Arcs in projective Hjelmslev planes

Abstract

The (£,/i)-arcs in projective Hjelmslev plane PHG(/?|) over a finite chain ring R are considered We prove general upper bounds on the cardinal!ty of such arcs and establish the maximum possible size of the projective (&, n)-arcs with n £{ (?,... rf + q-l}. Constructions of projective arcs in the Hjelmslev planes over the chain rings with 4 and 9 elements are also given.

Non-free extensions of the simplex codes over a chain ring with four elements

“…The number of all hyperplanes of type a = (a 0 , a 1 , a 2 ) ∈ N 3 0 is denoted by A a . The sequence {A a | a ∈ N 3 0 } is called the spectrum of K. For the current state of the research on arcs in projective Hjelmslev geometries, we refer to [6,7,10,11,14,20] and the surveys [12,13].…”

The maximal size of 6- and 7-arcs in projective Hjelmslev planes over chain rings of order 9

“…Initial work on this problem has been done in [12,28]. A complete solution for the two chain rings Z 4 , S 2 = F 2 [X]/(X 2 ) of order 4 and projective arcs (i.e., arcs without multiple points) appears already in [28].…”

“…A complete solution for the two chain rings Z 4 , S 2 = F 2 [X]/(X 2 ) of order 4 and projective arcs (i.e., arcs without multiple points) appears already in [28]. The recent extension to the non-projective case can be found in [27, Theorem 5].…”

“…Here, except for n = 2 and 9 n 12, which are comparatively easy and done in [28], progress has largely relied on computer searches providing constructions of n-arcs and in turn reasonable lower bounds for m n (R). The case n = 3 was finished in [3], the case n = 4 was finished in [21] and [8] by computer constructions matching the upper bound m 4 (R) 30 from [12, Theorem 5.3], and the cases n ∈ {5, 8} were finished in [9] by providing non-existence proofs matching the known constructive lower bounds for m 5 (R) resp.…”

Sets of Type (d 1,d 2) in Projective Hjelmslev Planes over Galois Rings

“…In this paper the ring R is always a Galois ring. Much more on projective Hjelmslev planes can be found in the work of Honold, Landjev and their coworkers [12,13,17,16]. A useful tool is the homomorphism φ : Z p s+1 → Z p s which maps an representing element from Z p s+1 to its remainder modulo p s .…”