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
References:
[1]

M. R. Best, Binary codes with a minimum distance of four,, IEEE Trans. Inform. Theory, 26 (1980), 738. doi: 10.1109/TIT.1980.1056269. Google Scholar

[2]

J. Borges, C. Fernández, J. Pujol, J. Rifà and M. Villanueva, $\mathbb Z_2\mathbb Z_4$-linear codes: generator matrices and duality,, Des. Codes Cryptogr., 54 (2010), 167. doi: 10.1007/s10623-009-9316-9. Google Scholar

[3]

J. Borges and J. Rifà, A characterization of 1-perfect additive codes,, IEEE Trans. Inform. Theory, 45 (1999), 1688. doi: 10.1109/18.771247. Google Scholar

[4]

J. Borges, J. Rifà and F. I. Solov'eva, On properties of propelinear and transitive binary codes,, in, 5 (2011), 65. Google Scholar

[5]

J. H. Conway and N. J. A. Sloane, Quaternary constructions for the binary single-error-correcting codes of Julin, Best and others,, Des. Codes Cryptogr., 4 (1994), 31. Google Scholar

[6]

M. Hall, Jr., "The Theory of Groups,'', The Macmillan Company, (1959). Google Scholar

[7]

A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbb Z_4$-linearity of Kerdock, Preparata, Goethals and related codes,, IEEE Trans. Inform. Theory, 40 (1994), 301. doi: 10.1109/18.312154. Google Scholar

[8]

D. S. Krotov, $\mathbb Z_4$-linear perfect codes (in Russian),, Discrete Analysis Oper. Res., 7 (2000), 78. Google Scholar

[9]

S. A. Malyugin, On equivalent classes of perfect binary codes of length 15 (in Russian),, Inst. of Mathematics of SB RAS, (2004). Google Scholar

[10]

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

[11]

M. Mollard, A generalized parity function and its use in the construction of perfect codes,, SIAM J. Alg. Disc. Meth., 7 (1986), 113. doi: 10.1137/0607013. Google Scholar

[12]

P. R. J. Östergå rd and O. Pottonen, The perfect binary one-error-correcting codes of length 15: Part I - Classification,, IEEE Trans. Inform. Theory, 55 (2009), 4657. 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]

K. T. Phelps and J. Rifà, On binary 1-perfect additive codes: some structural properties,, IEEE Trans. Inform. Theory, 48 (2002), 2587. doi: 10.1109/TIT.2002.801474. Google Scholar

[15]

V. N. Potapov, A lower bound for the number of transitive perfect codes,, J. Appl. Industrial Math., 1 (2007), 373. doi: 10.1134/S199047890703012X. Google Scholar

[16]

J. Rifà, J. M. Basart and L. Huguet, On completely regular propelinear codes,, in, (1989), 341. Google Scholar

[17]

J. Rifà and J. Pujol, Translation invariant propelinear codes,, IEEE Trans. Inform. Theory, 43 (1997), 590. doi: 10.1109/18.556115. Google Scholar

[18]

J. Rifà, J. Pujol and J. Borges, 1-perfect uniform and distance invariant partitions,, Appl. Algebra Engin. Commun. Comp., 11 (2001), 297. doi: 10.1007/PL00004224. Google Scholar

[19]

F. I. Solov'eva, On the construction of transitive codes,, Probl. Inform. Trans., 41 (2005), 204. doi: 10.1007/s11122-005-0025-3. Google Scholar

[20]

F. I. Solov'eva and S. T. Topalova, On automorphism groups of perfect binary codes and Steiner triple systems,, Probl. Inform. Trans., 36 (2000), 331. Google Scholar

[21]

Y. L. Vasil'ev, On nongroup close-packed codes,, Probl. Kybernetik, 8 (1962), 92. Google Scholar

show all references

References:
[1]

M. R. Best, Binary codes with a minimum distance of four,, IEEE Trans. Inform. Theory, 26 (1980), 738. doi: 10.1109/TIT.1980.1056269. Google Scholar

[2]

J. Borges, C. Fernández, J. Pujol, J. Rifà and M. Villanueva, $\mathbb Z_2\mathbb Z_4$-linear codes: generator matrices and duality,, Des. Codes Cryptogr., 54 (2010), 167. doi: 10.1007/s10623-009-9316-9. Google Scholar

[3]

J. Borges and J. Rifà, A characterization of 1-perfect additive codes,, IEEE Trans. Inform. Theory, 45 (1999), 1688. doi: 10.1109/18.771247. Google Scholar

[4]

J. Borges, J. Rifà and F. I. Solov'eva, On properties of propelinear and transitive binary codes,, in, 5 (2011), 65. Google Scholar

[5]

J. H. Conway and N. J. A. Sloane, Quaternary constructions for the binary single-error-correcting codes of Julin, Best and others,, Des. Codes Cryptogr., 4 (1994), 31. Google Scholar

[6]

M. Hall, Jr., "The Theory of Groups,'', The Macmillan Company, (1959). Google Scholar

[7]

A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbb Z_4$-linearity of Kerdock, Preparata, Goethals and related codes,, IEEE Trans. Inform. Theory, 40 (1994), 301. doi: 10.1109/18.312154. Google Scholar

[8]

D. S. Krotov, $\mathbb Z_4$-linear perfect codes (in Russian),, Discrete Analysis Oper. Res., 7 (2000), 78. Google Scholar

[9]

S. A. Malyugin, On equivalent classes of perfect binary codes of length 15 (in Russian),, Inst. of Mathematics of SB RAS, (2004). Google Scholar

[10]

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

[11]

M. Mollard, A generalized parity function and its use in the construction of perfect codes,, SIAM J. Alg. Disc. Meth., 7 (1986), 113. doi: 10.1137/0607013. Google Scholar

[12]

P. R. J. Östergå rd and O. Pottonen, The perfect binary one-error-correcting codes of length 15: Part I - Classification,, IEEE Trans. Inform. Theory, 55 (2009), 4657. 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]

K. T. Phelps and J. Rifà, On binary 1-perfect additive codes: some structural properties,, IEEE Trans. Inform. Theory, 48 (2002), 2587. doi: 10.1109/TIT.2002.801474. Google Scholar

[15]

V. N. Potapov, A lower bound for the number of transitive perfect codes,, J. Appl. Industrial Math., 1 (2007), 373. doi: 10.1134/S199047890703012X. Google Scholar

[16]

J. Rifà, J. M. Basart and L. Huguet, On completely regular propelinear codes,, in, (1989), 341. Google Scholar

[17]

J. Rifà and J. Pujol, Translation invariant propelinear codes,, IEEE Trans. Inform. Theory, 43 (1997), 590. doi: 10.1109/18.556115. Google Scholar

[18]

J. Rifà, J. Pujol and J. Borges, 1-perfect uniform and distance invariant partitions,, Appl. Algebra Engin. Commun. Comp., 11 (2001), 297. doi: 10.1007/PL00004224. Google Scholar

[19]

F. I. Solov'eva, On the construction of transitive codes,, Probl. Inform. Trans., 41 (2005), 204. doi: 10.1007/s11122-005-0025-3. Google Scholar

[20]

F. I. Solov'eva and S. T. Topalova, On automorphism groups of perfect binary codes and Steiner triple systems,, Probl. Inform. Trans., 36 (2000), 331. Google Scholar

[21]

Y. L. Vasil'ev, On nongroup close-packed codes,, Probl. Kybernetik, 8 (1962), 92. Google Scholar

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