Three Combinatorial Perspectives on Minimal Codes

Abstract

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Saturating systems and the rank covering radius

“…Similarly to saturating sets, one of the main problem in the theory of minimal codes is the construction of short families, that is equivalent to construct small strong blocking sets. Some recent results can be found in [3,4,26].…”

“…is a linear cutting blocking set, as shown in [2,Example 6.9]. So the [6,3] 256/2 system U with the same generator matrix is a rank 2-saturating system. It has the smallest F 2 -dimension.…”

“…Direct computations show that v = λ (1) u (1) + λ (2) u (2) + λ (3) u (3) . Since u (1) , u (2) ∈ F k q ⊆ U and u (3) ∈ F 3 q × F k−3 q 2 ⊆ U , we have that ρ rk (U ) ≤ 3.…”

“…then the process has already terminated in the Step (II) and it is enough to set λ (3) = 0 and u (3) = (0, . .…”

“…Let λ in F 16 such that λ 4 = λ + 1. The[6,3] 16/2 system with generator matrix G =   λ 4 λ 10 λ 8 λ 3 λ 9 λ 7 λ 14 λ 8 λ λ 8 0 λ 8 λ 10 0 λ 6 λ 5 λ 11 λ 3   .…”