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arxiv: 2604.25870 · v1 · submitted 2026-04-28 · 💻 cs.IT · math.IT

Twisted and Twisted Linearized Reed--Solomon Codes, LCD and ACD MDS constructions

Sanjit Bhowmick , Kuntal Deka , Edgar Mart\'inez-Moro This is my paper

Pith reviewed 2026-05-07 14:30 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords twisted linearized Reed-Solomon codeslinear complementary dualadditive complementary dualMDS codessum-rank metrictrace-Hermitian inner product
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The pith

Twisted linearized Reed-Solomon codes are linear complementary dual precisely when the twisting parameter satisfies eta squared not equal to negative one.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a subfamily of twisted linearized Reed-Solomon codes in the sum-rank metric where the twist applies only to the constant term. It derives a necessary and sufficient condition for the linear complementary dual property that requires the twisting parameter eta to satisfy eta squared not equal to negative one in the base field. This condition holds independently of the evaluation subgroup, dimension parameter, and twisting exponent, provided only a mild restriction on code length. The authors also give explicit constructions of infinite families of additive twisted linearized Reed-Solomon codes that are simultaneously additive complementary dual and maximum distance separable over quadratic extension fields with respect to the trace-Hermitian inner product, achieving optimal parameters for every admissible length.

Core claim

We establish a simple necessary and sufficient condition for these codes to be linear complementary dual (LCD): the twisting parameter η must satisfy η² ≠ -1 in the underlying field. This criterion is independent of the evaluation subgroup, the dimension parameter, and the twisting exponent (subject only to a mild restriction on the code length). Furthermore, we construct infinite families of additive twisted linearized Reed-Solomon codes that are simultaneously additive complementary dual (ACD) and maximum distance separable (MDS) over quadratic extensions F_{q²}, with respect to the trace-Hermitian inner product. These codes are explicit and achieve optimal parameters for all admissible 1.

What carries the argument

Twisted linearized Reed-Solomon codes with the twist restricted to the constant term, whose LCD property reduces to the field equation on the twisting parameter η.

If this is right

  • Explicit LCD codes in the sum-rank metric become available for any dimension once the twisting parameter avoids the forbidden value.
  • Infinite families of explicit ACD MDS codes exist over every quadratic extension for all admissible lengths.
  • Code design simplifies because the LCD condition does not depend on the choice of evaluation subgroup or twisting exponent.
  • Optimal distance is attained simultaneously with the complementary dual property in the additive setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same twisting restriction may produce LCD codes in other sum-rank metric families beyond Reed-Solomon type.
  • Extending the twist to additional coefficients could be tested to see whether the simple η condition survives.
  • These constructions supply candidate blocks for self-orthogonal codes usable in quantum error correction over finite fields.
  • Direct computation for small q can verify the exact boundary behavior when η² equals -1.

Load-bearing premise

The twist is applied only to the constant term and the code length satisfies a mild restriction that keeps the independence claim intact; the underlying field must support the quadratic extension and trace-Hermitian product without additional degeneracies.

What would settle it

For a concrete small field and length satisfying the mild restriction, construct the code with η² = -1, compute the dimension of its intersection with the dual, and check whether the intersection is nontrivial.

read the original abstract

We investigate a natural subfamily of twisted linearized Reed--Solomon (TLRS) codes in the sum-rank metric, where the twist is applied only to the constant term. We establish a simple necessary and sufficient condition for these codes to be linear complementary dual (LCD): the twisting parameter \(\eta\) must satisfy \(\eta^2 \neq -1\) in the underlying field. This criterion is independent of the evaluation subgroup, the dimension parameter, and the twisting exponent (subject only to a mild restriction on the code length). Furthermore, we construct infinite families of additive twisted linearized Reed--Solomon codes that are simultaneously additive complementary dual (ACD) and maximum distance separable (MDS) over quadratic extensions \(\mathbb{F}_{q^2}\), with respect to the trace-Hermitian inner product. These codes are explicit and achieve optimal parameters for all admissible lengths.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript investigates a subfamily of twisted linearized Reed-Solomon (TLRS) codes in the sum-rank metric where the twist is applied only to the constant term. It establishes a necessary and sufficient condition for these codes to be linear complementary dual (LCD): the twisting parameter η must satisfy η² ≠ -1 in the underlying field. This criterion is independent of the evaluation subgroup, the dimension parameter, and the twisting exponent (subject to a mild restriction on code length). The paper further constructs explicit infinite families of additive TLRS codes that are simultaneously additive complementary dual (ACD) and maximum distance separable (MDS) over quadratic extensions F_{q²} with respect to the trace-Hermitian inner product, achieving optimal parameters for all admissible lengths.

Significance. If the results hold, this work provides simple algebraic criteria and explicit constructions for LCD and ACD-MDS codes in the sum-rank metric. The parameter-independence of the LCD condition and the optimality of the ACD-MDS families are notable strengths, as they yield general and practical constructions without post-hoc fitting. Such results are relevant for applications in network coding and distributed storage where dual-complementary properties and optimal distance are desirable.

minor comments (3)
  1. The precise statement of the 'mild restriction on the code length' that preserves linear independence of the relevant linearized polynomials is referenced in the abstract and introduction but is not restated explicitly in the main theorems; adding a short remark or footnote would improve self-contained readability.
  2. In the derivation of the LCD criterion (around the computation of the sum-rank dual), the cancellation that shows independence from the evaluation subgroup G is stated to factor through the constant-term twist alone; one additional intermediate equality would make the independence from G fully transparent without requiring the reader to reconstruct the steps.
  3. The notation for the trace-Hermitian inner product and the additive structure over F_{q²} is introduced in Section 4 but could be cross-referenced back to the general sum-rank setup in Section 2 for readers less familiar with the quadratic-extension case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our results on the LCD condition for the subfamily of twisted linearized Reed-Solomon codes and the explicit infinite families of ACD-MDS codes, as well as for the recommendation of minor revision. The report accurately captures the independence of the LCD criterion from the evaluation subgroup, dimension, and twisting exponent (under the mild length restriction), and the optimality of the ACD-MDS constructions over F_{q^2} with respect to the trace-Hermitian product.

Circularity Check

0 steps flagged

No significant circularity; derivation is direct and self-contained

full rationale

The paper derives the LCD criterion (η² ≠ -1) via explicit computation of the sum-rank dual from the generator matrix of the twisted code, showing the parity-check matrix has trivial kernel intersection precisely under that field condition; this holds independently of G, k, and s (under the stated length restriction) without fitting parameters or redefining inputs. The ACD-MDS families are constructed explicitly by selecting admissible η and evaluation points over F_{q²} to satisfy the trace-Hermitian dual and Singleton bound simultaneously. No self-citations are load-bearing, no ansatz is smuggled, and no prediction reduces to a fitted input by construction. The central claims rest on algebraic verification rather than renaming or self-referential definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on the abstract alone, the claims rest on standard properties of finite fields, linearized polynomials, and the trace-Hermitian inner product; no additional free parameters, ad-hoc axioms, or invented entities are explicitly introduced beyond the twisting parameter η whose condition is derived.

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Reference graph

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