Saturating systems and the rank covering radius
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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_q^m^k 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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