American Institute of Mathematical Sciences

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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,  , ().  http://www.win.tue.nl/~aeb/codes/andw.html+" target="_new" title="Go to article in Google Scholar"> Google Scholar

[2]

A. Betten, R. Laue and A. Wassermann, A Steiner 5-design on 36 points, Des. Codes Crypt., 17 (1999), 181-186. 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), 1334-1380. doi: 10.1109/18.59932.  Google Scholar

[4]

R. H. F. Denniston, Sylvester's problem of the 15 school-girls, Discr. Math., 9 (1974), 229-233. 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), 263-267. doi: 10.1112/blms/8.3.263.  Google Scholar

[6]

T. Etzion, Optimal partitions for triples, J. Combin. Theory (A), 59 (1992), 161-176. 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), 269-284. doi: 10.1016/0097-3165(92)90069-7.  Google Scholar

[8]

T. Etzion, Partitions for quadruples, Ars Combin., 36 (1993), 296-308.  Google Scholar

[9]

T. Etzion and S. Bitan, On the chromatic number, colorings, and codes of the Johnson graph, Discr. Appl. Math., 70 (1996), 163-175. 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), 382-388. 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), 1324-1329. 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), 37-43. 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), 308-327. 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), 251-260. 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), 140-146, 147-155, 156-183, and IV, V, VI, ibid., 37 (1984), 136-163, 164-188, 189-192.  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), 5-95.  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), 1623-1630, 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), #A2.  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), 302-305. doi: 10.1016/0097-3165(91)90053-J.  Google Scholar

show all references

References:
[1]

, http://www.win.tue.nl/~aeb/codes/andw.html,  , ().  http://www.win.tue.nl/~aeb/codes/andw.html+" target="_new" title="Go to article in Google Scholar"> Google Scholar

[2]

A. Betten, R. Laue and A. Wassermann, A Steiner 5-design on 36 points, Des. Codes Crypt., 17 (1999), 181-186. 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), 1334-1380. doi: 10.1109/18.59932.  Google Scholar

[4]

R. H. F. Denniston, Sylvester's problem of the 15 school-girls, Discr. Math., 9 (1974), 229-233. 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), 263-267. doi: 10.1112/blms/8.3.263.  Google Scholar

[6]

T. Etzion, Optimal partitions for triples, J. Combin. Theory (A), 59 (1992), 161-176. 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), 269-284. doi: 10.1016/0097-3165(92)90069-7.  Google Scholar

[8]

T. Etzion, Partitions for quadruples, Ars Combin., 36 (1993), 296-308.  Google Scholar

[9]

T. Etzion and S. Bitan, On the chromatic number, colorings, and codes of the Johnson graph, Discr. Appl. Math., 70 (1996), 163-175. 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), 382-388. 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), 1324-1329. 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), 37-43. 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), 308-327. 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), 251-260. 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), 140-146, 147-155, 156-183, and IV, V, VI, ibid., 37 (1984), 136-163, 164-188, 189-192.  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), 5-95.  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), 1623-1630, 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), #A2.  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), 302-305. doi: 10.1016/0097-3165(91)90053-J.  Google Scholar

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