A -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 spanned by points of S. It is generally known that a -saturating set of PG(N, q) has size at least c • q N − +1 , with c > 1 3 a constant. Our main result is the discovery of a -saturating set of size roughly ( +1)( +2) 2 q N − +1 if q = (q ) +1 , with q an arbitrary prime power. The existence of such a set improves most known upper bounds on the smallest possible size of -saturating sets if < 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 -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 + 1, (q ) N . More precisely, these affine lines are the lines of the linear representation of a q -subgeometry PG( , q ) embedded in PG + 1, (q ) N .

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Let C n−1 (n, q) be the code arising from the incidence of points and hyperplanes in the Desarguesian projective space PG(n, q). Recently, Polverino and Zullo [12] proved that within this code, all non-zero code words of weight at most 2q n−1 are scalar multiples of either the incidence vector of one hyperplane, or the difference of the incidence vectors of two distinct hyperplanes. We improve this result, proving that when q > 17 and q / ∈ {25, 27, 29, 31, 32, 49, 121}, all code words of weight at most (4q − √ 8q − 33 2 )q n−2 are linear combinations of incidence vectors of hyperplanes through a fixed (n − 3)-space. Depending on the omitted value for q, we can lower the bound on the weight of c to obtain the same results.

This work focuses on higgledy-piggledy sets of $k$-subspaces in $\text{PG}(N,q)$, i.e. sets of projective subspaces that are 'well-spread-out'. More precisely, the set of intersection points of these $k$-subspaces with any $(N-k)$ subspace $\kappa$ of $\text{PG}(N,q)$ spans $\kappa$ itself. We highlight three methods to construct small higgledy-piggledy sets of $k$-subspaces and discuss, for $k\in\{1,N-2\}$, 'optimal' sets that cover the smallest possible number of points. Furthermore, we investigate small non-trivial higgledy-piggledy sets in $\text{PG}(N,q)$, $N\leqslant5$. Our main result is the existence of six lines of $\text{PG}(4,q)$ in higgledy-piggledy arrangement, two of which intersect. Exploiting the construction methods mentioned above, we also show the existence of six planes of $\text{PG}(4,q)$ in higgledy-piggledy arrangement, two of which maximally intersect, as well as the existence of two higgledy-piggledy sets in $\text{PG}(5,q)$ consisting of eight planes and seven solids, respectively. Finally, we translate these geometrical results to a coding- and graph-theoretical context.

<p style='text-indent:20px;'>Over the past few years, the codes <inline-formula><tex-math id="M1">\begin{document}$ {\mathcal{C}}_{n-1}(n,q) $\end{document}</tex-math></inline-formula> arising from the incidence of points and hyperplanes in the projective space <inline-formula><tex-math id="M2">\begin{document}$ {\rm{PG}}(n,q) $\end{document}</tex-math></inline-formula> attracted a lot of attention. In particular, small weight codewords of <inline-formula><tex-math id="M3">\begin{document}$ {\mathcal{C}}_{n-1}(n,q) $\end{document}</tex-math></inline-formula> are a topic of investigation. The main result of this work states that, if <inline-formula><tex-math id="M4">\begin{document}$ q $\end{document}</tex-math></inline-formula> is large enough and not prime, a codeword having weight smaller than roughly <inline-formula><tex-math id="M5">\begin{document}$ \frac{1}{2^{n-2}}q^{n-1}\sqrt{q} $\end{document}</tex-math></inline-formula> can be written as a linear combination of a few hyperplanes. Consequently, we use this result to provide a graph-theoretical sufficient condition for these codewords of small weight to be minimal.</p>