On blocking sets in projective Hjelmslev planes

A ( k, n)-blocking multiset in the projective Hjelmslev plane PHG(R-R(3)) is defined as a multiset k with k(P) = k, k(l) >= n for any line l and 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 vertical bar R vertical bar = q(m), R/rad R congruent to Fq, q = p(r), p prime. We prove that for a ( k, n)-blocking multiset with 1 <= n <= q, k >= n q(m-1)(q + 1). The image of a (n q(m-1)(q + 1), n)-blocking multiset with n < char R under the the canonical map pi((1)) is " sum of lines". In particular, the smallest (k, 1)-blocking set is the characteristic function of a line and its cardinality is k = q(m-1)( q + 1). We prove that if R has a subring S with root vertical bar R vertical bar elements that is a chain ring such that R is free over S then the subplane PHG(S-S(3)) is an irreducible 1-blocking set in PHG(R-R(3)). Corollaries are derived for chain rings with vertical bar R vertical bar = q(2), R/rad R congruent to F-q. In case of chain rings R with vertical bar R vertical bar = q(2), R/ rad R congruent to 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 epsilon {1,..., q + 1}. Finally, we discuss briefly the codes over F-q obtained from certain blocking sets.