Constructing saturating sets in projective spaces using subgeometries

• Published in: Designs, codes, and cryptography Vol. 90; no. 9; pp. 2113 - 2144
• Main Author: Denaux, Lins
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2022; Springer Nature B.V
• Subjects: Coding and Information Theory; Computer Science; Cryptology; Discrete Mathematics in Computer Science; Hyperplanes; Special Issue: The Art of Combinatorics – A Volume in Honour of Aart Blokhuis; Upper bounds; Covering codes; 51E20; 94B05; Affine spaces; Projective spaces; Saturating sets; 05B25; Linear representations; Subgeometries
• ISSN: 0925-1022; 1573-7586
• DOI: 10.1007/s10623-021-00951-y

A ϱ -saturating set of PG ( N , q ) is a point set S such that any point of PG ( N , q ) lies in a subspace of dimension at most ϱ spanned by points of S . It is generally known that a ϱ -saturating set of PG ( N , q ) has size at least c · ϱ q N - ϱ ϱ + 1 , with c > 1 3 a constant. Our main result is the discovery of a ϱ -saturating set of size roughly ( ϱ + 1 ) ( ϱ + 2 ) 2 q N - ϱ ϱ + 1 if q = ( q ′ ) ϱ + 1 , with q ′ an arbitrary prime power. The existence of such a set improves most known upper bounds on the smallest possible size of ϱ -saturating sets if ϱ < 2 N - 1 3 . As saturating sets have a one-to-one correspondence to linear covering codes, this result improves existing upper bounds on the length and covering density of such codes. To prove that this construction is a ϱ -saturating set, we observe that the affine parts of q ′ -subgeometries of PG ( N , q ) having a hyperplane in common, behave as certain lines of AG ( ϱ + 1 , ( q ′ ) N ) . More precisely, these affine lines are the lines of the linear representation of a q ′ -subgeometry PG ( ϱ , q ′ ) embedded in PG ( ϱ + 1 , ( q ′ ) N ) .