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# Saturating systems and the rank covering radius

Abstract : 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 $s_{q^m/q}(k,\rho)$, which is the minimum $\mathbb{F}_q$-dimension of a $q$-system in $\mathbb{F}_{q^m}^k$ which is rank $\rho$-saturating. This is equivalent to the covering problem in the rank metric. We obtain upper and lower bounds on $s_{q^m/q}(k,\rho)$ and evaluate it for certain values of $k$ and $\rho$. We give constructions of rank $\rho$-saturating systems suggested from geometry.
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https://hal.archives-ouvertes.fr/hal-03857719
Contributor : Martino Borello Connect in order to contact the contributor
Submitted on : Thursday, November 17, 2022 - 2:10:45 PM
Last modification on : Saturday, November 19, 2022 - 4:02:36 AM

### File

2206.14740.pdf
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### Identifiers

• HAL Id : hal-03857719, version 1
• ARXIV : 2206.14740

### Citation

Matteo Bonini, Martino Borello, Eimear Byrne. Saturating systems and the rank covering radius. 2022. ⟨hal-03857719⟩

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