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Canonical- systematic form for codes in hierarchical poset metrics
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 |
References:
[1] |
M. R. Best, Binary codes with a minimum distance of four, IEEE Trans. Inform. Theory, 26 (1980), 738-742.
doi: 10.1109/TIT.1980.1056269. |
[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), 167-179.
doi: 10.1007/s10623-009-9316-9. |
[3] |
J. Borges and J. Rifà, A characterization of 1-perfect additive codes, IEEE Trans. Inform. Theory, 45 (1999), 1688-1697.
doi: 10.1109/18.771247. |
[4] |
J. Borges, J. Rifà and F. I. Solov'eva, On properties of propelinear and transitive binary codes, in "3rd International Castle Meeting on Coding Theory and Applications (3ICMCTA),'' Servei de Publicacions UAB, 5 (2011), 65-70. |
[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), 31-42. |
[6] |
M. Hall, Jr., "The Theory of Groups,'' The Macmillan Company, New York, 1959. |
[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), 301-319.
doi: 10.1109/18.312154. |
[8] |
D. S. Krotov, $\mathbb Z_4$-linear perfect codes (in Russian), Discrete Analysis Oper. Res., 7 (2000), 78-90; English translation available at arXiv:0710.0198 |
[9] |
S. A. Malyugin, On equivalent classes of perfect binary codes of length 15 (in Russian), Inst. of Mathematics of SB RAS, Novosibirsk, 2004, 34 pp. |
[10] | |
[11] |
M. Mollard, A generalized parity function and its use in the construction of perfect codes, SIAM J. Alg. Disc. Meth., 7 (1986), 113-115.
doi: 10.1137/0607013. |
[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), 4657-4660.
doi: 10.1109/TIT.2009.2027525. |
[13] |
P. R. J. Östergå rd and O. Pottonen, The perfect binary one-error-correcting codes of length 15: Part I - Classification,, preprint, ().
|
[14] |
K. T. Phelps and J. Rifà, On binary 1-perfect additive codes: some structural properties, IEEE Trans. Inform. Theory, 48 (2002), 2587-2592.
doi: 10.1109/TIT.2002.801474. |
[15] |
V. N. Potapov, A lower bound for the number of transitive perfect codes, J. Appl. Industrial Math., 1 (2007), 373-379.
doi: 10.1134/S199047890703012X. |
[16] |
J. Rifà, J. M. Basart and L. Huguet, On completely regular propelinear codes, in "Proc. 6th Int. Conference, AAECC-6,'' (1989), 341-355. |
[17] |
J. Rifà and J. Pujol, Translation invariant propelinear codes, IEEE Trans. Inform. Theory, 43 (1997), 590-598.
doi: 10.1109/18.556115. |
[18] |
J. Rifà, J. Pujol and J. Borges, 1-perfect uniform and distance invariant partitions, Appl. Algebra Engin. Commun. Comp., 11 (2001), 297-311.
doi: 10.1007/PL00004224. |
[19] |
F. I. Solov'eva, On the construction of transitive codes, Probl. Inform. Trans., 41 (2005), 204-211.
doi: 10.1007/s11122-005-0025-3. |
[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), 331-335. |
[21] |
Y. L. Vasil'ev, On nongroup close-packed codes, Probl. Kybernetik, 8 (1962), 92-95. |
show all references
References:
[1] |
M. R. Best, Binary codes with a minimum distance of four, IEEE Trans. Inform. Theory, 26 (1980), 738-742.
doi: 10.1109/TIT.1980.1056269. |
[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), 167-179.
doi: 10.1007/s10623-009-9316-9. |
[3] |
J. Borges and J. Rifà, A characterization of 1-perfect additive codes, IEEE Trans. Inform. Theory, 45 (1999), 1688-1697.
doi: 10.1109/18.771247. |
[4] |
J. Borges, J. Rifà and F. I. Solov'eva, On properties of propelinear and transitive binary codes, in "3rd International Castle Meeting on Coding Theory and Applications (3ICMCTA),'' Servei de Publicacions UAB, 5 (2011), 65-70. |
[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), 31-42. |
[6] |
M. Hall, Jr., "The Theory of Groups,'' The Macmillan Company, New York, 1959. |
[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), 301-319.
doi: 10.1109/18.312154. |
[8] |
D. S. Krotov, $\mathbb Z_4$-linear perfect codes (in Russian), Discrete Analysis Oper. Res., 7 (2000), 78-90; English translation available at arXiv:0710.0198 |
[9] |
S. A. Malyugin, On equivalent classes of perfect binary codes of length 15 (in Russian), Inst. of Mathematics of SB RAS, Novosibirsk, 2004, 34 pp. |
[10] | |
[11] |
M. Mollard, A generalized parity function and its use in the construction of perfect codes, SIAM J. Alg. Disc. Meth., 7 (1986), 113-115.
doi: 10.1137/0607013. |
[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), 4657-4660.
doi: 10.1109/TIT.2009.2027525. |
[13] |
P. R. J. Östergå rd and O. Pottonen, The perfect binary one-error-correcting codes of length 15: Part I - Classification,, preprint, ().
|
[14] |
K. T. Phelps and J. Rifà, On binary 1-perfect additive codes: some structural properties, IEEE Trans. Inform. Theory, 48 (2002), 2587-2592.
doi: 10.1109/TIT.2002.801474. |
[15] |
V. N. Potapov, A lower bound for the number of transitive perfect codes, J. Appl. Industrial Math., 1 (2007), 373-379.
doi: 10.1134/S199047890703012X. |
[16] |
J. Rifà, J. M. Basart and L. Huguet, On completely regular propelinear codes, in "Proc. 6th Int. Conference, AAECC-6,'' (1989), 341-355. |
[17] |
J. Rifà and J. Pujol, Translation invariant propelinear codes, IEEE Trans. Inform. Theory, 43 (1997), 590-598.
doi: 10.1109/18.556115. |
[18] |
J. Rifà, J. Pujol and J. Borges, 1-perfect uniform and distance invariant partitions, Appl. Algebra Engin. Commun. Comp., 11 (2001), 297-311.
doi: 10.1007/PL00004224. |
[19] |
F. I. Solov'eva, On the construction of transitive codes, Probl. Inform. Trans., 41 (2005), 204-211.
doi: 10.1007/s11122-005-0025-3. |
[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), 331-335. |
[21] |
Y. L. Vasil'ev, On nongroup close-packed codes, Probl. Kybernetik, 8 (1962), 92-95. |
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