On blocking sets in projective Hjelmslev planes

A $(k, n)$-blocking multiset in the projective Hjelmslev plane PHG($R^3_R$) is defined as a multiset $\mathfrak K$ with $\mathfrak K(\mathcal P) = k$, $\mathfrak K(l) \geq n$ for any line $l$ and $\mathfrak K(l_0) = n$ for at least one line $l_0$. In this paper, we investigate blocking sets in projective Hjelmslev planes over chain rings $R$ with $|R| = q^m, R$∕rad$R \cong \mathbb F_q, q = p^r, p$ prime. We prove that for a $(k, n)$-blocking multiset with $1 \leq n \leq q, k \geq n$ q m-1 $(q+1)$. The image of a $(n$ q m-1 $(q +1), n)$-blocking multiset with $n q m-1 $(q + 1)$. We prove that if $R$ has a subring $S$ with $\sqrt R$ elements that is a chain ring such that $R$ is free over $S$ then the subplane PHG($S^3_S$) is an irreducible $1$-blocking set in PHG($R^3_R$). Corollaries are derived for chain rings with $|R| = q^2, R$∕rad$R \cong \mathbb F_q$. In case of chain rings $R$ with $|R| = q^2, R$∕rad$R \cong \mathbb F_q$ and $n = 1$, we prove that the size of the second smallest irreducible $(k, 1)$-blocking set is $q^2 + q + 1$. We classify all blocking sets with this cardinality. It turns out that if char$R = p$ there exist (up to isomorphism) two such sets; if char$R = p^2$ the irreducible $(q^2 + q + 1, 1)$-blocking set is unique. We introduce a class of irreducible $(q^2 + q + s, 1)$ blocking sets for every $s \in {1, \cdots , q + 1}$. Finally, we discuss briefly the codes over $\mathbb F_q$ obtained from certain blocking sets.