The dual construction for arcs in projective Hjelmslev spaces
Thomas Honold Ivan Landjev
In this paper, we present a duality construction for multiarcs in projective Hjelmslev geometries over chain rings of nilpotency index 2. We compute the parameters of the resulting arcs and discuss some examples.
keywords: dual space Finite chain ring Galois ring projective Hjelmslev space duality construction multiarc.
The non-existence of $(104,22;3,5)$-arcs
Ivan Landjev Assia Rousseva
Using some recent results about multiple extendability of arcs and codes, we prove the nonexistence of $(104,22)$-arcs in $PG(3,5)$. This implies the non-existence of Griesmer $[104,4,82]_5$-codes and settles one of the four remaining open cases for the main problem of coding theory for $q=5,k=4,d=82$.
keywords: the Griesmer bound. extendable arcs Projective geometries optimal linear codes quasidivisible arcs
On blocking sets in projective Hjelmslev planes
Ivan Landjev
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 $1 \leq n \leq q, k \geq n$qm-1$(q+1)$. The image of a $(n$qm-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 $k =$qm-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.
keywords: blocking set affine Hjelmslev plane chain rings linear codes over finite chain rings. arcs Projective Hjelmslev plane Galois rings
Characterization of some optimal arcs
Ivan Landjev Assia Rousseva
In this paper, we prove the nonexistence of arcs with parameters $(398,101)$, $(464,117)$, and $(467,118)$ in PG$(4,4)$. The proof relies on the geometric characterization of $(117,30)$- and $(118,30)$-arcs in PG$(3,4)$. This settles the problem of finding the exact value of $n_4(5,d)$ for eight values of $d$: $297,298,347,348,349,...,352$.
keywords: projective geometries over nite elds optimal codes minihypers. duality construction blocking sets arcs Griesmer codes dual space

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