Extended Han-Zhang Codes: MDS/NMDS Dichotomy

• Extended Han-Zhang codes are defined by a one-coordinate polynomial extension that omits the x^(k-1) term, resulting in a clear MDS/NMDS dichotomy based on the evaluation set.
• They employ ℓ-error-correcting pairs to realize a polynomial-time decoding algorithm that corrects up to roughly half the minimum distance with O(n^3) complexity.
• Deep hole constructions are used to generate larger non-GRS MDS codes and to analyze monomial equivalences with Roth-Lempel codes, revealing the codes’ rich geometric structure.

Searching arXiv for papers on extended Han-Zhang codes and related constructions. Extended Han-Zhang codes are a class of linear codes defined by a one-coordinate extension of a constrained polynomial-evaluation space. In the formulation studied in 2025, each code in the family is either a non-generalized Reed-Solomon maximum distance separable code or a near-MDS code, and the family is analyzed through $\ell$-error-correcting pairs, covering radii, and deep holes. The same line of work also shows how deep holes can be used to construct further non-GRS MDS codes and to clarify when the resulting extensions are monomially equivalent to Roth-Lempel codes (Li et al., 26 Aug 2025).

1. Definition and algebraic form

Let $$\mathcal{A}=\{a_1,\ldots,a_n\}\subseteq \mathbb{F}_q$$ be an evaluation-point sequence and let $\mathbf{v}=(v_1,\ldots,v_n)\in (\mathbb{F}_q^*)^n$. The extended Han-Zhang code is

$$\mathcal{C}_k(\mathcal{A}, \mathbf{v}, \infty) = \big\{ (v_1 f(a_1), \ldots, v_n f(a_n), f_k) : f(x)\in \mathcal{V}_k \big\},$$

where

$$\mathcal{V}_k = \left\{ f(x)=\sum_{i=0}^{k-2} f_i x^i + f_k x^k : f_i\in \mathbb{F}_q \right\}.$$

Its length is $n+1$, and its dimension is $k$ (Li et al., 26 Aug 2025).

The defining feature is the omission of the $x^{k-1}$ term from the message space together with the addition of the coefficient $f_k$ as the final coordinate. This places the family close to evaluation-code constructions while preventing it from collapsing to a standard GRS representation. The associated structural question is not whether the code is evaluation-based, but whether the evaluation set $\mathcal{A}$ satisfies the combinatorial condition governing the MDS/NMDS transition.

A central organizing principle is whether $$\mathcal{A}=\{a_1,\ldots,a_n\}\subseteq \mathbb{F}_q$$0 is $$\mathcal{A}=\{a_1,\ldots,a_n\}\subseteq \mathbb{F}_q$$1-zero-sum free or contains a $$\mathcal{A}=\{a_1,\ldots,a_n\}\subseteq \mathbb{F}_q$$2-zero-sum subset. In the literature on this family, that dichotomy completely determines whether the code lies on the MDS side or on the NMDS side of the construction.

2. MDS/NMDS dichotomy and non-GRS status

The basic classification is explicit. If $$\mathcal{A}=\{a_1,\ldots,a_n\}\subseteq \mathbb{F}_q$$3 is $$\mathcal{A}=\{a_1,\ldots,a_n\}\subseteq \mathbb{F}_q$$4-zero-sum free, then $$\mathcal{A}=\{a_1,\ldots,a_n\}\subseteq \mathbb{F}_q$$5 is an $$\mathcal{A}=\{a_1,\ldots,a_n\}\subseteq \mathbb{F}_q$$6 non-GRS MDS code. If $$\mathcal{A}=\{a_1,\ldots,a_n\}\subseteq \mathbb{F}_q$$7 contains a $$\mathcal{A}=\{a_1,\ldots,a_n\}\subseteq \mathbb{F}_q$$8-zero-sum subset, then it is an $$\mathcal{A}=\{a_1,\ldots,a_n\}\subseteq \mathbb{F}_q$$9 NMDS code. The same source states that all extended Han-Zhang codes are not monomially equivalent to GRS codes (Li et al., 26 Aug 2025).

This dichotomy is unusually sharp. The family does not interpolate among many distance regimes: it lands exactly in one of two classes, MDS or NMDS. For code classification, this is significant because non-GRS MDS families are comparatively rare, while NMDS families are structurally close to the Singleton boundary and often admit explicit weight and dual-structure analysis.

A plausible implication is that the family is useful as a controlled testbed for studying the transition between strict Singleton optimality and Singleton defect one. That interpretation is consistent with the later use of covering-radius and deep-hole arguments, which rely on the extension preserving strong distance structure rather than merely improving minimum distance heuristically.

3. Error-correcting pairs and polynomial-time decoding

The 2025 decoding work studies extended Han-Zhang codes via $\mathbf{v}=(v_1,\ldots,v_n)\in (\mathbb{F}_q^*)^n$0-error-correcting pairs. For an $\mathbf{v}=(v_1,\ldots,v_n)\in (\mathbb{F}_q^*)^n$1 linear code $\mathbf{v}=(v_1,\ldots,v_n)\in (\mathbb{F}_q^*)^n$2, an $\mathbf{v}=(v_1,\ldots,v_n)\in (\mathbb{F}_q^*)^n$3-ECP is a pair of length-$\mathbf{v}=(v_1,\ldots,v_n)\in (\mathbb{F}_q^*)^n$4 codes $\mathbf{v}=(v_1,\ldots,v_n)\in (\mathbb{F}_q^*)^n$5 satisfying

1. $\mathbf{v}=(v_1,\ldots,v_n)\in (\mathbb{F}_q^*)^n$6,
2. $\mathbf{v}=(v_1,\ldots,v_n)\in (\mathbb{F}_q^*)^n$7,
3. $\mathbf{v}=(v_1,\ldots,v_n)\in (\mathbb{F}_q^*)^n$8,
4. $\mathbf{v}=(v_1,\ldots,v_n)\in (\mathbb{F}_q^*)^n$9, with $$\mathcal{C}_k(\mathcal{A}, \mathbf{v}, \infty) = \big\{ (v_1 f(a_1), \ldots, v_n f(a_n), f_k) : f(x)\in \mathcal{V}_k \big\},$$0 denoting the coordinate-wise Schur product. The decoding radius is taken as

$$\mathcal{C}_k(\mathcal{A}, \mathbf{v}, \infty) = \big\{ (v_1 f(a_1), \ldots, v_n f(a_n), f_k) : f(x)\in \mathcal{V}_k \big\},$$1

The existence and precise form of the relevant ECPs depend on both the MDS/NMDS status and the parity of $$\mathcal{C}_k(\mathcal{A}, \mathbf{v}, \infty) = \big\{ (v_1 f(a_1), \ldots, v_n f(a_n), f_k) : f(x)\in \mathcal{V}_k \big\},$$2 (Li et al., 26 Aug 2025).

When $$\mathcal{C}_k(\mathcal{A}, \mathbf{v}, \infty) = \big\{ (v_1 f(a_1), \ldots, v_n f(a_n), f_k) : f(x)\in \mathcal{V}_k \big\},$$3 is MDS and $$\mathcal{C}_k(\mathcal{A}, \mathbf{v}, \infty) = \big\{ (v_1 f(a_1), \ldots, v_n f(a_n), f_k) : f(x)\in \mathcal{V}_k \big\},$$4, the code has only $$\mathcal{C}_k(\mathcal{A}, \mathbf{v}, \infty) = \big\{ (v_1 f(a_1), \ldots, v_n f(a_n), f_k) : f(x)\in \mathcal{V}_k \big\},$$5-ECPs, not full $$\mathcal{C}_k(\mathcal{A}, \mathbf{v}, \infty) = \big\{ (v_1 f(a_1), \ldots, v_n f(a_n), f_k) : f(x)\in \mathcal{V}_k \big\},$$6-ECPs. In that case one such pair is

$$\mathcal{C}_k(\mathcal{A}, \mathbf{v}, \infty) = \big\{ (v_1 f(a_1), \ldots, v_n f(a_n), f_k) : f(x)\in \mathcal{V}_k \big\},$$7

If $$\mathcal{C}_k(\mathcal{A}, \mathbf{v}, \infty) = \big\{ (v_1 f(a_1), \ldots, v_n f(a_n), f_k) : f(x)\in \mathcal{V}_k \big\},$$8, then the code admits an $$\mathcal{C}_k(\mathcal{A}, \mathbf{v}, \infty) = \big\{ (v_1 f(a_1), \ldots, v_n f(a_n), f_k) : f(x)\in \mathcal{V}_k \big\},$$9-ECP of the form

$$\mathcal{V}_k = \left\{ f(x)=\sum_{i=0}^{k-2} f_i x^i + f_k x^k : f_i\in \mathbb{F}_q \right\}.$$0

$$\mathcal{V}_k = \left\{ f(x)=\sum_{i=0}^{k-2} f_i x^i + f_k x^k : f_i\in \mathbb{F}_q \right\}.$$1

When $$\mathcal{V}_k = \left\{ f(x)=\sum_{i=0}^{k-2} f_i x^i + f_k x^k : f_i\in \mathbb{F}_q \right\}.$$2 is NMDS and $$\mathcal{V}_k = \left\{ f(x)=\sum_{i=0}^{k-2} f_i x^i + f_k x^k : f_i\in \mathbb{F}_q \right\}.$$3, there exists an $$\mathcal{V}_k = \left\{ f(x)=\sum_{i=0}^{k-2} f_i x^i + f_k x^k : f_i\in \mathbb{F}_q \right\}.$$4-ECP with

$$\mathcal{V}_k = \left\{ f(x)=\sum_{i=0}^{k-2} f_i x^i + f_k x^k : f_i\in \mathbb{F}_q \right\}.$$5

If $$\mathcal{V}_k = \left\{ f(x)=\sum_{i=0}^{k-2} f_i x^i + f_k x^k : f_i\in \mathbb{F}_q \right\}.$$6, the same $$\mathcal{V}_k = \left\{ f(x)=\sum_{i=0}^{k-2} f_i x^i + f_k x^k : f_i\in \mathbb{F}_q \right\}.$$7-ECP form as in the MDS even case applies.

These ECPs lead to an explicit decoding algorithm, presented as Algorithm 1, that corrects up to $$\mathcal{V}_k = \left\{ f(x)=\sum_{i=0}^{k-2} f_i x^i + f_k x^k : f_i\in \mathbb{F}_q \right\}.$$8 errors in polynomial time, with $$\mathcal{V}_k = \left\{ f(x)=\sum_{i=0}^{k-2} f_i x^i + f_k x^k : f_i\in \mathbb{F}_q \right\}.$$9 about half of the minimum distance. The algorithm computes the syndrome, sets up the matrices prescribed by the relevant ECP, solves a linear system to obtain an error-locator vector, extracts the error support from its zeros, reconstructs the error values through syndrome equations, and then corrects the received word. Its complexity is $n+1$0 (Li et al., 26 Aug 2025).

The decoding theory is important for two reasons. First, it gives the first explicit decoding algorithms for extended Han-Zhang codes. Second, it shows that the family is not only structurally interesting as a source of non-GRS MDS and NMDS codes, but also algorithmically accessible through standard algebraic-decoding primitives adapted to its non-GRS geometry.

4. Covering radius and deep holes

The covering radius of the family is completely determined: $n+1$1 This holds for all instances, MDS or NMDS (Li et al., 26 Aug 2025).

Deep holes are characterized through an extension criterion. A vector $n+1$2 is a deep hole if and only if the code generated by adding $n+1$3 as a new row to the generator matrix of $n+1$4 is an $n+1$5 MDS code (Li et al., 26 Aug 2025). This criterion converts a nearest-codeword extremal problem into an MDS-extension problem.

Two explicit deep-hole classes are given. For

$n+1$6

the vector

$n+1$7

is a deep hole if and only if either $n+1$8, or $n+1$9 and $k$0 is an $k$1-set. For

$k$2

the criterion is expressed through the absence of certain parameter values in a set constructed from the elementary symmetric polynomials of subsets of $k$3 (Li et al., 26 Aug 2025).

These results connect extended Han-Zhang codes directly to maximum-likelihood decoding. Deep holes are the worst-case points for MLD, so their explicit determination gives structural information about the extremal geometry of the code, not merely its unique-decoding region.

5. Construction of larger non-GRS MDS codes and Roth-Lempel relations

The deep-hole characterization yields a constructive mechanism. By augmenting the generator matrix with a deep hole, one obtains a new code with parameters $k$4, and this new code is non-GRS MDS (Li et al., 26 Aug 2025).

The resulting codes are not uniform with respect to equivalence. In some cases, notably when the deep hole is associated to $k$5 and the parameter fits, the new codes are monomially equivalent to Roth-Lempel codes

$k$6

In other explicit cases they are not monomially equivalent to Roth-Lempel codes, so they furnish genuinely new non-GRS MDS examples (Li et al., 26 Aug 2025).

This aspect places extended Han-Zhang codes at a junction between extension theory and the classification of non-GRS MDS families. The deep-hole method does not simply repackage a known RL construction; depending on the parameter regime, it either recovers an RL-equivalent code or escapes the RL equivalence class. That distinction matters in classification problems, where monomial equivalence is the operative notion of sameness.

6. Related constructions and broader coding-theoretic context

Several contemporaneous papers place extended Han-Zhang codes within a broader landscape of generalized Roth-Lempel and twisted-evaluation constructions. In the study of $k$7-TGRS and $k$8-ETGRS codes, every such code is either MDS or NMDS, and all $k$9 MDS ETGRS codes are non-GRS; the same work also gives an explicit $x^{k-1}$0 decoding algorithm for ETGRS codes and determines covering radii and a class of deep holes for the duals of TGRS codes, including Han-Zhang codes (Li et al., 4 Aug 2025). This suggests a close structural affinity between extended Han-Zhang codes and other twisted evaluation families that are governed by the same MDS/NMDS dichotomy.

Generalized Roth-Lempel work enlarges the corresponding hook-matrix viewpoint. One paper replaces the fixed matrix

$x^{k-1}$1

with any invertible $x^{k-1}$2 matrix $x^{k-1}$3, obtaining two classes of NMDS codes from generalized RL generator matrices and completely determining their weight distributions. In that framework, setting $x^{k-1}$4 recovers Han and Fan or Zhang and Zheng codes (Liang et al., 24 Jun 2025). A subsequent paper defines extended generalized Roth-Lempel codes and shows that, for $x^{k-1}$5, $x^{k-1}$6, $x^{k-1}$7, $x^{k-1}$8, $x^{k-1}$9, and

$f_k$0

the resulting code is monomially equivalent to Han-Zhang’s 2023 construction; it also establishes MDS/AMDS criteria and completely determines the weight distribution of a class of NMDS EGRL codes (Liang et al., 17 Aug 2025).

A different but related usage appears in the literature on generalized and extended product codes. There, Extended Han-Zhang codes are described as optimal codes that maximize erasure correction with two or a few extra global parities added to product codes, and can be viewed as optimal EPC codes in the $f_k$1 format, matching the minimum-distance upper bound and using field sizes at least $f_k$2 (Blaum et al., 2016). This does not duplicate the polynomial-evaluation definition above, but it shows that the term has also been used in a parity-augmented product-code setting.

Taken together, these results position extended Han-Zhang codes within a dense cluster of non-GRS MDS, NMDS, RL-type, and extended-evaluation constructions. Their distinctive features are the rigid MDS/NMDS bifurcation, explicit ECP-based decoding, complete covering-radius determination, and the use of deep holes as a mechanism for producing larger non-GRS MDS codes.