American Institute of Mathematical Sciences

Advanced
November  2018, 12(4): 629-639. doi: 10.3934/amc.2018037

Characterization of extended Hamming and Golay codes as perfect codes in poset block spaces

B. K. Dass 1, , Namita Sharma 1,, and  Rashmi Verma 2,

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

* Corresponding author: Namita Sharma

Received  April 2015 Revised  March 2017 Published  September 2018

Fund Project: The first author is supported by a R & D grant of University of Delhi 2014-15. The second author is supported by Junior Research Fellowship Grant AA/139/F-177 of University Grants Commission

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.

Keywords: Poset block metric, extended binary Hamming code, extended Golay codes.
Mathematics Subject Classification: Primary: 94B05; Secondary: 94B60.
Citation: B. K. Dass, Namita Sharma, Rashmi Verma. Characterization of extended Hamming and Golay codes as perfect codes in poset block spaces. Advances in Mathematics of Communications, 2018, 12 (4) : 629-639. doi: 10.3934/amc.2018037
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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

O. Heden, Maximal partial spreads and two-weight codes, Discrete Math., 62 (1986), 277-293. doi: 10.1016/0012-365X(86)90215-3. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, 2003. doi: 10.1017/CBO9780511807077. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

B. Lindström, Group partitions and mixed perfect codes, Canad. Math. Bull., 18 (1975), 57-60. doi: 10.4153/CMB-1975-011-2. Google Scholar

[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. Google Scholar

[20]

H. Niederreiter, Point sets and sequences with small discrepancy, Monatsh. Math., 104 (1987), 273-337. doi: 10.1007/BF01294651. Google Scholar

[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. Google Scholar

[22]

P. Udomkavanich and S. Jitman, Bounds and modifications on linear error-block codes, Int. Math. Forum, 5 (2010), 35-50. Google Scholar

[23]

M. Yu. Rosenbloom and M. A. Tsfasman, Codes for the m-metric, Problemy Peredachi Informatsii, 33 (1997), 45-52. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

O. Heden, Maximal partial spreads and two-weight codes, Discrete Math., 62 (1986), 277-293. doi: 10.1016/0012-365X(86)90215-3. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, 2003. doi: 10.1017/CBO9780511807077. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

B. Lindström, Group partitions and mixed perfect codes, Canad. Math. Bull., 18 (1975), 57-60. doi: 10.4153/CMB-1975-011-2. Google Scholar

[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. Google Scholar

[20]

H. Niederreiter, Point sets and sequences with small discrepancy, Monatsh. Math., 104 (1987), 273-337. doi: 10.1007/BF01294651. Google Scholar

[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. Google Scholar

[22]

P. Udomkavanich and S. Jitman, Bounds and modifications on linear error-block codes, Int. Math. Forum, 5 (2010), 35-50. Google Scholar

[23]

M. Yu. Rosenbloom and M. A. Tsfasman, Codes for the m-metric, Problemy Peredachi Informatsii, 33 (1997), 45-52. Google Scholar

[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]

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
[6]

M.T. Boudjelkha. Extended Riemann Bessel functions. Conference Publications, 2005, 2005 (Special) : 121-130. doi: 10.3934/proc.2005.2005.121
[7]

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
[8]

Anna M. Barry, Esther WIdiasih, Richard Mcgehee. Nonsmooth frameworks for an extended Budyko model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2447-2463. doi: 10.3934/dcdsb.2017125
[9]

Zari Dzalilov, Iradj Ouveysi, Alexander Rubinov. An extended lifetime measure for telecommunication network. Journal of Industrial & Management Optimization, 2008, 4 (2) : 329-337. doi: 10.3934/jimo.2008.4.329
[10]

Michael Baake, John A. G. Roberts, Reem Yassawi. Reversing and extended symmetries of shift spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 835-866. doi: 10.3934/dcds.2018036
[11]

T. Zolezzi. Extended wellposedness of optimal control problems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 547-553. doi: 10.3934/dcds.1995.1.547
[12]

Siniša Slijepčević. Extended gradient systems: Dimension one. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 503-518. doi: 10.3934/dcds.2000.6.503
[13]

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
[14]

Zari Dzalilov, Iradj Ouveysi, Tolga Bektaş. An extended lifetime measure for telecommunications networks: Improvements and implementations. Journal of Industrial & Management Optimization, 2012, 8 (3) : 639-649. doi: 10.3934/jimo.2012.8.639
[15]

Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439
[16]

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
[17]

Domingo Gomez-Perez, Ana-Isabel Gomez, Andrew Tirkel. Arrays composed from the extended rational cycle. Advances in Mathematics of Communications, 2017, 11 (2) : 313-327. doi: 10.3934/amc.2017024
[18]

Fabio Della Rossa, Carlo D’Angelo, Alfio Quarteroni. A distributed model of traffic flows on extended regions. Networks & Heterogeneous Media, 2010, 5 (3) : 525-544. doi: 10.3934/nhm.2010.5.525
[19]

Weihua Liu, Andrew Klapper. AFSRs synthesis with the extended Euclidean rational approximation algorithm. Advances in Mathematics of Communications, 2017, 11 (1) : 139-150. doi: 10.3934/amc.2017008
[20]

Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (155)
  • HTML views (340)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences