New Constructions of Additive MDS TRS Codes

Abstract: Additive codes over finite fields generalize linear codes, and additive MDS codes provide a natural extension of linear MDS codes. In this article, we study additive twisted Reed--Solomon (TRS) codes and obtain new constructions of additive MDS codes. First, for additive TRS codes with twist $t=2$ and an arbitrary hook, we establish necessary and sufficient conditions for the codes to be additive MDS, thereby generalizing the results in Section 3 of [Jiayu Ma et al., New families of additive non-Reed-Solomon MDS codes]. In particular, we show that the existence of an additive MDS TRS code with $t=2$ and hook $h=0$ yields codes of larger lengths than those obtained for $t=2$ and $h=k-1$ in [Jiayu Ma et al., New families of additive non-Reed-Solomon MDS codes]. Next, we consider additive TRS codes with twist vector $\mathbf{t}=(1,2)$ and hook vector $\mathbf{h}=(0,0)$, and derive necessary and sufficient conditions for them to be additive MDS. We further establish the existence of such codes. Using the Schur square technique, we obtain mild conditions under which the constructed families are inequivalent to additive Reed--Solomon (RS) codes. Finally, we determine parity-check matrices for both families of additive MDS codes considered in this article.

• The paper demonstrates methods for constructing new additive MDS twisted Reed–Solomon (ATRS) codes, specifically addressing single-twist with arbitrary hooks and doubly twisted codes, and corrects a parity-check formula error in prior work.
• The study provides precise criteria for MDS conditions through detailed analysis of traces and norms of evaluation points, requiring mathematical induction and assumptions like char$(F_q)$ ⧵ r.
• The authors use unique vector characteristics and Schur square analysis to prove ineffectiveness to additive RS codes, concluding these codes exist for certain residue classes and parameters.

Context and contribution

This paper concerns additive twisted Reed–Solomon (ATRS) codes over extension fields $\mathbb{F}_{q^r}$, i.e., $\mathbb{F}_q$-linear codes of length $n$ in $\mathbb{F}_{q^r}^n$ obtained by evaluating $k$-dimensional spaces of "twisted" polynomials over $\mathbb{F}_q$. Building on the additive generalized Reed–Solomon (AGRS) framework of Yadav–Sharma [sharma2024mds] and the single-twist analysis ($t=2$, hook $h=k-1$) of Ma et al. [JiayuMa2026], the authors make three contributions: (i) necessary and sufficient conditions for ATRS codes with twist $t=2$ and an arbitrary hook to be additive MDS; (ii) the first treatment of doubly twisted additive codes, with twist vector $\bm{t}=(1,2)$ and hook vector $\mathbb{F}_q$0; and (iii) a corrected, systematic method for computing parity-check matrices, along the way identifying an error in the parity-check formula of [JiayuMa2026]. Throughout, evaluation points are drawn from $\mathbb{F}_q$1, one representative per Frobenius orbit of elements with exactly $\mathbb{F}_q$2 distinct conjugates.

Single-twist codes with arbitrary hook

The first family is $\mathbb{F}_q$3, spanned over $\mathbb{F}_q$4 by evaluations of polynomials of degree less than $\mathbb{F}_q$5 in which the coefficient $\mathbb{F}_q$6 contributes an extra term $\mathbb{F}_q$7. Since this space is contained in that of $\mathbb{F}_q$8, a distance lower bound shows the code is always either additive MDS or additive almost MDS; it is automatically additive MDS whenever $\mathbb{F}_q$9. For the two exceptional residue classes, the paper gives exact criteria: for $n$0 the code is additive MDS if and only if $n$1 for every subset $n$2 with $n$3, where $n$4 is an explicit expression built from elementary symmetric functions of the conjugates $n$5 and traces $n$6; for $n$7, the code is additive almost MDS if and only if $n$8 and $n$9 for some $\mathbb{F}_{q^r}^n$0 of size $\mathbb{F}_{q^r}^n$1. These results strictly generalize [JiayuMa2026], which handles only $\mathbb{F}_{q^r}^n$2.

The case $\mathbb{F}_{q^r}^n$3 admits a substantially simpler criterion involving norms rather than general symmetric functions: MDS-ness is equivalent to $\mathbb{F}_{q^r}^n$4 for all $\mathbb{F}_{q^r}^n$5. This simplification enables existence results via explicit counting of special field elements:

• Trace-zero family: assuming $\mathbb{F}_{q^r}^n$6, the paper proves $\mathbb{F}_{q^r}^n$7, where $\mathbb{F}_{q^r}^n$8 collects trace-zero representatives of full-length orbits. The key lemma shows $\mathbb{F}_{q^r}^n$9 has $k$0 distinct conjugates iff $k$1 lies in no proper subfield, using an induction on $k$2 that requires $k$3 — an assumption on which the resulting existence claim genuinely depends. Taking $k$4 yields additive MDS codes $k$5 for every nonzero $k$6, verified computationally, e.g., a $k$7 code over $k$8.
• Norm-one family: analogously, $k$9 counts norm-one representatives, giving additive MDS codes for all $\mathbb{F}_q$0 and every $\mathbb{F}_q$1 when $\mathbb{F}_q$2, e.g., a $\mathbb{F}_q$3 code over $\mathbb{F}_q$4.

The authors emphasize that these lengths exceed those attainable at $\mathbb{F}_q$5 in prior work — the principal payoff of allowing arbitrary hooks.

Inequivalence to additive RS codes via Schur squares

Monomial equivalence preserves the dimension of the Schur square, while $\mathbb{F}_q$6. The paper exhibits, in each family, an explicit set of $\mathbb{F}_q$7 linearly independent Schur products lying in $\mathbb{F}_q$8, so that $\mathbb{F}_q$9 whenever $t=2$0 (the independence argument reduces to showing a polynomial of degree $t=2$1 cannot vanish on all $t=2$2 conjugates). Consequently both families contain additive MDS codes not monomially equivalent to any ARS code under the mild condition $t=2$3. The same technique covers the double-twist family.

Double-twisted codes

For $t=2$4, $t=2$5, the twisted space is spanned by $t=2$6. As before, such codes are additive MDS or almost MDS, and are MDS for all $t=2$7. For the exceptional classes, the criteria become: MDS iff $t=2$8 for all $t=2$9; almost-MDS iff $h=k-1$0 and $h=k-1$1 for some $h=k-1$2 of size $h=k-1$3. Existence follows on the same special point sets: over $h=k-1$4 (trace zero), the code is MDS for all pairs $h=k-1$5 when $h=k-1$6; over $h=k-1$7 (norm one), it is MDS whenever $h=k-1$8 and $h=k-1$9. For $t=2$0, a counting argument gives MDS codes whenever $t=2$1. Examples include a $t=2$2 code and an $t=2$3 code over $t=2$4.

Parity-check matrices and correction to prior work

The dual construction uses a vector $t=2$5 with $t=2$6, where $t=2$7 is the full Vandermonde-type matrix up to exponent $t=2$8; the product $t=2$9 is lower-triangular Toeplitz with entries $\bm{t}=(1,2)$0. Applying elementary row/column operations eliminates the offending block, yielding an explicit parity-check matrix when $\bm{t}=(1,2)$1 for both families. When $\bm{t}=(1,2)$2, the paper shows $\bm{t}=(1,2)$3 as well (via the minimal polynomial relation), and the dual collapses cleanly: $\bm{t}=(1,2)$4.

Notably, the paper demonstrates by counterexample that Theorem 3.6(a) of [JiayuMa2026] is incorrect: applying its formula to a concrete $\bm{t}=(1,2)$5 code over $\bm{t}=(1,2)$6 produces a matrix $\bm{t}=(1,2)$7 with $\bm{t}=(1,2)$8, violating the defining property of a parity-check matrix. The corrected procedure here resolves this defect generally.

Limitations and open questions

Several restrictions qualify the results. The automatic MDS dichotomy leaves the residue classes $\bm{t}=(1,2)$9 as the only cases requiring conditions, and the counting-based existence arguments need $\mathbb{F}_q$00. The trace-zero existence result requires $\mathbb{F}_q$01, and the norm-one family requires $\mathbb{F}_q$02; whether analogous constructions exist outside these hypotheses is left open. The inequivalence theorem holds only for $\mathbb{F}_q$03, so short and long codes in these families are not covered. The double-twist analysis is restricted to hooks $\mathbb{F}_q$04; the general two-hook problem remains untreated, as does the determination of parity-check matrices for arbitrary twist/hook vectors. Finally, the paper notes that extending the additive framework to other TGRS variants may yield further additive MDS families, but does not carry this out.

Conclusion

The paper extends the theory of additive twisted Reed–Solomon codes in two directions: arbitrary hooks for the single twist $\mathbb{F}_q$05, and a first study of the double-twist case $\mathbb{F}_q$06. It supplies sharp MDS criteria in terms of traces and norms of evaluation points, constructive existence theorems based on explicit enumeration of trace-zero and norm-one orbit representatives (yielding lengths beyond those previously known for $\mathbb{F}_q$07), Schur-square proofs of inequivalence to ARS codes, and correct parity-check matrices — including a demonstrated correction to an error in the existing literature [JiayuMa2026].