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

Codes from hall planes of odd order

J. D. Key , T. P. McDonough and  V. C. Mavron

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.

Keywords: Non-desarguesian planes, Hamada-Sachar conjecture, codes.
Mathematics Subject Classification: Primary: 94B05, 51A35; Secondary: 05B05.
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.  Google Scholar
[2]

W. Bosma, J. 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.  Google Scholar

[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. Google Scholar

[4]

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

[5]

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

[6]

D. Ghinelli, M. 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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

J. D. Key, T. 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.  Google Scholar

[10]

J. D. Key, T. 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.  Google Scholar

[11]

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

[12]

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

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.  Google Scholar
[2]

W. Bosma, J. 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.  Google Scholar

[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. Google Scholar

[4]

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

[5]

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

[6]

D. Ghinelli, M. 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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

J. D. Key, T. 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.  Google Scholar

[10]

J. D. Key, T. 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.  Google Scholar

[11]

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

[12]

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

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