August  2012, 6(3): 329-346. doi: 10.3934/amc.2012.6.329

Structural properties of binary propelinear codes

1. 

Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193-Bellaterra

2. 

Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, Russian Federation, Russian Federation

3. 

Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193-Bellaterra, Cerdanyola del Vallès, Spain

Received  September 2011 Revised  March 2012 Published  August 2012

The paper deals with some structural properties of propelinear binary codes, in particular propelinear perfect binary codes. We consider the connection of transitive codes with propelinear codes and show that there exists a binary code, the Best code of length 10, size 40 and minimum distance 4, which is transitive but not propelinear. We propose several constructions of propelinear codes and introduce a new large class of propelinear perfect binary codes, called normalized propelinear perfect codes. Finally, based on the different values for the rank and the dimension of the kernel, we give a lower bound on the number of nonequivalent propelinear perfect binary codes.
Citation: Joaquim Borges, Ivan Yu. Mogilnykh, Josep Rifà, Faina I. Solov'eva. Structural properties of binary propelinear codes. Advances in Mathematics of Communications, 2012, 6 (3) : 329-346. doi: 10.3934/amc.2012.6.329
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show all references

References:
[1]

IEEE Trans. Inform. Theory, 26 (1980), 738-742. doi: 10.1109/TIT.1980.1056269.  Google Scholar

[2]

Des. Codes Cryptogr., 54 (2010), 167-179. doi: 10.1007/s10623-009-9316-9.  Google Scholar

[3]

IEEE Trans. Inform. Theory, 45 (1999), 1688-1697. doi: 10.1109/18.771247.  Google Scholar

[4]

in "3rd International Castle Meeting on Coding Theory and Applications (3ICMCTA),'' Servei de Publicacions UAB, 5 (2011), 65-70. Google Scholar

[5]

Des. Codes Cryptogr., 4 (1994), 31-42.  Google Scholar

[6]

The Macmillan Company, New York, 1959.  Google Scholar

[7]

IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154.  Google Scholar

[8]

Discrete Analysis Oper. Res., 7 (2000), 78-90; English translation available at arXiv:0710.0198  Google Scholar

[9]

Inst. of Mathematics of SB RAS, Novosibirsk, 2004, 34 pp. Google Scholar

[10]

S. A. Malyugin, Private communication,, 2004., ().   Google Scholar

[11]

SIAM J. Alg. Disc. Meth., 7 (1986), 113-115. doi: 10.1137/0607013.  Google Scholar

[12]

IEEE Trans. Inform. Theory, 55 (2009), 4657-4660. doi: 10.1109/TIT.2009.2027525.  Google Scholar

[13]

P. R. J. Östergå rd and O. Pottonen, The perfect binary one-error-correcting codes of length 15: Part I - Classification,, preprint, ().   Google Scholar

[14]

IEEE Trans. Inform. Theory, 48 (2002), 2587-2592. doi: 10.1109/TIT.2002.801474.  Google Scholar

[15]

J. Appl. Industrial Math., 1 (2007), 373-379. doi: 10.1134/S199047890703012X.  Google Scholar

[16]

in "Proc. 6th Int. Conference, AAECC-6,'' (1989), 341-355.  Google Scholar

[17]

IEEE Trans. Inform. Theory, 43 (1997), 590-598. doi: 10.1109/18.556115.  Google Scholar

[18]

Appl. Algebra Engin. Commun. Comp., 11 (2001), 297-311. doi: 10.1007/PL00004224.  Google Scholar

[19]

Probl. Inform. Trans., 41 (2005), 204-211. doi: 10.1007/s11122-005-0025-3.  Google Scholar

[20]

Probl. Inform. Trans., 36 (2000), 331-335.  Google Scholar

[21]

Probl. Kybernetik, 8 (1962), 92-95. Google Scholar

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