American Institute of Mathematical Sciences

Advanced
August  2011, 5(3): 417-424. doi: 10.3934/amc.2011.5.417

Some new distance-4 constant weight codes

Andries E. Brouwer 1, and  Tuvi Etzion 2,

1. 

Dept. of Math., Techn. Univ. Eindhoven, P. O. Box 513, 5600MB Eindhoven, Netherlands
2. 

Department of Computer Science, Technion, Haifa 32000

Received  March 2010 Revised  February 2011 Published  August 2011

Improved binary constant weight codes with minimum distance 4 are constructed. A table with bounds on the chromatic number of small Johnson graphs is given.
Keywords: Binary constant weight code..
Mathematics Subject Classification: Primary: 94B60; Secondary: 05E30, 05C1.
Citation: Andries E. Brouwer, Tuvi Etzion. Some new distance-4 constant weight codes. Advances in Mathematics of Communications, 2011, 5 (3) : 417-424. doi: 10.3934/amc.2011.5.417
References:
[1]

, http://www.win.tue.nl/~aeb/codes/andw.html,  , ().   Google Scholar

[2]

A. Betten, R. Laue and A. Wassermann, A Steiner 5-design on 36 points,, Des. Codes Crypt., 17 (1999), 181.  doi: 10.1023/A:1026427226213.  Google Scholar

[3]

A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, A new table of constant weight codes,, IEEE Trans. Inform. Theory, 36 (1990), 1334.  doi: 10.1109/18.59932.  Google Scholar

[4]

R. H. F. Denniston, Sylvester's problem of the 15 school-girls,, Discr. Math., 9 (1974), 229.  doi: 10.1016/0012-365X(74)90004-1.  Google Scholar

[5]

R. H. F. Denniston, Some new 5-designs,, Bull. London Math. Soc., 8 (1976), 263.  doi: 10.1112/blms/8.3.263.  Google Scholar

[6]

T. Etzion, Optimal partitions for triples,, J. Combin. Theory (A), 59 (1992), 161.  doi: 10.1016/0097-3165(92)90062-Y.  Google Scholar

[7]

T. Etzion, Partitions of triples into optimal packings,, J. Combin. Theory (A), 59 (1992), 269.  doi: 10.1016/0097-3165(92)90069-7.  Google Scholar

[8]

T. Etzion, Partitions for quadruples,, Ars Combin., 36 (1993), 296.   Google Scholar

[9]

T. Etzion and S. Bitan, On the chromatic number, colorings, and codes of the Johnson graph,, Discr. Appl. Math., 70 (1996), 163.  doi: 10.1016/0166-218X(96)00104-7.  Google Scholar

[10]

T. Etzion and P. R. J. Östergård, Greedy and heuristic algorithms for codes and colorings,, IEEE Trans. Inform. Theory, 44 (1998), 382.  doi: 10.1109/18.651069.  Google Scholar

[11]

T. Etzion and C. L. M. van Pul, New lower bounds for constant weight codes,, IEEE Trans. Inform. Theory, 35 (1989), 1324.  doi: 10.1109/18.45293.  Google Scholar

[12]

R. L. Graham and N. J. A. Sloane, Lower bounds for constant weight codes,, IEEE Trans. Inform. Theory, 26 (1980), 37.  doi: 10.1109/TIT.1980.1056141.  Google Scholar

[13]

L. J. Ji, A new existence proof for large sets of disjoint Steiner triple systems,, J. Combin. Theory (A), 112 (2005), 308.  doi: 10.1016/j.jcta.2005.06.005.  Google Scholar

[14]

L. J. Ji, Partition of triples of order $6k+5$ into $6k+3$ optimal packings and one packing of size $8k+4$,, Graphs Combin., 22 (2006), 251.  doi: 10.1007/s00373-005-0632-1.  Google Scholar

[15]

J. X. Lu, On large sets of disjoint Steiner triple systems I, II, III,, J. Combin. Theory (A), 34 (1983), 140.   Google Scholar

[16]

N. S. Mendelsohn and S. H. Y. Hung, On the Steiner systems $S(3,4,14)$ and $S(4,5,15)$,, Utilitas Math., 1 (1972), 5.   Google Scholar

[17]

K. J. Nurmela, M. K. Kaikkonen and P. R. J. Östergård, New constant weight codes from linear permutation groups,, IEEE Trans. Inform. Theory, 43 (1997), 1623.  doi: 10.1109/18.623163.  Google Scholar

[18]

D. H. Smith, L. A. Hughes and S. Perkins, A new table of constant weight codes of length greater than 28,, Electronic J. Combin., 13 (2006).   Google Scholar

[19]

L. Teirlinck, A completion of Lu's determination of the spectrum of large sets of disjoint Steiner Triple systems,, J. Combin. Theory (A), 57 (1991), 302.  doi: 10.1016/0097-3165(91)90053-J.  Google Scholar

show all references

References:
[1]

, http://www.win.tue.nl/~aeb/codes/andw.html,  , ().   Google Scholar

[2]

A. Betten, R. Laue and A. Wassermann, A Steiner 5-design on 36 points,, Des. Codes Crypt., 17 (1999), 181.  doi: 10.1023/A:1026427226213.  Google Scholar

[3]

A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, A new table of constant weight codes,, IEEE Trans. Inform. Theory, 36 (1990), 1334.  doi: 10.1109/18.59932.  Google Scholar

[4]

R. H. F. Denniston, Sylvester's problem of the 15 school-girls,, Discr. Math., 9 (1974), 229.  doi: 10.1016/0012-365X(74)90004-1.  Google Scholar

[5]

R. H. F. Denniston, Some new 5-designs,, Bull. London Math. Soc., 8 (1976), 263.  doi: 10.1112/blms/8.3.263.  Google Scholar

[6]

T. Etzion, Optimal partitions for triples,, J. Combin. Theory (A), 59 (1992), 161.  doi: 10.1016/0097-3165(92)90062-Y.  Google Scholar

[7]

T. Etzion, Partitions of triples into optimal packings,, J. Combin. Theory (A), 59 (1992), 269.  doi: 10.1016/0097-3165(92)90069-7.  Google Scholar

[8]

T. Etzion, Partitions for quadruples,, Ars Combin., 36 (1993), 296.   Google Scholar

[9]

T. Etzion and S. Bitan, On the chromatic number, colorings, and codes of the Johnson graph,, Discr. Appl. Math., 70 (1996), 163.  doi: 10.1016/0166-218X(96)00104-7.  Google Scholar

[10]

T. Etzion and P. R. J. Östergård, Greedy and heuristic algorithms for codes and colorings,, IEEE Trans. Inform. Theory, 44 (1998), 382.  doi: 10.1109/18.651069.  Google Scholar

[11]

T. Etzion and C. L. M. van Pul, New lower bounds for constant weight codes,, IEEE Trans. Inform. Theory, 35 (1989), 1324.  doi: 10.1109/18.45293.  Google Scholar

[12]

R. L. Graham and N. J. A. Sloane, Lower bounds for constant weight codes,, IEEE Trans. Inform. Theory, 26 (1980), 37.  doi: 10.1109/TIT.1980.1056141.  Google Scholar

[13]

L. J. Ji, A new existence proof for large sets of disjoint Steiner triple systems,, J. Combin. Theory (A), 112 (2005), 308.  doi: 10.1016/j.jcta.2005.06.005.  Google Scholar

[14]

L. J. Ji, Partition of triples of order $6k+5$ into $6k+3$ optimal packings and one packing of size $8k+4$,, Graphs Combin., 22 (2006), 251.  doi: 10.1007/s00373-005-0632-1.  Google Scholar

[15]

J. X. Lu, On large sets of disjoint Steiner triple systems I, II, III,, J. Combin. Theory (A), 34 (1983), 140.   Google Scholar

[16]

N. S. Mendelsohn and S. H. Y. Hung, On the Steiner systems $S(3,4,14)$ and $S(4,5,15)$,, Utilitas Math., 1 (1972), 5.   Google Scholar

[17]

K. J. Nurmela, M. K. Kaikkonen and P. R. J. Östergård, New constant weight codes from linear permutation groups,, IEEE Trans. Inform. Theory, 43 (1997), 1623.  doi: 10.1109/18.623163.  Google Scholar

[18]

D. H. Smith, L. A. Hughes and S. Perkins, A new table of constant weight codes of length greater than 28,, Electronic J. Combin., 13 (2006).   Google Scholar

[19]

L. Teirlinck, A completion of Lu's determination of the spectrum of large sets of disjoint Steiner Triple systems,, J. Combin. Theory (A), 57 (1991), 302.  doi: 10.1016/0097-3165(91)90053-J.  Google Scholar

[1]

Denis S. Krotov, Patric R. J.  Östergård, Olli Pottonen. Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code. Advances in Mathematics of Communications, 2016, 10 (2) : 393-399. doi: 10.3934/amc.2016013
[2]

Nicolay M. Tanushev, Luminita Vese. A piecewise-constant binary model for electrical impedance tomography. Inverse Problems & Imaging, 2007, 1 (2) : 423-435. doi: 10.3934/ipi.2007.1.423
[3]

Masaaki Harada, Ethan Novak, Vladimir D. Tonchev. The weight distribution of the self-dual $[128,64]$ polarity design code. Advances in Mathematics of Communications, 2016, 10 (3) : 643-648. doi: 10.3934/amc.2016032
[4]

Andreas Klein, Leo Storme. On the non-minimality of the largest weight codewords in the binary Reed-Muller codes. Advances in Mathematics of Communications, 2011, 5 (2) : 333-337. doi: 10.3934/amc.2011.5.333
[5]

Daniel Heinlein, Michael Kiermaier, Sascha Kurz, Alfred Wassermann. A subspace code of size $ \bf{333} $ in the setting of a binary $ \bf{q} $-analog of the Fano plane. Advances in Mathematics of Communications, 2019, 13 (3) : 457-475. doi: 10.3934/amc.2019029
[6]

Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1
[7]

Irene Márquez-Corbella, Edgar Martínez-Moro, Emilio Suárez-Canedo. On the ideal associated to a linear code. Advances in Mathematics of Communications, 2016, 10 (2) : 229-254. doi: 10.3934/amc.2016003
[8]

Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45
[9]

Fengwei Li, Qin Yue, Fengmei Liu. The weight distributions of constacyclic codes. Advances in Mathematics of Communications, 2017, 11 (3) : 471-480. doi: 10.3934/amc.2017039
[10]

Tim Alderson, Alessandro Neri. Maximum weight spectrum codes. Advances in Mathematics of Communications, 2019, 13 (1) : 101-119. doi: 10.3934/amc.2019006
[11]

Olof Heden. The partial order of perfect codes associated to a perfect code. Advances in Mathematics of Communications, 2007, 1 (4) : 399-412. doi: 10.3934/amc.2007.1.399
[12]

Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889
[13]

Sascha Kurz. The $[46, 9, 20]_2$ code is unique. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020074
[14]

Miriam Manoel, Patrícia Tempesta. Binary differential equations with symmetries. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1957-1974. doi: 10.3934/dcds.2019082
[15]

Lluís Alsedà, Michał Misiurewicz. Semiconjugacy to a map of a constant slope. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3403-3413. doi: 10.3934/dcdsb.2015.20.3403
[16]

Chengju Li, Sunghan Bae, Shudi Yang. Some two-weight and three-weight linear codes. Advances in Mathematics of Communications, 2019, 13 (1) : 195-211. doi: 10.3934/amc.2019013
[17]

Petr Lisoněk, Layla Trummer. Algorithms for the minimum weight of linear codes. Advances in Mathematics of Communications, 2016, 10 (1) : 195-207. doi: 10.3934/amc.2016.10.195
[18]

Alexander Barg, Arya Mazumdar, Gilles Zémor. Weight distribution and decoding of codes on hypergraphs. Advances in Mathematics of Communications, 2008, 2 (4) : 433-450. doi: 10.3934/amc.2008.2.433
[19]

Diana M. Thomas, Ashley Ciesla, James A. Levine, John G. Stevens, Corby K. Martin. A mathematical model of weight change with adaptation. Mathematical Biosciences & Engineering, 2009, 6 (4) : 873-887. doi: 10.3934/mbe.2009.6.873
[20]

Joaquim Borges, Ivan Yu. Mogilnykh, Josep Rifà, Faina I. Solov'eva. Structural properties of binary propelinear codes. Advances in Mathematics of Communications, 2012, 6 (3) : 329-346. doi: 10.3934/amc.2012.6.329

2019 Impact Factor: 0.734

Tools

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences