Linear cutting blocking sets and minimal codes in the rank metric

Alfarano, Gianira N.; Borello, Martino; Neri, Alessandro; Ravagnani, Alberto

doi:10.1016/j.jcta.2022.105658

Cited by 28 publications

(82 citation statements)

References 36 publications

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“…It is proved in [Proposition 3.2 in [2]] that C is (rank-)nondegenerate if and only if for every A ∈ GL n (q), the code C • A is (Hamming-)nondegenerate. Note that, as already observed in [28, Corollary 6.5], nondegenerate rank-metric [n, k] q m /q code may exist only if n ≤ mk.…”

Section: Linear Codesmentioning

confidence: 99%

“…Linear cutting blocking sets were introduced recently in [2], in connection with minimal codes in the rank-metric. In order to define these last, we need to introduce the notion of rank-support: for a word c ∈ F 1×n q m and an ordered basis Γ = {γ 1 , .…”

Section: Construction With Linear Cutting Blocking Setsmentioning

confidence: 99%

“…The geometry of rank-metric codes has recently been investigated [2,32]: rank-metric codes correspond to q-systems and linear sets. In this paper, we exploit this relationship further: we introduce the notion of a rank saturating system in correspondence with a rank-metric covering code.…”

Section: Introductionmentioning

confidence: 99%

“…Such codes, as considered in this paper, are F q m -linear subspaces for which the ambient space F 1×n q m is endowed with the rank distance function. While there exists a more general description of rank-metric codes simply as linear spaces of matrices, the restriction to the F q m -linear subspaces of F 1×n q m has more immediate connections to finite geometry [2,5]. In this paper, we introduce the notion of an [n, k] q m /q rank ρ-saturating system.…”

Section: Introductionmentioning

confidence: 99%

“…In our analysis, we will use the notion of an [n, k] q m /q system, which is simply an n-dimensional F q -subspace of F k q m whose F q m -span is the full space and is a q-analogue of a projective system. Such q-systems have been used already in [2] and [32] to describe geometric aspects of rankmetric codes. Then an [n, k] q m /q rank ρ-saturating system is one whose associated linear set is a (ρ − 1)-saturating set in PG(k − 1, q m ).…”

Section: Introductionmentioning

confidence: 99%

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Saturating systems and the rank covering radius

Bonini¹,

Borello²,

Byrne³

2022

Preprint

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We introduce the concept of a rank saturating system and outline its correspondence to a rank-metric code with a given covering radius. We consider the problem of finding the value of s q m /q (k, ρ), which is the minimum F q -dimension of a q-system in F k q m which is rank ρ-saturating. This is equivalent to the covering problem in the rank metric. We obtain upper and lower bounds on s q m /q (k, ρ) and evaluate it for certain values of k and ρ. We give constructions of rank ρ-saturating systems suggested from geometry.

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Section: Linear Codesmentioning

confidence: 99%

Section: Construction With Linear Cutting Blocking Setsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Saturating systems and the rank covering radius

Bonini¹,

Borello²,

Byrne³

2022

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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Generalised evasive subspaces

Gruica,

Ravagnani,

Sheekey

et al. 2024

J of Combinatorial Designs

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We introduce and explore a new concept of evasive subspace with respect to a collection of subspaces sharing a common dimension, most notably partial spreads. We show that this concept generalises known notions of subspace scatteredness and evasiveness. We establish various upper bounds for the dimension of an evasive subspace with respect to arbitrary partial spreads, obtaining improvements for the Desarguesian ones. We also establish existence results for evasive spaces in a nonconstructive way, using a graph theory approach. The upper and lower bounds we derive have a precise interpretation as bounds for the critical exponent of certain combinatorial geometries. Finally, we investigate connections between the notion of evasive space we introduce and the theory of rank‐metric codes, obtaining new results on the covering radius and on the existence of minimal vector rank‐metric codes.

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Linear cutting blocking sets and minimal codes in the rank metric

Cited by 28 publications

References 36 publications

Saturating systems and the rank covering radius

Saturating systems and the rank covering radius

Geometric dual and sum‐rank minimal codes

Generalised evasive subspaces

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