We prove new upper bounds on the smallest size of affine blocking sets, that is, sets of points in a finite affine space that intersect every affine subspace of a fixed codimension. We show an equivalence between affine blocking sets with respect to codimension-2 subspaces that are generated by taking a union of lines through the origin, and strong blocking sets in the corresponding projective space, which in turn are equivalent to minimal codes. Using this equivalence, we improve the current best upper bounds on the smallest size of a strong blocking set in finite projective spaces over fields of size at least 3. Furthermore, using coding theoretic techniques, we improve the current best lower bounds on strong blocking set.Over the finite field F 3 , we prove that minimal codes are equivalent to linear trifferent codes. Using this equivalence, we show that any linear trifferent code of length n has size at most 3 n/4.55 , improving the recent upper bound of Pohoata and Zakharov. Moreover, we show the existence of linear trifferent codes of length n and size at least 1 3 (9/5) n/4 , thus (asymptotically) matching the best lower bound on trifferent codes.We also give explicit constructions of affine blocking sets with respect to codimension-2 subspaces that are a constant factor bigger than the best known lower bound. By restricting to F 3 , we obtain linear trifferent codes of size at least 3 7n/240 , improving the current best explicit construction that has size 3 n/112 .
Let $V$ be a $(d+1)$-dimensional vector space over a field $\mathbb{F}$. Sesquilinear forms over $V$ have been largely studied when they are reflexive and hence give rise to a (possibly degenerate) polarity of the $d$-dimensional projective space $\mathrm{PG}(V)$. Everything is known in this case for both degenerate and non-degenerate reflexive forms if $\mathbb{F}$ is either ${\mathbb R}$, ${\mathbb C}$ or a finite field ${\mathbb F}_q$. In this paper we consider degenerate, non-reflexive sesquilinear forms of $V=\mathbb{F}_{q^n}^3$. We will see that these forms give rise to degenerate correlations of $\mathrm{PG}(2,q^n)$ whose set of absolute points are, besides cones, the (possibly degenerate) $C_F^m$-sets studied by Donati and Durante in 2014. In the final section we collect some results from the huge work of B.C. Kestenband regarding what is known for the set of the absolute points of correlations in $\mathrm{PG}(2,q^n)$ induced by a non-degenerate, non-reflexive sesquilinear form of $V=\mathbb{F}_{q^n}^3$.
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