-
Previous Article
$ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes
- AMC Home
- This Issue
- Next Article
Characterization of extended Hamming and Golay codes as perfect codes in poset block spaces
1. | Department of Mathematics, University of Delhi, Delhi-110 007, India |
2. | Mata Sundri College for Women (University of Delhi), Mata Sundri Lane, Delhi-110 002, India |
Alves, Panek and Firer (Error-block codes and poset metrics, Adv. Math. Commun., 2 (2008), 95-111) classified all poset block structures which turn the [8,4,4] extended binary Hamming code into a 1-perfect poset block code. However, the proof needs corrections that are supplied in this paper. We provide a counterexample to show that the extended binary Golay code is not 1-perfect for the proposed poset block structures. All poset block structures turning the extended binary and ternary Golay codes into 1-perfect codes are classified.
References:
[1] |
J. Ahn, H. K. Kim, J. S. Kim and M. Kim,
Classification of perfect linear codes with crown poset structure, Discrete Math., 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, Adv. Math. Commun., 2 (2008), 95-111.
doi: 10.3934/amc.2008.2.95. |
[3] |
A. Blokhuis, A. E. Brouwer and H. A. Wilbrink,
Heden's bound on maximal partial spreads, Discrete Math., 74 (1989), 335-339.
doi: 10.1016/0012-365X(89)90148-9. |
[4] |
R. Brualdi, J. S. Graves and M. Lawrence,
Codes with a poset metric, Discrete Math., 147 (1995), 57-72.
doi: 10.1016/0012-365X(94)00228-B. |
[5] |
R. G. L. D'Oliveira and M. Firer,
The packing radius of a code and partitioning problems: The case for poset metrics over finite vector spaces, Discrete Math., 338 (2015), 2143-2167.
doi: 10.1016/j.disc.2015.05.011. |
[6] |
L. V. Felix and M. Firer,
Canonical-systematic form for codes in hierarchical poset metrics, Adv. Math. Commun., 6 (2012), 315-328.
doi: 10.3934/amc.2012.6.315. |
[7] |
K. Feng, L. Xu and F. J. Hickernell,
Linear error-block codes, Finite Fields Appl., 12 (2006), 638-652.
doi: 10.1016/j.ffa.2005.03.006. |
[8] |
M. Firer, L. Panek and L. L. R. Rifo, Coding and decoding schemes tailor made for image transmission, in 2013 IEEE Information Theory and Applications Workshop (ITA), (2013), 1–8.
doi: 10.1109/ITA.2013.6502935. |
[9] |
M. Firer and J. A. Pinheiro, Bounds for the complexity of syndrome decoding for poset metrics, in 2015 IEEE Information Theory Workshop (ITW), (2015), 1–5.
doi: 10.1109/ITW.2015.7133130. |
[10] |
O. Heden,
Maximal partial spreads and two-weight codes, Discrete Math., 62 (1986), 277-293.
doi: 10.1016/0012-365X(86)90215-3. |
[11] |
O. Heden, A survey of the different types of vector space partitions,
Discrete Math. Algorithm. Appl., 4 (2012), 1250001, 14 pp.
doi: 10.1142/S1793830912500012. |
[12] |
M. Herzog and J. Schönheim,
Linear and nonlinear single-error correcting perfect mixed codes, Information and Control, 18 (1971), 364-368.
doi: 10.1016/S0019-9958(71)90464-5. |
[13] |
M. Herzog and J. Schönheim,
Group partition, factorization and the vector covering problem, Canad. Math. Bull., 15 (1972), 207-214.
doi: 10.4153/CMB-1972-038-x. |
[14] |
W. C. Huffman and V. Pless,
Fundamentals of Error-Correcting Codes, Cambridge University Press, 2003.
doi: 10.1017/CBO9780511807077. |
[15] |
J. Y. Hyun and H. K. Kim,
The poset structures admitting the extended binary Hamming code to be a perfect code, Discrete Math., 288 (2004), 37-47.
doi: 10.1016/j.disc.2004.07.010. |
[16] |
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 Math., 308 (2008), 4057-4068.
doi: 10.1016/j.disc.2007.07.111. |
[17] |
Y. Lee,
Projective systems and perfect codes with a poset metric, Finite Fields Appl., 10 (2004), 105-112.
doi: 10.1016/S1071-5797(03)00046-7. |
[18] |
B. Lindström,
Group partitions and mixed perfect codes, Canad. Math. Bull., 18 (1975), 57-60.
doi: 10.4153/CMB-1975-011-2. |
[19] |
S. Ling and F. Özbudak,
Constructions and bounds on linear error-block codes, Des. Codes Cryptogr., 45 (2007), 297-316.
doi: 10.1007/s10623-007-9119-9. |
[20] |
H. Niederreiter,
Point sets and sequences with small discrepancy, Monatsh. Math., 104 (1987), 273-337.
doi: 10.1007/BF01294651. |
[21] |
H. Niederreiter,
A combinatorial problem for vector spaces over finite fields, Discrete Math., 96 (1991), 221-228.
doi: 10.1016/0012-365X(91)90315-S. |
[22] |
P. Udomkavanich and S. Jitman,
Bounds and modifications on linear error-block codes, Int. Math. Forum, 5 (2010), 35-50.
|
[23] |
M. Yu. Rosenbloom and M. A. Tsfasman,
Codes for the m-metric, Problemy Peredachi Informatsii, 33 (1997), 45-52.
|
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 Math., 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, Adv. Math. Commun., 2 (2008), 95-111.
doi: 10.3934/amc.2008.2.95. |
[3] |
A. Blokhuis, A. E. Brouwer and H. A. Wilbrink,
Heden's bound on maximal partial spreads, Discrete Math., 74 (1989), 335-339.
doi: 10.1016/0012-365X(89)90148-9. |
[4] |
R. Brualdi, J. S. Graves and M. Lawrence,
Codes with a poset metric, Discrete Math., 147 (1995), 57-72.
doi: 10.1016/0012-365X(94)00228-B. |
[5] |
R. G. L. D'Oliveira and M. Firer,
The packing radius of a code and partitioning problems: The case for poset metrics over finite vector spaces, Discrete Math., 338 (2015), 2143-2167.
doi: 10.1016/j.disc.2015.05.011. |
[6] |
L. V. Felix and M. Firer,
Canonical-systematic form for codes in hierarchical poset metrics, Adv. Math. Commun., 6 (2012), 315-328.
doi: 10.3934/amc.2012.6.315. |
[7] |
K. Feng, L. Xu and F. J. Hickernell,
Linear error-block codes, Finite Fields Appl., 12 (2006), 638-652.
doi: 10.1016/j.ffa.2005.03.006. |
[8] |
M. Firer, L. Panek and L. L. R. Rifo, Coding and decoding schemes tailor made for image transmission, in 2013 IEEE Information Theory and Applications Workshop (ITA), (2013), 1–8.
doi: 10.1109/ITA.2013.6502935. |
[9] |
M. Firer and J. A. Pinheiro, Bounds for the complexity of syndrome decoding for poset metrics, in 2015 IEEE Information Theory Workshop (ITW), (2015), 1–5.
doi: 10.1109/ITW.2015.7133130. |
[10] |
O. Heden,
Maximal partial spreads and two-weight codes, Discrete Math., 62 (1986), 277-293.
doi: 10.1016/0012-365X(86)90215-3. |
[11] |
O. Heden, A survey of the different types of vector space partitions,
Discrete Math. Algorithm. Appl., 4 (2012), 1250001, 14 pp.
doi: 10.1142/S1793830912500012. |
[12] |
M. Herzog and J. Schönheim,
Linear and nonlinear single-error correcting perfect mixed codes, Information and Control, 18 (1971), 364-368.
doi: 10.1016/S0019-9958(71)90464-5. |
[13] |
M. Herzog and J. Schönheim,
Group partition, factorization and the vector covering problem, Canad. Math. Bull., 15 (1972), 207-214.
doi: 10.4153/CMB-1972-038-x. |
[14] |
W. C. Huffman and V. Pless,
Fundamentals of Error-Correcting Codes, Cambridge University Press, 2003.
doi: 10.1017/CBO9780511807077. |
[15] |
J. Y. Hyun and H. K. Kim,
The poset structures admitting the extended binary Hamming code to be a perfect code, Discrete Math., 288 (2004), 37-47.
doi: 10.1016/j.disc.2004.07.010. |
[16] |
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 Math., 308 (2008), 4057-4068.
doi: 10.1016/j.disc.2007.07.111. |
[17] |
Y. Lee,
Projective systems and perfect codes with a poset metric, Finite Fields Appl., 10 (2004), 105-112.
doi: 10.1016/S1071-5797(03)00046-7. |
[18] |
B. Lindström,
Group partitions and mixed perfect codes, Canad. Math. Bull., 18 (1975), 57-60.
doi: 10.4153/CMB-1975-011-2. |
[19] |
S. Ling and F. Özbudak,
Constructions and bounds on linear error-block codes, Des. Codes Cryptogr., 45 (2007), 297-316.
doi: 10.1007/s10623-007-9119-9. |
[20] |
H. Niederreiter,
Point sets and sequences with small discrepancy, Monatsh. Math., 104 (1987), 273-337.
doi: 10.1007/BF01294651. |
[21] |
H. Niederreiter,
A combinatorial problem for vector spaces over finite fields, Discrete Math., 96 (1991), 221-228.
doi: 10.1016/0012-365X(91)90315-S. |
[22] |
P. Udomkavanich and S. Jitman,
Bounds and modifications on linear error-block codes, Int. Math. Forum, 5 (2010), 35-50.
|
[23] |
M. Yu. Rosenbloom and M. A. Tsfasman,
Codes for the m-metric, Problemy Peredachi Informatsii, 33 (1997), 45-52.
|
[1] |
Olof Heden, Fabio Pasticci, Thomas Westerbäck. On the existence of extended perfect binary codes with trivial symmetry group. Advances in Mathematics of Communications, 2009, 3 (3) : 295-309. doi: 10.3934/amc.2009.3.295 |
[2] |
Marcelo Muniz S. Alves, Luciano Panek, Marcelo Firer. Error-block codes and poset metrics. Advances in Mathematics of Communications, 2008, 2 (1) : 95-111. doi: 10.3934/amc.2008.2.95 |
[3] |
Olof Heden, Fabio Pasticci, Thomas Westerbäck. On the symmetry group of extended perfect binary codes of length $n+1$ and rank $n-\log(n+1)+2$. Advances in Mathematics of Communications, 2012, 6 (2) : 121-130. doi: 10.3934/amc.2012.6.121 |
[4] |
Yujuan Li, Guizhen Zhu. On the error distance of extended Reed-Solomon codes. Advances in Mathematics of Communications, 2016, 10 (2) : 413-427. doi: 10.3934/amc.2016015 |
[5] |
Xiang Wang, Wenjuan Yin. New nonexistence results on perfect permutation codes under the hamming metric. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021058 |
[6] |
Luciano Panek, Jerry Anderson Pinheiro, Marcelo Muniz Alves, Marcelo Firer. On perfect poset codes. Advances in Mathematics of Communications, 2020, 14 (3) : 477-489. doi: 10.3934/amc.2020061 |
[7] |
Heeralal Janwa, Fernando L. Piñero. On parameters of subfield subcodes of extended norm-trace codes. Advances in Mathematics of Communications, 2017, 11 (2) : 379-388. doi: 10.3934/amc.2017032 |
[8] |
Yanyan Gao, Qin Yue, Xinmei Huang, Yun Yang. Two classes of cyclic extended double-error-correcting Goppa codes. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022003 |
[9] |
Anna-Lena Horlemann-Trautmann, Kyle Marshall. New criteria for MRD and Gabidulin codes and some Rank-Metric code constructions. Advances in Mathematics of Communications, 2017, 11 (3) : 533-548. doi: 10.3934/amc.2017042 |
[10] |
M.T. Boudjelkha. Extended Riemann Bessel functions. Conference Publications, 2005, 2005 (Special) : 121-130. doi: 10.3934/proc.2005.2005.121 |
[11] |
Anna M. Barry, Esther WIdiasih, Richard Mcgehee. Nonsmooth frameworks for an extended Budyko model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2447-2463. doi: 10.3934/dcdsb.2017125 |
[12] |
Zari Dzalilov, Iradj Ouveysi, Alexander Rubinov. An extended lifetime measure for telecommunication network. Journal of Industrial and Management Optimization, 2008, 4 (2) : 329-337. doi: 10.3934/jimo.2008.4.329 |
[13] |
Michael Baake, John A. G. Roberts, Reem Yassawi. Reversing and extended symmetries of shift spaces. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 835-866. doi: 10.3934/dcds.2018036 |
[14] |
Siniša Slijepčević. Extended gradient systems: Dimension one. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 503-518. doi: 10.3934/dcds.2000.6.503 |
[15] |
T. Zolezzi. Extended wellposedness of optimal control problems. Discrete and Continuous Dynamical Systems, 1995, 1 (4) : 547-553. doi: 10.3934/dcds.1995.1.547 |
[16] |
Joaquim Borges, Josep Rifà, Victor Zinoviev. Completely regular codes by concatenating Hamming codes. Advances in Mathematics of Communications, 2018, 12 (2) : 337-349. doi: 10.3934/amc.2018021 |
[17] |
Jianying Fang. 5-SEEDs from the lifted Golay code of length 24 over Z4. Advances in Mathematics of Communications, 2017, 11 (1) : 259-266. doi: 10.3934/amc.2017017 |
[18] |
Zari Dzalilov, Iradj Ouveysi, Tolga Bektaş. An extended lifetime measure for telecommunications networks: Improvements and implementations. Journal of Industrial and Management Optimization, 2012, 8 (3) : 639-649. doi: 10.3934/jimo.2012.8.639 |
[19] |
Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439 |
[20] |
Francisco Crespo, Sebastián Ferrer. On the extended Euler system and the Jacobi and Weierstrass elliptic functions. Journal of Geometric Mechanics, 2015, 7 (2) : 151-168. doi: 10.3934/jgm.2015.7.151 |
2021 Impact Factor: 1.015
Tools
Metrics
Other articles
by authors
[Back to Top]