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New families of additive non-RS MDS codes and ACD codes from twisted Reed-Solomon codes
Xia Li, Yu Wang, Yansheng Wu
TL;DR
Additive MDS codes generalize linear MDS codes, and the paper addresses their construction from single-twist additive GTRS codes. It derives MDS conditions, establishes non-equivalence to additive RS codes and dual closure for a class, and gives an ACD existence result under stated field-size and characteristic conditions.
Problem
The paper addresses the need for additive MDS constructions beyond previously established GTRS cases with restricted hook and twist parameters.
Source: §1 · p. 2Method
The paper analyzes single-twist additive GTRS codes using MDS conditions and Schur-square dimensions, and studies a class whose duals remain additive GTRS codes.
Source: §3 · pp. 6, 10, §5 · p. 24Results
The paper establishes necessary and sufficient MDS conditions for general hook and twist parameters and proves the resulting additive MDS codes are not monomially equivalent to additive RS codes.
Source: §3 · pp. 6, 10, §5 · p. 24Takeaways & Limitations
For fixed admissible n, r, k, and t, sufficiently large prime powers q with characteristic outside a finite set admit choices yielding an additive GTRS ACD code.
Source: §4 · p. 22The stated ACD existence result uses h = k − 1 and requires q to be sufficiently large with characteristic avoiding a finite set.
Source: §4 · pp. 19, 21, 22
Abstract
from arXiv · showhide
Additive codes generalize linear codes and play an important role in the construc?tion of quantum codes. As generalizations of linear MDS codes, additive MDS codes have received much research attention. As an important subclass of additive codes, ACD codes are widely used for quantum error correction. In this paper, we study single-twist additive generalized twisted Reed-Solomon (GTRS) codes over finite fields. We derive necessary and sufficient conditions for these codes to be MDS for arbitrary hook and twist parameters. Then we employ Schur squares to prove that such additive MDS codes are not monomially equivalent to additive Reed-Solomon codes. We also identify a class of additive GTRS codes whose duals remain additive GTRS codes, and show that, for all sufficiently large prime powers q whose characteristic avoids a finite set, such GTRS codes yield ACD codes.
1. Introduction
The introduction motivates additive MDS and ACD codes through their coding-theoretic and quantum applications, then positions additive GTRS codes as a route to broader constructions. It identifies the paper’s focus as MDS conditions for general hook and twist parameters and outlines its further results.
1. Introduction
- Additive codes generalize linear codes while retaining algebraic structure and can be described by a basis.Source: §1 · p. 1
- ACD codes with complementary duals are motivated as countermeasures against passive and active side-channel analysis in embedded cryptosystems.The introduction also notes their use in quantum error correction and constructions over finite fields.Source: §1 · p. 2
- Twisted Reed-Solomon research includes MDS subclasses, codes not monomially equivalent to classical RS codes, and GTRS-based MDS LCD constructions.The introduction also notes that ordinary TRS duals usually leave the TRS family, while a special type is closed under duality.Source: §1 · p. 2
- Earlier additive GTRS results treated t = 1 with 0 ≤ h ≤ k − 1 or t = 2 with h = k − 1, motivating the general h and t case.The paper focuses on constructing additive MDS codes beyond those parameter regimes.Source: §1 · p. 2
- The paper proceeds from foundational notions to additive non-RS MDS constructions and dual closure, then to ACD constructions and a conclusion.Source: §1 · p. 2
2. Preliminary
This section establishes the paper’s notation and foundational code concepts, including additive codes, GRS and GTRS constructions, duality, and Schur products. These definitions and cited criteria support the later MDS and ACD arguments.
2. Preliminary
- An additive code is an F_q-linear subspace of F_(q^r)^n, with dimension taken over F_q and minimum distance defined using Hamming weight.Source: §2 · p. 3
- The trace inner product defines code duals, and an additive code is ACD when its intersection with its dual is only the zero vector.Source: §2 · p. 3
- Additive GRS codes are formed by evaluating low-degree polynomials at distinct points and multiplying the evaluations by nonzero column multipliers.When all multipliers equal 1, the construction is called an additive RS code; injectivity ensures dimension k.Source: §2 · p. 4
- Additive GTRS codes use polynomial spaces with twist terms specified by hook, twist, and coefficient parameters; setting all multipliers to 1 gives additive TRS codes.Source: §2 · p. 4
- The Schur product is formed coordinatewise, and its span yields a dimension criterion used to rule out monomial equivalence to additive RS codes.The cited argument compares Schur-square dimensions, which are invariant under monomial equivalence.Source: §2 · pp. 4–5
3. Additive MDS codes from additive GTRS codes over finite fields
This section studies single-twist additive GTRS codes, giving conditions for MDS status and examining when they are not monomially equivalent to additive Reed–Solomon codes. It also identifies a family whose duals remain additive GTRS codes and provides a construction for an all-one multiplier vector.
3. Additive MDS codes from additive GTRS codes over finite fields
- Theorem 3.2 characterizes when T_k(α, t, h, η) is additive MDS: it is MDS if and only if (η, α) ∉ Λ.When b ≠ 0 and 1 ≤ t ≤ r − b, Λ is empty, so every η ∈ F_q* gives an additive MDS code.Source: §3 · p. 10
- The MDS criterion extends prior necessary-and-sufficient conditions for t = 2, h = k − 1 and for t = 1 with 0 ≤ h ≤ k − 1.The paper attributes those earlier cases to Ma et al. and Yadav and Sharma, respectively.Source: §3 · p. 13
- The examples give additive MDS codes T_8(α, 3, 7, z^8) with parameters over F_4096 and T_6(α, 4, 5, z^9) with parameters over F_256.In both examples, the chosen η satisfies (η, α) ∉ Λ.Source: §3 · pp. 13–14
- For h = k − 1, Theorem 3.3 constructs an additive GTRS code whose dual is also additive GTRS, with the dual form determined by D = S_{nr−1} + ηS_{nr−1+t}.The construction assumes n ≤ |B|, 1 ≤ k ≤ nr − 1, 1 ≤ t ≤ nr − k, η ≠ 0, and evaluation elements with disjoint conjugacy classes of size r.Source: §3 · pp. 14, 16, 18
- The stated sums satisfy S_m = 0 for 0 ≤ m ≤ nr − 2, and the terminal sum satisfies S_{nr−1} = 1.Source: §3 · pp. 15–16
- In both cases D ≠ 0 and D = 0, the constructed code and its proposed dual are shown orthogonal with dimensions summing to nr, so they are duals.The proof checks trace inner products across generator-matrix row blocks; when D = 0, the alternate dual construction is used.Source: §3 · pp. 18–19
4. ACD codes from additive GTRS codes over finite fields
The section gives conditions for additive GTRS codes to be ACD and constructs such codes, including additive MDS ACD codes. It proves existence for all sufficiently large prime powers whose characteristic avoids a finite set.
4. ACD codes from additive GTRS codes over finite fields
- The additive GTRS code Tk(α, 1, t, k −1, η) is an ACD code if and only if ∆ ≠ 0.Here ∆ = T2k−2 + 2ηT2k−2+t + η2T2k−2+2t − w⊤A−1w.Source: §4 · p. 19
- The construction establishes existence by choosing evaluation points with nonzero matrix determinants, then selecting a twist parameter for which ∆ does not vanish.The determinant polynomials are nonzero outside finitely many characteristics, and the resulting ∆(η) is a nonzero polynomial of degree at most two.Source: §4 · pp. 21–23
- For every sufficiently large prime power q whose characteristic avoids a finite set, suitable evaluation points and η yield an additive GTRS ACD code.The construction selects evaluation points with det(A) and det(B0) nonzero, then chooses η outside the roots of a nonzero polynomial of degree at most two.Source: §4 · pp. 20–23
- If (η, α) /∈ Λ and ∆ ≠ 0, Tk(α, 1, t, k −1, η) is an additive MDS ACD code.When S = ∅, Λ = ∅, so the condition reduces to ∆ ≠ 0 for any nonzero η.Source: §4 · p. 23
5. Conclusion
The conclusion recaps the paper’s constructions and results for additive GTRS codes.
5. Conclusion
- The paper constructs additive non-RS MDS codes from twisted Reed-Solomon codes and establishes necessary and sufficient MDS conditions.It also identifies additive GTRS codes whose duals remain in that family and constructs ACD codes from additive GTRS codes.Source: §5 · p. 24
- A class of additive GTRS codes has duals that remain additive GTRS codes.Source: §5 · p. 24
- The paper establishes a construction of ACD codes from additive GTRS codes.Source: §5 · p. 24