February 2017, 11(1): 179-185. doi: 10.3934/amc.2017011

Codes from hall planes of odd order

Institute of Mathematics, Physics and Computer Science, Aberystwyth University, Aberystwyth SY23 3BZ, UK

Received  August 2015 Published  February 2017

We show explicitly that the dimension of the ternary code of the Hall plane of order 9 is greater than the dimension of the ternary code of the desarguesian plane of order 9. The proof requires finding a word with some defined properties in the dual ternary code of the desarguesian plane of order 9. The idea can be generalised for other orders, provided that words in the dual code of the desarguesian projective plane that have the specified properties can be found.

Citation: J. D. Key, T. P. McDonough, V. C. Mavron. Codes from hall planes of odd order. Advances in Mathematics of Communications, 2017, 11 (1) : 179-185. doi: 10.3934/amc.2017011
References:
[1] E. F. Assmus Jr and J. D. Key, Designs and Their Codes, Cambridge Univ. Press, Cambridge, 1992. doi: 10.1017/CBO9781316529836.
[2]

W. BosmaJ. Cannon and C. Playoust, The Magma algebra system Ⅰ: the user language, J. Symb. Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.

[3]

J. Cannon, A. Steel and G. White, Linear codes over finite fields, in Handbook of Magma Functions (eds. J. Cannon and W. Bosma), Computational Algebra Group, Dep. Math. , Univ. Sydney, 2006, V2. 13,3951-4023.

[4]

P. Dembowski, Finite geometries, in Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, 1968.

[5]

The GAP Group, GAP -Groups, Algorithms, and Programming, Version 4. 4. 12,2008, available at http://www.gap-system.org.

[6]

D. GhinelliM. J. de Resmini and J. D. Key, Minimum words of codes from affine planes, J. Geom., 91 (2008), 43-51. doi: 10.1007/s00022-008-2096-y.

[7]

D. R. Hughes and F. C. Piper, Projective Planes, Springer-Verlag, New York, 1973.

[8]

J. D. Key and M. J. de Resmini, Small sets of even type and codewords, J. Geom., 61 (1998), 83-104. doi: 10.1007/BF01237498.

[9]

J. D. KeyT. P. McDonough and V. C. Mavron, An upper bound for the minimum weight of the dual codes of desarguesian planes, Europ. J. Combin., 30 (2009), 220-229. doi: 10.1016/j.ejc.2008.01.003.

[10]

J. D. KeyT. P. McDonough and V. C. Mavron, Codes from Hall planes of even order, J. Geom., 105 (2014), 33-41. doi: 10.1007/s00022-013-0189-8.

[11]

H. Lüneburg, Translation Planes, Springer-Verlag, New York, 1980.

[12]

T. G. Ostrom, Finite Translation Planes, Springer-Verlag, 1970.

show all references

References:
[1] E. F. Assmus Jr and J. D. Key, Designs and Their Codes, Cambridge Univ. Press, Cambridge, 1992. doi: 10.1017/CBO9781316529836.
[2]

W. BosmaJ. Cannon and C. Playoust, The Magma algebra system Ⅰ: the user language, J. Symb. Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.

[3]

J. Cannon, A. Steel and G. White, Linear codes over finite fields, in Handbook of Magma Functions (eds. J. Cannon and W. Bosma), Computational Algebra Group, Dep. Math. , Univ. Sydney, 2006, V2. 13,3951-4023.

[4]

P. Dembowski, Finite geometries, in Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, 1968.

[5]

The GAP Group, GAP -Groups, Algorithms, and Programming, Version 4. 4. 12,2008, available at http://www.gap-system.org.

[6]

D. GhinelliM. J. de Resmini and J. D. Key, Minimum words of codes from affine planes, J. Geom., 91 (2008), 43-51. doi: 10.1007/s00022-008-2096-y.

[7]

D. R. Hughes and F. C. Piper, Projective Planes, Springer-Verlag, New York, 1973.

[8]

J. D. Key and M. J. de Resmini, Small sets of even type and codewords, J. Geom., 61 (1998), 83-104. doi: 10.1007/BF01237498.

[9]

J. D. KeyT. P. McDonough and V. C. Mavron, An upper bound for the minimum weight of the dual codes of desarguesian planes, Europ. J. Combin., 30 (2009), 220-229. doi: 10.1016/j.ejc.2008.01.003.

[10]

J. D. KeyT. P. McDonough and V. C. Mavron, Codes from Hall planes of even order, J. Geom., 105 (2014), 33-41. doi: 10.1007/s00022-013-0189-8.

[11]

H. Lüneburg, Translation Planes, Springer-Verlag, New York, 1980.

[12]

T. G. Ostrom, Finite Translation Planes, Springer-Verlag, 1970.

[1]

Mostafa Fazly, Nassif Ghoussoub. On the Hénon-Lane-Emden conjecture. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2513-2533. doi: 10.3934/dcds.2014.34.2513

[2]

Todd A. Drumm and William M. Goldman. Crooked planes. Electronic Research Announcements, 1995, 1: 10-17.

[3]

Cristina García Pillado, Santos González, Victor Markov, Consuelo Martínez, Alexandr Nechaev. New examples of non-abelian group codes. Advances in Mathematics of Communications, 2016, 10 (1) : 1-10. doi: 10.3934/amc.2016.10.1

[4]

Dubi Kelmer. Quantum ergodicity for products of hyperbolic planes. Journal of Modern Dynamics, 2008, 2 (2) : 287-313. doi: 10.3934/jmd.2008.2.287

[5]

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

[6]

Olof Heden, Faina I. Solov’eva. Partitions of $\mathbb F$n into non-parallel Hamming codes. Advances in Mathematics of Communications, 2009, 3 (4) : 385-397. doi: 10.3934/amc.2009.3.385

[7]

Olof Heden, Denis S. Krotov. On the structure of non-full-rank perfect $q$-ary codes. Advances in Mathematics of Communications, 2011, 5 (2) : 149-156. doi: 10.3934/amc.2011.5.149

[8]

Yael Ben-Haim, Simon Litsyn. A new upper bound on the rate of non-binary codes. Advances in Mathematics of Communications, 2007, 1 (1) : 83-92. doi: 10.3934/amc.2007.1.83

[9]

Chris Johnson, Martin Schmoll. Pseudo-Anosov eigenfoliations on Panov planes. Electronic Research Announcements, 2014, 21: 89-108. doi: 10.3934/era.2014.21.89

[10]

Michael Hutchings, Frank Morgan, Manuel Ritore and Antonio Ros. Proof of the double bubble conjecture. Electronic Research Announcements, 2000, 6: 45-49.

[11]

G. A. Swarup. On the cut point conjecture. Electronic Research Announcements, 1996, 2: 98-100.

[12]

Janos Kollar. The Nash conjecture for threefolds. Electronic Research Announcements, 1998, 4: 63-73.

[13]

Roman Shvydkoy. Lectures on the Onsager conjecture. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 473-496. doi: 10.3934/dcdss.2010.3.473

[14]

Joel Hass, Michael Hutchings and Roger Schlafly. The double bubble conjecture. Electronic Research Announcements, 1995, 1: 98-102.

[15]

Alex Eskin, Gregory Margulis and Shahar Mozes. On a quantitative version of the Oppenheim conjecture. Electronic Research Announcements, 1995, 1: 124-130.

[16]

Uri Shapira. On a generalization of Littlewood's conjecture. Journal of Modern Dynamics, 2009, 3 (3) : 457-477. doi: 10.3934/jmd.2009.3.457

[17]

Vitali Kapovitch, Anton Petrunin, Wilderich Tuschmann. On the torsion in the center conjecture. Electronic Research Announcements, 2018, 25: 27-35. doi: 10.3934/era.2018.25.004

[18]

Kamil Otal, Ferruh Özbudak. Explicit constructions of some non-Gabidulin linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 589-600. doi: 10.3934/amc.2016028

[19]

Andreas Klein, Leo Storme. On the non-minimality of the largest weight codewords in the binary Reed-Muller codes. Advances in Mathematics of Communications, 2011, 5 (2) : 333-337. doi: 10.3934/amc.2011.5.333

[20]

Daniele Bartoli, Leo Storme. On the functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric. Advances in Mathematics of Communications, 2014, 8 (3) : 271-280. doi: 10.3934/amc.2014.8.271

2017 Impact Factor: 0.564

Metrics

  • PDF downloads (2)
  • HTML views (6)
  • Cited by (0)

Other articles
by authors

[Back to Top]