Sum-rank product codes and bounds on the minimum distance

Alfarano, Gianira N.; Lobillo, F. J.; Neri, Alessandro; Wachter-Zeh, Antonia

doi:10.1016/j.ffa.2022.102013

Cited by 8 publications

(8 citation statements)

References 56 publications

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“…Thus, equivalence of sum-rank metric codes is defined through isometries of the whole space. Moreover, in that paper codes were exclusively considered to be L-linear and hence only L-linear isometries were considered; see also [2,Theorem 3.8] In the following, we aim to characterize the K-linear and K-semilinear isometries, deriving a similar result in the space L[X; θ]/(H Λ (X)).…”

Section: Remark 415 Observe That For Any Set Of Elements Amentioning

confidence: 99%

“…Since we have that L/K is a cyclic Galois extension of degree n, then for every divisor s of n there exists an intermediate field K ⊆ E ⊆ L such that [L : E] = s and E/K is Galois with Gal(E/K) = θ s . 6 Moreover, we will write K (2) to denote the multiplicative subgroup of K * consisting of all the squares, that is…”

Section: The Analogue Of Trombetti-zhou Construction and New Mds Codesmentioning

confidence: 99%

“…Let n = 2t and let E be an intermediate extension such that [L : E] = 2. Let γ ∈ L * be such that N L/K (γ) / ∈ K (2) . Moreover, assume that Λ ⊆ K (2) .…”

Section: The Analogue Of Trombetti-zhou Construction and New Mds Codesmentioning

confidence: 99%

“…Proof. By definition, for the existence of an element γ such that N L/K (γ) / ∈ K (2) , we need that K (2) K * , and this is possible if and only if char(K) = 2.…”

Section: The Analogue Of Trombetti-zhou Construction and New Mds Codesmentioning

confidence: 99%

“…Let 1 ≤ k ≤ ℓ − 1 be positive integers. Let γ ∈ F q 2 \ (F q 2 ) (2) , and let Λ ⊆ (F q 2 ) (2) with |Λ| = ℓ. The code…”

Section: The Analogue Of Trombetti-zhou Construction and New Mds Codesmentioning

confidence: 99%

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Twisted Linearized Reed-Solomon Codes: A Skew Polynomial Framework

Neri¹

2021

Preprint

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We provide an algebraic description for sum-rank metric codes, as quotient space of a skew polynomial ring. This approach generalizes at the same time the skew group algebra setting for rank-metric codes and the polynomial setting for codes in the Hamming metric. This allows to construct twisted linearized Reed-Solomon codes, a new family of maximum sum-rank distance codes extending at the same time Sheekey's twisted Gabidulin codes in the rank metric and twisted Reed-Solomon codes in the Hamming metric. Furthermore, we provide an analogue in the sum-rank metric of Trombetti-Zhou construction, which also provides a family of maximum sum-rank distance codes. As a byproduct, in the extremal case of the Hamming metric, we obtain a new family of additive MDS codes over quadratic fields.

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Section: Remark 415 Observe That For Any Set Of Elements Amentioning

confidence: 99%

Section: The Analogue Of Trombetti-zhou Construction and New Mds Codesmentioning

confidence: 99%

“…Let n = 2t and let E be an intermediate extension such that [L : E] = 2. Let γ ∈ L * be such that N L/K (γ) / ∈ K (2) . Moreover, assume that Λ ⊆ K (2) .…”

Section: The Analogue Of Trombetti-zhou Construction and New Mds Codesmentioning

confidence: 99%

“…Proof. By definition, for the existence of an element γ such that N L/K (γ) / ∈ K (2) , we need that K (2) K * , and this is possible if and only if char(K) = 2.…”

Section: The Analogue Of Trombetti-zhou Construction and New Mds Codesmentioning

confidence: 99%

“…Let 1 ≤ k ≤ ℓ − 1 be positive integers. Let γ ∈ F q 2 \ (F q 2 ) (2) , and let Λ ⊆ (F q 2 ) (2) with |Λ| = ℓ. The code…”

Section: The Analogue Of Trombetti-zhou Construction and New Mds Codesmentioning

confidence: 99%

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Twisted Linearized Reed-Solomon Codes: A Skew Polynomial Framework

Neri¹

2021

Preprint

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Geometric dual and sum‐rank minimal codes

Borello,

Zullo

2024

J of Combinatorial Designs

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The main purpose of this paper is to further study the structure, parameters and constructions of the recently introduced minimal codes in the sum‐rank metric. These objects form a bridge between the classical minimal codes in the Hamming metric, the subject of intense research over the past three decades partly because of their cryptographic properties, and the more recent rank‐metric minimal codes. We prove some bounds on their parameters, existence results, and, via a tool that we name geometric dual, we manage to construct minimal codes with few weights. A generalization of the celebrated Ashikhmin–Barg condition is proved and used to ensure the minimality of certain constructions.

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