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On perfect poset codes
1. | Center of Exact Sciences and Engineering, State University of West Paraná, Foz do Iguaçu, Paraná, Brazil |
2. | ILACVN - Federal University for Latin American Integration, Foz do Iguaçu, Paraná, Brazil |
3. | Polytechnic Center - Department of Mathematics, Federal University of Paraná, Curitiba, Paraná, Brazil |
4. | IMECC - Department of Mathematics, University of Campinas, Campinas, São Paulo, Brazil |
We consider on $ \mathbb{F}_{q}^{n} $ metrics determined by posets and classify the parameters of $ 1 $-perfect poset codes in such metrics. We show that a code with same parameters of a $ 1 $-perfect poset code is not necessarily perfect, however, we give necessary and sufficient conditions for this to be true. Furthermore, we characterize the unique way up to a labeling on the poset, considering some conditions, to extend an $ r $-perfect poset code over $ \mathbb{F}_q^n $ to an $ r $-perfect poset code over $ \mathbb{F}_q^{n+m} $.
References:
[1] |
J. Ahn, H. K. Kim, J. S. Kim and M. Kim,
Classification of perfect linear codes with crown poset structure, Discrete Mathematics, 268 (2003), 21-30.
doi: 10.1016/S0012-365X(02)00679-9. |
[2] |
M. M. S. Alves, L. Panek and M. Firer,
Error-block codes and poset metrics, Advances in Mathematics of Communications, 2 (2008), 95-111.
doi: 10.3934/amc.2008.2.95. |
[3] |
A. Barg, L. V. Felix, M. Firer and M. V. P. Spreafico,
Linear codes on posets with extension property, Discrete Mathematics, 317 (2014), 1-13.
doi: 10.1016/j.disc.2013.11.001. |
[4] |
A. Barg and W. Park,
On linear ordered codes, Moscow Mathematical J., 15 (2015), 679-702.
doi: 10.17323/1609-4514-2015-15-4-679-702. |
[5] |
R. Brualdi, J. S. Graves and M. Lawrence,
Codes with a poset metric, Discrete Mathematics, 147 (1995), 57-72.
doi: 10.1016/0012-365X(94)00228-B. |
[6] |
B. K. Dass, N. Sharma and R. Verma,
Perfect codes in poset spaces and poset block spaces, Finite Fields and Their Applications, 46 (2017), 90-106.
doi: 10.1016/j.ffa.2017.02.003. |
[7] |
L. V. Felix and M. Firer,
Canonical-systematic form for codes in hierarchical poset metrics, Advances in Mathematics of Communications, 6 (2012), 315-328.
doi: 10.3934/amc.2012.6.315. |
[8] |
M. Firer, M. M. S. Alves, J. A. Pinheiro and L. Panek, Poset Codes: Partial Orders, Metrics and Coding Theory, SpringerBriefs in Mathematics. Springer, Cham, 2018.
doi: 10.1007/978-3-319-93821-9. |
[9] |
M. Firer and J. A. Pinheiro, Bounds for complexity of syndrome decoding for poset metrics, 2015 IEEE Information Theory Workshop, (2015), 1–5.
doi: 10.1109/ITW.2015.7133130. |
[10] |
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511807077.![]() ![]() ![]() |
[11] |
J. Y. Hyun and H. K. Kim,
The poset structures admitting the extended binary Hamming code to be a perfect code, Discrete Mathematics, 288 (2004), 37-47.
doi: 10.1016/j.disc.2004.07.010. |
[12] |
C. Jang, H. K. Kim, D. Y. Oh and Y. Rho,
The poset structures admitting the extended binary Golay code to be a perfect code, Discrete Mathematics, 308 (2008), 4057-4068.
doi: 10.1016/j.disc.2007.07.111. |
[13] |
Y. Jang and J. Park,
On a MacWilliams type identity and a perfectness for a binary linear $(n, n-1, j)$-poset code, Discrete Mathematics, 265 (2003), 85-104.
doi: 10.1016/S0012-365X(02)00624-6. |
[14] |
H. K. Kim and D. Y. Oh,
On the nonexistence of triple-error-correcting perfect binary linear codes with a crown poset structure, Discrete Mathematics, 297 (2005), 174-181.
doi: 10.1016/j.disc.2005.03.018. |
[15] |
H. K. Kim and D. S. Krotov,
The poset metrics that allow binary codes of codimension $m$ to be $m$-, $m-1$-, or $m-2$-perfect, IEEE Transactions on Information Theory, 54 (2008), 5241-5246.
doi: 10.1109/TIT.2008.929972. |
[16] |
J. G. Lee,
Perfect codes on some ordered sets, Bulletin of the Korean Mathematical Society, 43 (2006), 293-297.
doi: 10.4134/BKMS.2006.43.2.293. |
[17] |
Y. Lee,
Projective systems and perfect codes with a poset metric, Finite Fields and Their Applications, 10 (2004), 105-112.
doi: 10.1016/S1071-5797(03)00046-7. |
[18] |
R. A. Machado, J. A. Pinheiro and M. Firer,
Characterization of metrics induced by hierarquical posets, IEEE Transactions on Information Theory, 63 (2017), 3630-3640.
doi: 10.1109/TIT.2017.2691763. |
[19] |
R. G. L. D'Oliveira and M. Firer,
The packing radius of a code and partitioning problems: The case for poset metrics on finite vector spaces, Discrete Mathematics, 338 (2015), 2143-2167.
doi: 10.1016/j.disc.2015.05.011. |
[20] |
L. Panek, M. Firer, H. K. Kim and J. Y. Hyun,
Groups of linear isometries on poset structures, Discrete Mathematics, 308 (2008), 4116-4123.
doi: 10.1016/j.disc.2007.08.001. |
show all references
References:
[1] |
J. Ahn, H. K. Kim, J. S. Kim and M. Kim,
Classification of perfect linear codes with crown poset structure, Discrete Mathematics, 268 (2003), 21-30.
doi: 10.1016/S0012-365X(02)00679-9. |
[2] |
M. M. S. Alves, L. Panek and M. Firer,
Error-block codes and poset metrics, Advances in Mathematics of Communications, 2 (2008), 95-111.
doi: 10.3934/amc.2008.2.95. |
[3] |
A. Barg, L. V. Felix, M. Firer and M. V. P. Spreafico,
Linear codes on posets with extension property, Discrete Mathematics, 317 (2014), 1-13.
doi: 10.1016/j.disc.2013.11.001. |
[4] |
A. Barg and W. Park,
On linear ordered codes, Moscow Mathematical J., 15 (2015), 679-702.
doi: 10.17323/1609-4514-2015-15-4-679-702. |
[5] |
R. Brualdi, J. S. Graves and M. Lawrence,
Codes with a poset metric, Discrete Mathematics, 147 (1995), 57-72.
doi: 10.1016/0012-365X(94)00228-B. |
[6] |
B. K. Dass, N. Sharma and R. Verma,
Perfect codes in poset spaces and poset block spaces, Finite Fields and Their Applications, 46 (2017), 90-106.
doi: 10.1016/j.ffa.2017.02.003. |
[7] |
L. V. Felix and M. Firer,
Canonical-systematic form for codes in hierarchical poset metrics, Advances in Mathematics of Communications, 6 (2012), 315-328.
doi: 10.3934/amc.2012.6.315. |
[8] |
M. Firer, M. M. S. Alves, J. A. Pinheiro and L. Panek, Poset Codes: Partial Orders, Metrics and Coding Theory, SpringerBriefs in Mathematics. Springer, Cham, 2018.
doi: 10.1007/978-3-319-93821-9. |
[9] |
M. Firer and J. A. Pinheiro, Bounds for complexity of syndrome decoding for poset metrics, 2015 IEEE Information Theory Workshop, (2015), 1–5.
doi: 10.1109/ITW.2015.7133130. |
[10] |
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511807077.![]() ![]() ![]() |
[11] |
J. Y. Hyun and H. K. Kim,
The poset structures admitting the extended binary Hamming code to be a perfect code, Discrete Mathematics, 288 (2004), 37-47.
doi: 10.1016/j.disc.2004.07.010. |
[12] |
C. Jang, H. K. Kim, D. Y. Oh and Y. Rho,
The poset structures admitting the extended binary Golay code to be a perfect code, Discrete Mathematics, 308 (2008), 4057-4068.
doi: 10.1016/j.disc.2007.07.111. |
[13] |
Y. Jang and J. Park,
On a MacWilliams type identity and a perfectness for a binary linear $(n, n-1, j)$-poset code, Discrete Mathematics, 265 (2003), 85-104.
doi: 10.1016/S0012-365X(02)00624-6. |
[14] |
H. K. Kim and D. Y. Oh,
On the nonexistence of triple-error-correcting perfect binary linear codes with a crown poset structure, Discrete Mathematics, 297 (2005), 174-181.
doi: 10.1016/j.disc.2005.03.018. |
[15] |
H. K. Kim and D. S. Krotov,
The poset metrics that allow binary codes of codimension $m$ to be $m$-, $m-1$-, or $m-2$-perfect, IEEE Transactions on Information Theory, 54 (2008), 5241-5246.
doi: 10.1109/TIT.2008.929972. |
[16] |
J. G. Lee,
Perfect codes on some ordered sets, Bulletin of the Korean Mathematical Society, 43 (2006), 293-297.
doi: 10.4134/BKMS.2006.43.2.293. |
[17] |
Y. Lee,
Projective systems and perfect codes with a poset metric, Finite Fields and Their Applications, 10 (2004), 105-112.
doi: 10.1016/S1071-5797(03)00046-7. |
[18] |
R. A. Machado, J. A. Pinheiro and M. Firer,
Characterization of metrics induced by hierarquical posets, IEEE Transactions on Information Theory, 63 (2017), 3630-3640.
doi: 10.1109/TIT.2017.2691763. |
[19] |
R. G. L. D'Oliveira and M. Firer,
The packing radius of a code and partitioning problems: The case for poset metrics on finite vector spaces, Discrete Mathematics, 338 (2015), 2143-2167.
doi: 10.1016/j.disc.2015.05.011. |
[20] |
L. Panek, M. Firer, H. K. Kim and J. Y. Hyun,
Groups of linear isometries on poset structures, Discrete Mathematics, 308 (2008), 4116-4123.
doi: 10.1016/j.disc.2007.08.001. |
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Luciano Viana Felix, Marcelo Firer. Canonical- systematic form for codes in hierarchical poset metrics. Advances in Mathematics of Communications, 2012, 6 (3) : 315-328. doi: 10.3934/amc.2012.6.315 |
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