twisted GRS Codes and Their Extensions” (Li et al., 4 Aug 2025)
• “The $\eta$8-extended twisted generalized Reed-Solomon code” (Zhu et al., 2022)