Arcs and blocking sets in projective Hjelmslev planes
by
Ivan N. Landjev
New Bulgarian University, Bulgaria

Abstract. The geometric nature of certain problems in coding theory has been long known. The geometric approach to such problems is based on the equivalence between linear codes with no coordinate identically zero and multisets of points in projective coordinate geometries over appropriate algebraic structures.

In the last twenty years, linear codes over various classes of rings (notably finite chain rings and Frobenius rings) came into the focus of the coding theory resarch. These codes admit a geometric treatment as (multi)sets of points in a special class of geometries known as Hjelmslev geometries.

In this talk, we give a survey of the recent results on arcs and blocking sets in the projective planes of index of nilpotency 2. These sets admit an immediate interpretation as linear codes, but are also interesting in their own right.

Let R be a finite chain ring with \lvert R\rvert=q², r/\rad R ≅ F_q. A (k, n)-arc \PHG(R_R³) is defined as a multiset \mathfrakK with \mathfrakK(P)=k, \mathfrakK(l) ≤ n for any line l and \mathfrakK(l₀)=n for at least one line l₀. We consider (k, n)-arcs in projective Hjelmslev planes \PHG(R_R³), where R is a finite chain ring. We introduce general upper bounds on the cardinality of such arcs and establish the maximum possible size of projective (k, n)-arcs with n ∈ {q², ..., q²+q-1}. Constructions of projective arcs in the Hjelmslev planes over the small chain rings R with 4, 9, 16 and 25 elements are also given. Further we prove that maximal (q²+q+1, 2)-arcs in \PHG(R_R³) exist if and only if \Char R=4.

A (k, n)-blocking multiset in the projective Hjelmslev plane \PHG(R_R³) is defined as a multiset \mathfrakK with \mathfrakK(P)=k, \mathfrakK(l) ≥ n for any line l and \mathfrakK(l₀)=n for at least one line l₀. We consider blocking sets in projective Hjelmslev planes over chain rings R with \lvert R\rvert=q^m, R/\rad R ≅ F_q, q=p^r, p prime. We prove that for a (k, n)-blocking multiset with 1 ≤ n ≤ q, k ≥ nq^m-1(q+1). The image of a (nq^m-1(q+1), n)-blocking multiset with n < \Char R under the the canonical map p⁽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 √{\lvert R\rvert} elements that is a chain ring such that R is free over S then the subplane \PHG(S_S³) is an irreducible 1-blocking set in \PHG(R_R³).

In case of chain rings R with \lvert R\rvert=q², R/\rad R ≅ F_q and n=1, we prove that the size of the second smallest irreducible (k, 1)-blocking set is q²+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=p² the irreducible (q²+q+1, 1)-blocking set is unique. We introduce a class of irreducible (q²+q+s, 1) blocking sets for every s ∈ {1, ..., q+1}.

Date received: February 29, 2008