A(n,4,w)

n\w		5	6	7	8	9	10	11	12	13	14	15
15			399
16		322	616
17
18		544		2042
19
20						10048
21		1113		6161	10767	17177	20654
22				8338	16527	25902	37127	40624
23				11696	23467	41413	58659	76233
24						59904	98852	118422	151484
25			7787	21220	47265	89742	142373	198387	231530
26				27050	66352	129708	222775	320584	401937	431724
27				35874	88604	188561	334859	518014	686164	791461
28				44915	122685	263008	508952	819041	1167909	1420920	1535756
29		4423	17710	57943	157734	365699	728330	1266026	1895939	2499311	2870880
30		4901	21931	73853	214545	514015	1085000	1977548	3143989	4325235	5313399	5697080
n/w		5	6	7	8	9	10	11	12	13	14	15

[aeb & T. Etzion, Nov-Dec 2009]

The standard reference for bounds on binary constant weight codes of length at most 28 is

A. E. Brouwer, James B. Shearer, N. J. A. Sloane & Warren D. Smith,
A new table of constant weight codes,
IEEE Trans. Inf. Th. 36 (1990) 1334-1380.

Tuvi Etzion & Sara Bitan,
On the chromatic number, colorings, and codes of the Johnson graph,
Discr. Applied Math. 70 (1996) 163-175,

Kari J. Nurmela, Markku K. Kaikkonen & Patric R. J. Östergård,
New constant weight codes from linear permutation groups,
IEEE Trans. Inf. Th. 43 (1997) 1623-1630,

Tuvi Etzion & Patric R. J. Östergård,
Greedy and heuristic algorithms for codes and colorings,
IEEE Trans. Inf. Th. 44 (1998) 382-388.

Tables for larger word length were given in

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.

R. Montemanni and D.H. Smith,
Heuristic Algorithms for Constructing Binary Constant Weight Codes,
IEEE Transactions on Information Theory 55 (2009) 4651–4656.

Using partition methods one can improve the bounds given there for d=4. A small table with lower bounds for A(n,4,5):

29	30	31	32	33	34	35	36	37	38	39	40
4423	5148s	6138g	6582	7656s	8976.s	10472.s	10948	12473	13471	15010	17119
41	42	43	44	45	46	47	48	49	50	51	52
19258	20671	22728	25564	28413s	31878.s	35673.s	36809	40560	42920	46612	51420
53	54	55	56	57	58	59	60	61	62	63	64
56251	59293	63973	69931	75550	79330	85728	93206	100527	105472	112457	121902

The existence of the Steiner systems S(5,6,36) and S(5,6,48) implies A(36,4,6) = 62832, A(35,4,5) = 10472, A(35,4,6) = 52360, A(34,4,5) = 8976, A(33,4,5) ≥ 7656. A(48,4,6) = 285384, A(47,4,5) = 35673, A(47,4,6) = 249711, A(46,4,5) = 31878, A(45,4,5) ≥ 28413.

The bound A(31,4,5) ≥ 6138 is found by taking a union of ASL(1,31)-orbits.

Partitions

n	w	#	norm	part sizes	link
10	4	9	5574	30 30 30 28 26 23 22 20 1	d
10	5	9	7854	36 36 34 32 32 29 25 24 4	n
10	5	9	7344	36 36 36 24 24 24 24 24 24	o
10	5	9	7734	36 33 33 33 31 31 31 12 12	p
11	4	10	10948	35 35 35 35 35 34 31 31 31 28	ac
11	4	10	10946	35 35 35 35 34 33 32 32 32 27	y
11	4	11	10910	35 35 35 35 35 34 34 34 28 23 2	c
11	4	11	10894	35 35 35 35 34 33 33 33 29 27 1	z
11	4	11	10884	35 35 35 35 35 34 34 34 29 21 3	d
11	4	13	10782	35 35 35 34 34 34 34 34 34 12 3 3 3	t
11	4	11	10764	35 35 35 35 35 35 35 32 27 20 6	m
11	4	12	10746	35 35 35 35 35 35 35 32 29 16 7 1	x
11	5	9	25968	66 66 60 55 55 55 54 39 12	e

See the subdirectories part10.4, part10.5, part11.4, part11.5, part12.4, part12.5, part12.6, part13.4, part13.5, part13.6, part14.4, part14.5, part14.6, part14.7 for machine-readable descriptions of the partitions. (Improvements are welcome. Mail aeb@cwi.nl .)

Lexicographically best constant starts known:

n\w		3	4	5	6	7	8
6		44
7		72
8		87	142
9		127	185
10		139	305	363
11		179	357	662
12		2011	516	804	1322
13		2611	655	1234	1663
14		2813	914	1694	2782	3252
15		3513	1058	2372	3991	5851
16		3715	1408	3221	6151	8361	11701

The entries here give the size of the largest constant weight code known with given n, w, and for each such code the maximal known number of pairwise disjoint such codes. This is the most greedy start known for a (n,w) partition. Of course there need not be a completion of a greedy start to a good partition.

A(n,10,6)

n		11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37
b		2	2	2	2	3	3	3	4	4	5	7	7	8	9	10	13	14	16	20	25	31	31	31-32	33-34	35	37	37-43
N		1	2	2	2	1	3	4	1	3	1	1	3	2	2	10	2	2	1	1	1	1	7	?	?	?	2	?

In particular, A(36,10,6) = 37.

Comments

One needs 6, 6+5=11, 11+4=15, 15+3=18, 18+2=20, 20+1=21, 21+0=21 points to have 1,2,3,4,5,6,7 lines, respectively. This gives A(n,10,6) for n ≤ 21. The unique system of 7 lines on 21 points has group Sym(7) of order 5040.

The two systems of 8 lines on 23 points have groups of orders 48, 144. The nicer system is the dual of the partial linear space with 8 points and 23 lines obtained by taking two disjoint triples and the 21 pairs not in a triple and not disjoint from both.

The two systems of 9 lines on 24 points have groups of orders 12, 72. The nicer system is the dual of the linear space with 9 points and 24 lines obtained by taking six triples forming a 3x3 grid, and the 18 remaining edges.

The ten systems of 10 lines on 25 points have groups of orders 2, 3, 3, 4, 4, 6, 10, 12, 24, 120. The nicest system is the dual of the linear space with 10 points and 25 lines obtained by taking ten triples, three on each point, and the 15 remaining edges. If the 10 points are given as pairs ab from a 5-set, then the 10 triples are the triples abc from that 5-set, with the natural incidence. The group is Sym(5).

The two systems of 13 lines on 26 points are the duals of the two Steiner triple systems STS(13) (so have 3 lines on each point). They have groups of orders 6 and 39.

The two systems of 14 lines on 27 points have groups of orders 6, 12. Both systems are the dual of a partial linear space obtained by deleting the 2-line from the linear space with 14 points and 28 lines (1 2-line, 24 3-lines, 3 4-lines; where 4-lines and 2-line partition the point set). Cf. A.E. Brouwer, The linear spaces on 15 points, Ars Combinatoria 12 (1981) 3-35.

The unique system of 16 lines on 28 points is obtained by puncturing PG(2,5) thrice (in a noncollinear triple).

The unique system of 20 lines on 29 points is obtained by puncturing PG(2,5) twice.

The unique system of 25 lines on 30 points is the punctured PG(2,5).

The unique system of 31 lines on 31 points is the projective plane PG(2,5) of order 5.

There is no system of 32 lines on 32 points, since that would be a GD[6,1,2;32] and Schellenberg (1975) proved that none exists. (See also BCN, p. 24.)

There is no system of 33 lines on 33 points (as one sees by exhaustive search). In particular, there is no GD[6,1,3;33].

There are precisely two systems of 37 lines on 36 points. Both have a cyclic group of order 30, acting 30+6 on the points, and 30+6+1 on the blocks. (That no larger system exists follows by exhaustive search.)