American Institute of Mathematical Sciences

Advanced
February  2007, 1(1): 65-81. doi: 10.3934/amc.2007.1.65

On blocking sets in projective Hjelmslev planes

Ivan Landjev 1,

1. 

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev str. bl. 8, Sofia 1113, Bulgaria

Received  April 2006 Revised  August 2006 Published  January 2007

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.
Mathematics Subject Classification: Primary: 51E26, 51E21, 51E22; Secondary: 94B0.
Citation: Ivan Landjev. On blocking sets in projective Hjelmslev planes. Advances in Mathematics of Communications, 2007, 1 (1) : 65-81. doi: 10.3934/amc.2007.1.65
[1]

Aicha Batoul, Kenza Guenda, T. Aaron Gulliver. Some constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2016, 10 (4) : 683-694. doi: 10.3934/amc.2016034
[2]

Somphong Jitman, San Ling, Patanee Udomkavanich. Skew constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2012, 6 (1) : 39-63. doi: 10.3934/amc.2012.6.39
[3]

Michael Kiermaier, Matthias Koch, Sascha Kurz. $2$-arcs of maximal size in the affine and the projective Hjelmslev plane over $\mathbb Z$25. Advances in Mathematics of Communications, 2011, 5 (2) : 287-301. doi: 10.3934/amc.2011.5.287
[4]

Ferruh Özbudak, Patrick Solé. Gilbert-Varshamov type bounds for linear codes over finite chain rings. Advances in Mathematics of Communications, 2007, 1 (1) : 99-109. doi: 10.3934/amc.2007.1.99
[5]

Delphine Boucher, Patrick Solé, Felix Ulmer. Skew constacyclic codes over Galois rings. Advances in Mathematics of Communications, 2008, 2 (3) : 273-292. doi: 10.3934/amc.2008.2.273
[6]

Zihui Liu, Dajian Liao. Higher weights and near-MDR codes over chain rings. Advances in Mathematics of Communications, 2018, 12 (4) : 761-772. doi: 10.3934/amc.2018045
[7]

Anderson Silva, C. Polcino Milies. Cyclic codes of length $ 2p^n $ over finite chain rings. Advances in Mathematics of Communications, 2020, 14 (2) : 233-245. doi: 10.3934/amc.2020017
[8]

Kanat Abdukhalikov. On codes over rings invariant under affine groups. Advances in Mathematics of Communications, 2013, 7 (3) : 253-265. doi: 10.3934/amc.2013.7.253
[9]

David Grant, Mahesh K. Varanasi. The equivalence of space-time codes and codes defined over finite fields and Galois rings. Advances in Mathematics of Communications, 2008, 2 (2) : 131-145. doi: 10.3934/amc.2008.2.131
[10]

Sergio R. López-Permouth, Steve Szabo. On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings. Advances in Mathematics of Communications, 2009, 3 (4) : 409-420. doi: 10.3934/amc.2009.3.409
[11]

Minjia Shi, Daitao Huang, Lin Sok, Patrick Solé. Double circulant self-dual and LCD codes over Galois rings. Advances in Mathematics of Communications, 2019, 13 (1) : 171-183. doi: 10.3934/amc.2019011
[12]

Eimear Byrne. On the weight distribution of codes over finite rings. Advances in Mathematics of Communications, 2011, 5 (2) : 395-406. doi: 10.3934/amc.2011.5.395
[13]

Nuh Aydin, Yasemin Cengellenmis, Abdullah Dertli, Steven T. Dougherty, Esengül Saltürk. Skew constacyclic codes over the local Frobenius non-chain rings of order 16. Advances in Mathematics of Communications, 2020, 14 (1) : 53-67. doi: 10.3934/amc.2020005
[14]

Nabil Bennenni, Kenza Guenda, Sihem Mesnager. DNA cyclic codes over rings. Advances in Mathematics of Communications, 2017, 11 (1) : 83-98. doi: 10.3934/amc.2017004
[15]

Thomas Westerbäck. Parity check systems of nonlinear codes over finite commutative Frobenius rings. Advances in Mathematics of Communications, 2017, 11 (3) : 409-427. doi: 10.3934/amc.2017035
[16]

Thomas Honold, Ivan Landjev. The dual construction for arcs in projective Hjelmslev spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 11-21. doi: 10.3934/amc.2011.5.11
[17]

M. DeDeo, M. Martínez, A. Medrano, M. Minei, H. Stark, A. Terras. Spectra of Heisenberg graphs over finite rings. Conference Publications, 2003, 2003 (Special) : 213-222. doi: 10.3934/proc.2003.2003.213
[18]

Steven T. Dougherty, Esengül Saltürk, Steve Szabo. Codes over local rings of order 16 and binary codes. Advances in Mathematics of Communications, 2016, 10 (2) : 379-391. doi: 10.3934/amc.2016012
[19]

Steven T. Dougherty, Abidin Kaya, Esengül Saltürk. Cyclic codes over local Frobenius rings of order 16. Advances in Mathematics of Communications, 2017, 11 (1) : 99-114. doi: 10.3934/amc.2017005
[20]

Hai Q. Dinh, Hien D. T. Nguyen. On some classes of constacyclic codes over polynomial residue rings. Advances in Mathematics of Communications, 2012, 6 (2) : 175-191. doi: 10.3934/amc.2012.6.175

2019 Impact Factor: 0.734

Tools

Metrics

  • PDF downloads (65)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences