Constructing saturating sets in projective spaces using subgeometries

Abstract

A $\varrho$-saturating set of $\text{PG}(N,q)$ is a point set $\mathcal{S}$ such that any point of $\text{PG}(N,q)$ lies in a subspace of dimension at most $\varrho$ spanned by points of $\mathcal{S}$. It is generally known that a $\varrho$-saturating set of $\text{PG}(N,q)$ has size at least $c\cdot \varrho q^{\frac{N-\varrho }{\varrho +1}}$, with $c>\frac{1}{3}$ a constant. Our main result is the discovery of a $\varrho$-saturating set of size roughly $\frac{(\varrho +1)(\varrho +2)}{2}{q}^{\frac{N-\varrho }{\varrho +1}}$ if $q=(q^{\prime }{)}^{\varrho +1}$, with $q^{\prime }$ an arbitrary prime power. The existence of such a set improves most known upper bounds on the smallest possible size of $\varrho$-saturating sets if $\varrho <\frac{2N-1}{3}$. 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 $\varrho$-saturating set, we observe that the affine parts of $q^{\prime }$-subgeometries of $\text{PG}(N,q)$ having a hyperplane in common, behave as certain lines of $\text{AG}(\varrho +1,(q^{\prime }{)}^N)$. More precisely, these affine lines are the lines of the linear representation of a $q^{\prime }$-subgeometry $\text{PG}(\varrho ,q^{\prime })$ embedded in $\text{PG}(\varrho +1,(q^{\prime }{)}^N)$.

[v1] 25 pages, 1 figure [v2] 30 pages, 1 figure: added translation of the main results to the coding theoretical context and made a more thorough comparison with the existing literature [v3] 30 pages, 1 figure: fixed some details and minor grammar and spelling mistakes

Cited by in corpus (1)

• Higgledy-piggledy sets in projective spaces of small dimension