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

B. K. Dass; Namita Sharma; Rashmi Verma

doi:10.3934/amc.2018037

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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.
[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.

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