Some new distance-4 constant weight codes

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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

Abstract
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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, , ().
[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.
[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.
[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.
[5]	R. H. F. Denniston, Some new 5-designs, Bull. London Math. Soc., 8 (1976), 263-267. doi: 10.1112/blms/8.3.263.
[6]	T. Etzion, Optimal partitions for triples, J. Combin. Theory (A), 59 (1992), 161-176. doi: 10.1016/0097-3165(92)90062-Y.
[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.
[8]	T. Etzion, Partitions for quadruples, Ars Combin., 36 (1993), 296-308.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

show all references

References:

[1]	, http://www.win.tue.nl/~aeb/codes/andw.html, , ().
[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.
[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.
[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.
[5]	R. H. F. Denniston, Some new 5-designs, Bull. London Math. Soc., 8 (1976), 263-267. doi: 10.1112/blms/8.3.263.
[6]	T. Etzion, Optimal partitions for triples, J. Combin. Theory (A), 59 (1992), 161-176. doi: 10.1016/0097-3165(92)90062-Y.
[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.
[8]	T. Etzion, Partitions for quadruples, Ars Combin., 36 (1993), 296-308.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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