On Saturating Sets in Small Projective Geometries

Abstract

A set of points, S ⊆ P G (r, q), is said to be -saturating if, for any point x ∈ P G(r, q), there exist + 1 points in S that generate a subspace in which x lies. The cardinality of a smallest possible set S with this property is denoted by k(r, q, ). We give a short survey of what is known about k(r, q, 1) and present new results for k(r, q, 2) for small values of r and q. One construction presented proves that k(5, q, 2) ≤ 3q + 1 for q = 2, q ≥ 4. We further give an upper bound on k ( + 1, p m , ).

Linear codes with covering radius 3

“…The results in the table follow from the present work and earlier results published in [1,6,8,11,14,19]. Proper references are given for all entries.…”

Complete caps in projective spaces PG (n, q)

“…We write a cap as a set of points. (1,0,0,0), (0,1,7,15), (1,9,5,2), (0, 1,2,9), (1,2,12,12), (1,6,15,14), (1,4,15,8), (1,8,16,1), (1,12,6,1), (1,5,11,11), (1,16,14,11), (1,11,8,9), (1,3,7,7), (1,9,6,12), (0, 1,9,12), (1,6,5,…”

“…(1,0,0,0), (0,1,3,4), (1,10,5,13), (1,9,14,4), (1,12,11,10), (1,17,3,8), (1,6,0,4), (1,4,9,3), (1,13,10,10), (1,13,1,6), (0, 1,15,14), (1,13,4,11), (1,15,18,17), (1,5,11,10), (1,9,13,4), (1,4,8,1...…”

“…(1,0,0,0), (0,1,0,13), (1,1,22,6), (1,2,4,1), (1,2,0,25), (1,4,12,23), (1,9,15,16), (1,19,26,24), (1,16,22,1), (1,26,21,17), (1,5,13,12), (1,8,15,7), (1,0,24,25), (1,14,16,15), (1,2,4,2), (1,21,1,21), (1,3,7,20), (1,0,7,1), (1,3,24,12), (1,14,13,12), (1,19,10,…”

“…(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1), (1,1,3,19), (1,6,21,25), (1,3,13,20), (1,7,19,23), (1,14,25,14), (1,23,24,9), (1,16,7,27), (1,11,26,21), (1,13,14,10), (1,28,4,17), (1,20,27,5), (1,19,14,4), ( (1,16,17,9), (1,12,13,26), …”