Bounds for binary constant weight codes

Jump to d=4, d=6, d=8, d=10, d=12, d=14, d=16, d=18.

Sources

BSSS: A. E. Brouwer, J. B. Shearer, N. J. A. Sloane & W. D. Smith,
A new table of constant weight codes,
IEEE Trans. Inform. Theory 36 (1990) 1334-1380.

A. E. Brouwer & T. Etzion,
Some new distance-4 constant weight codes,
Advances in Mathematics of Communications 5 (2011) 417-424. (PDF) (web)

Unmarked upper bounds (for n ≤ 28) are taken from

AVZ: Erik Agrell, Alexander Vardy & Kenneth Zeger,
Upper bounds for constant weight codes,
IEEE Trans. Inform. Theory 46 (2000) 2373-2395. (PDF) (web)

S: Alexander Schrijver,
New code upper bounds from the Terwilliger algebra and semidefinite programming,
IEEE Trans. Inform. Theory 51 (2005) 2859-2866. (PDF)

For n > 28 unmarked upper bounds are from the Johnson bound. In most cases unmarked lower bounds for n > 28 are codes first constructed here. For n ≤ 28, the lower bounds first given here are shown with a yellow background.

All other sources are explicitly indicated.

Tables for 29 ≤ n ≤ 63 and (d,w) one of (6,5), (6,6), (8,5), (8,6), (8,7), (10,6), (10,7), (10,8), (12,7), (12,8), (14,8) were given in

SHP: 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. (PDF)

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

MS2: R. Montemanni and D.H. Smith,
Some constant weight codes from primitive permutation groups, Electr. J. Combin. 19 (2012) Issue 4, P4.

Various bounds derived by taking a slice from a general binary code:

CXY: Yeow Meng Chee, Chaoping Xing and Sze Ling Yeo,
New Constant-Weight Codes From Propagation Rules
IEEE Transactions on Information Theory 56 (2010) 1596-1599.

Improvements to the upper bounds were derived by

KKT: Byung Gyun Kang, Hyun Kwang Kim & Phan Thanh Toan,
Improved linear programming bounds on sizes of constant-weight codes,
arXiv:1108.5104v1, Aug 2011.

KT: Hyun Kwang Kim & Phan Thanh Toan,
Improved Semidefinite Programming Bound on Sizes of Codes
arXiv:1212.3467v1, Dec 2012.

Po: Sven Polak,
Semidefinite programming bounds for constant weight codes,
arXiv:1703.05171, Mar 2017.

Lost codes

There used to be 13 lost codes, but [BJLÖ] removed three items from the list by proving A(24,6,9) ≥ 3080, A(24,6,11) ≥ 5376, A(25,6,10) ≥ 6600. They also showed A(24,6,12) ≥ 5558. And [R] removed one item by proving A(26,8,11) ≥ 2030. And [L] removed two by proving A(23,6,10) ≥ 2979 and A(24,6,10) ≥ 4214. He also showed A(23,6,11) ≥ 3539, A(24,6,8) ≥ 1855. Polishing the former removed one more by proving A(23,6,11) ≥ 3588. And [E] removed five items for codes A(n,6,8), starting with the Steiner system S(5,7,28).

Earlier, in

M.R. Best, A.E. Brouwer, F.J. MacWilliams, A.M. Odlyzko & N.J.A. Sloane,
Bounds for binary codes of length less than 25,
IEEE Trans. Inf. Th. 24 (1978) 81-93.

Errata

There is some freedom in "Construction B" as described in [BSSS], and the partitions given in [BSSS], Table VI are not the best ones can obtain using this construction.

In [BSSS], Section XI, description for A(16,4,8) ≥ 1170, there are two typos: replace 07F and E40 by 407F and FE40.

In [BSSS], Table XV, description for A(18,6,9) ≥ 304, delete the "(17,18)" from the third generator. (Below we show A(18,6,9) ≥ 320.)

In [BSSS], Table I-E, description for A(26,12,12) ≥ 54 and A(26,12,13) ≥ 58, the label z9 refers to Table XVI, but these codes are not given there.

Constructions

c: circulant, or code with cyclic group.

d: doubling (special case of j: A(2n,2d,2w) ≥ A(n,d,w)).

g: code with specified group of automorphisms.

j: juxtaposition.

p: using partitions.

s: shortened code (from code of length n+1 and weight w or w+1).
sb: shortened code (from code of length n+1 and weight w), using explicit inspection.
sd: shortened code (from code of length n+1 and weight w+1), using explicit inspection.

l: lengthened code (from code of length n–1 and weight w or w–1).

x: lexmin code.

H: Hadamard matrix ([BSSS], Thm 10): A(4m,2m,2m) = 8m–2 if a Hadamard matrix of order 4m exists.

OA (orthogonal array): If k ≤ q+1, where q is a prime power, then an e-OA(k,q) exists for any e with 0 ≤ e ≤ k. It has q^e blocks (transversals) of size k on qk points partitioned into k groups of size q, and shows that A(qk,2(k–e+1),k) ≥ q^e. Usually many blocks can be added.

Upper bounds

Note on the Johnson bound

References for labels

ACL: Hui Kheng Aw, Yeow Meng Chee & Alan C. H. Ling, Six New Constant Weight Binary Codes, Ars Comb. 67 (2003) 313-318, show A(18,6,5) ≥ 69. Also A(27,8,5) ≥ 31, A(29,8,5) ≥ 34, A(30,8,5) ≥ 36, A(33,8,5) ≥ 44, A(34,8,5) ≥ 47.

BJi: Jingjun Bao & Lijun Ji, The completion of optimal (3,4)-packings, preprint, 2014, show A(23,4,4) = 419, etc.

BJLÖ: Michael Braun, Joshua Humpich, Antti Laaksonen & Patric R. J. Östergård, New lower bounds on binary constant weight error-correcting codes, J. of Combinatorial Math. and Combinatorial Computing 111 (2019) 213-224 , shows A(16,4,6) ≥ 624, A(22,6,6) ≥ 343, etc.

Bl: Iliya Bluskov, New constant weight codes and packing numbers, preprint, 2017 (see also Electronic Notes in Discrete Math. 65 (2018) 31-36), shows A(29,8,5) ≥ 36, A(30,8,5) ≥ 41, A(32,8,5) = 44.

Ch: Yeow Meng Chee, A new lower bound for A(17,6,6), Ars Comb. 83 (2007) 361-363, shows A(17,6,6) ≥ 113.

E: William Echols, email 2026-07-29 (with 126 new codes) and email 2026-08-08 (with 6 new codes, removing five items from the list of lost codes).

EB: Tuvi Etzion & Sara Bitan, On the chromatic number, colorings, and codes of the Johnson graph, Discrete Appl. Math. 70 (1996) 163-175, show A(24,4,7) ≥ 15656.

Ex: Geoffrey Exoo, email 2014-08-27, shows A(15,4,5) ≥ 242.

Ho: Iiro Honkala, Heikki Hämäläinen & Markku Kaikkonen, Some lower bounds for constant weight codes, Dicr. Appl. Math. 18 (1987) 95-98, show A(25,12,11) ≥ 36, A(26,12,12) ≥ 54, A(26,12,13) ≥ 58, A(27,12,9) = 39. (A more direct construction: take the planes in AG(3,3).)

Ji: L. Ji, Asymptotic Determination of the Last Packing Number of Quadruples, Designs, Codes and Cryptography 38 (2006) 83-95, shows A(17,4,4) = 157, A(29,4,4) = 877, etc.

JX: Lingfei Jin & Chaoping Xing, New binary codes from rational function fields, preprint, 2014, show A(27,6,13) ≥ 27600, A(28,6,12) ≥ 40237, A(28,6,13) ≥ 49510, A(28,6,14) ≥ 53046.

Ko: Klaus-Uwe Koschnick, Some new constant weight codes, IEEE Trans. Info. Theory 37 (1991) 370-372, shows A(23,10,7) ≥ 20.

L: Christian Lysenstøen, New lower bounds for binary constant-weight codes: A(23,6,10) ≥ 2979 and A(24,6,10) ≥ 4214, arXiv:2607.19550, constructs four improved codes. After polishing, these became A(23,6,10) ≥ 2992, A(23,6,11) ≥ 3588, A(24,6,8) ≥ 1859, A(24,6,10) ≥ 4237.

MEL: Beniamin Mounits, Tuvi Etzion & Simon Litsyn, New upper bounds on A(n,d), arXiv:cs/0508107, Aug 2005, show A(23,10,9) ≤ 78, A(24,10,9) ≤ 116, A(25,10,9) ≤ 157, A(27,10,9) ≤ 293, A(28,10,10) ≤ 785.

Mi: Moshe Milshtein, A new binary (17,4,5) constant weight code, Cryptography and Communications 15 (2023) 443–453, shows A(17,4,5) ≥ 444.

Mo: B. Mounits, T. Etzion & S. Litsyn, New upper bounds on codes via association schemes and linear programming, Advances of Math. in Communications 1 (2007) 173-195, show A(19,6,7) ≤ 519, A(22,6,11) ≤ 5033, A(26,6,11) ≤ 42017.

Nu: Kari J. Nurmela, Markku K. Kaikkonen & Patric R. J. Östergård, New constant weight codes from linear permutation groups, IEEE Trans. Info. Theory 43 (1997) 1623-1630, show A(16,4,5) ≥ 315 and A(20,4,6) ≥ 2304, with shortened codes showing A(19,4,5) ≥ 692 and A(19,4,6) ≥ 1620. Also A(18,6,8) ≥ 260, A(18,6,9) ≥ 304 (both also in [BSSS]), A(21,6,5) ≥ 108, A(28,6,5) ≥ 280, A(28,6,11) ≥ 29484. Also A(26,8,13) ≥ 3588, A(27,8,13) ≥ 4094, A(28,8,10) ≥ 1820, A(28,8,12) ≥ 4916, A(28,8,14) ≥ 6090. Also A(22,10,10) ≥ 46, A(23,10,11) ≥ 65, A(24,10,11) ≥ 95, A(24,10,12) ≥ 122, A(25,10,12) ≥ 132, A(26,10,9) ≥ 91, A(27,10,9) ≥ 118, A(27,10,10) ≥ 162, A(27,10,11) ≥ 222, A(28,10,10) ≥ 210, A(28,10,11) ≥ 286, A(28,10,12) ≥ 365, A(28,10,13) ≥ 756, A(28,10,14) ≥ 790. Also A(26,12,10) ≥ 33, A(28,12,10) ≥ 49, A(28,12,11) ≥ 65, A(28,12,12) ≥ 103, A(28,12,13) ≥ 99, A(28,12,14) ≥ 172.

Ö: Patric R. J. Östergård, Classification of binary constant weight codes, IEEE Trans. Infor. Theory 56 (2010) 3779-3785, shows A(12,4,5) = 80. Also A(15,6,7) = 69 and A(17,6,6) ≤ 124. Also A(18,8,7) = 33, A(19,8,7) = 52, A(19,8,8) = 78, A(20,8,8) = 130 and A(18,8,8) ≤ 49, A(18,8,9) ≤ 58.

R: Christopher D. Rosin, Automated discovery of improved constant weight binary codes, arXiv:2603.00174, shows A(18,6,6) ≥ 133, A(22,8,9) ≥ 292, etc.

Sa: Ankan Sadhu, email 2026-07-25, shows A(38,8,5) ≥ 66, A(39,8,5) ≥ 68, A(47,8,5) ≥ 101, A(48,8,5) ≥ 103.

SWY: D.R. Stinson, R.Wei & J. Yin,
Packings, pp. 550-556 in: Handbook of Combinatorial Designs – 2nd ed. (C.J. Colbourn and J.H. Dinitz eds.), CRC Press, 2007.

To: V.D. Tonchev, Maximum disjoint bases and constant weight codes, IEEE Trans. Infor. Theory 44 (1998) 333-334, shows A(17,4,5) ≥ 441.

Jump to d=4, d=6, d=8, d=10, d=12, d=14, d=16, d=18.

Bounds on A(n,4,w)

p: using partitions, see [BSSS] and here. Most of these bounds are from the Brouwer-Etzion paper.

Values of A(n,4,3) and A(n,4,4)

For w=3 and w=4 all values are known ([BSSS], Theorems 4, 5; [Ji]; [BJi]). A table for n ≤ 64:

Bounds on A(n,4,5)

S: The Steiner systems S(5,6,36) and S(5,6,48) yield A(36,4,6) = 62832 and A(48,4,6) = 285384 with derived designs showing A(35,4,5) = 10472, A(35,4,6) = 52360, A(47,4,5) = 35673, A(47,4,6) = 249711.

g: A(31,4,5) ≥ 6138 and A(32,4,5) ≥ 6758 use ASL(1,31) of order 465.

The group codes A(32,4,5) ≥ 6758^g, A(42,4,5) ≥ 21320^g, A(43,4,5) ≥ 23478^g, A(50,4,5) ≥ 42924^g, A(55,4,5) ≥ 62964^g, A(63,4,5) ≥ 113337^g were found by [MS2].

Hamming codes

gap> LoadPackage("guava");;
gap> WeightDistribution(ExtendedCode(HammingCode(4,GF(2))));
[ 1, 0, 0, 0, 140, 0, 448, 0, 870, 0, 448, 0, 140, 0, 0, 0, 1 ]
gap> WeightDistribution(ExtendedCode(HammingCode(5,GF(2))));
[ 1, 0, 0, 0, 1240, 0, 27776, 0, 330460, 0, 2011776, 0, 7063784, 0, 14721280, 
  0, 18796230, 0, 14721280, 0, 7063784, 0, 2011776, 0, 330460, 0, 27776, 0, 
  1240, 0, 0, 0, 1 ]
gap> WeightDistribution(ExtendedCode(HammingCode(6,GF(2))));
[ 1, 0, 0, 0, 10416, 0, 1166592, 0, 69194232, 0, 2366570752, 0, 51316746768, 
  0, 747741998592, 0, 7633243745820, 0, 56276359749120, 0, 306558278858160, 
  0, 1255428754917120, 0, 3916392495228360, 0, 9399341113166592, 0, 
  17480786291963792, 0, 25316999607653376, 0, 28634752793916486, 0, 
  25316999607653376, 0, 17480786291963792, 0, 9399341113166592, 0, 
  3916392495228360, 0, 1255428754917120, 0, 306558278858160, 0, 
  56276359749120, 0, 7633243745820, 0, 747741998592, 0, 51316746768, 0, 
  2366570752, 0, 69194232, 0, 1166592, 0, 10416, 0, 0, 0, 1 ]

These are not best possible: slightly better results follow by the Kløve bound found in

Torleiv Kløve,
A lower bound for A (n,4,w),
IEEE Trans. Inf. Th. IT-27 (1981) 257-258.

Jump to d=4, d=6, d=8, d=10, d=12, d=14, d=16, d=18.

Bounds on A(n,6,w)

One has A(n–2,d–2,w–1) ≥ A(n,d,w) (Honkala et al., cf. [BSSS], Thm 21), and this yields A(22,6,7) ≥ 759.

G: from the Golay code.

t: using the Nordstrom-Robinson code (really, the Golay code), possibly with tails.

gs: using the Graham-Sloane construction and an S₂ set (B₂ sequence, modified modular Golomb ruler), that is, a subset of an abelian group where all sums of two (distinct) elements are different. One divides all words of weight w into M classes, each of min dist at least 6. This means that A(n,d,w) ≥ (n choose w)/M where, for d = 6, M = 624, 651, 728, 757, 840, 871 for n = 25, 26, 27, 28, 29, 30, respectively. The largest class found may be slightly larger than this average. These codes are not always maximal, and one finds A(26,6,12) ≥ 14852+2, A(29,6,10) ≥ 23870+7 by adding 2 (resp. 7) words to a gs-code.

A code showing A(23,6,5) ≥ 147 is found by taking the 759 octads on a 24-set where the last 8 positions support an octad, and selecting the 140 octads with weight 3+1 in the last 7+1 positions. Delete the last position, and replace the triple by the i-th shift of 1000000 when the triple equals or is disjoint from the i-th shift of 1101000. Now add 7 words, say in lexmin order.

E: six codes giving lower bounds for A(n,6,8) were constructed from S(5,7,28) by extending with a 1 bit, shortening to length n, and using a maximum independent set solver to append compatible blocks.

A code showing A(20,6,6) ≥ 232 is found by taking the 112 octads with weight 1+1 in the last 7+1 positions, and 120 of the 140 octads with weight 3+1 there, namely those with 6 of the 7 shifts. Replace the 8-bit tail by a 4-bit one of weight 2, different for different shifts.

Several of the above ugly codes were obtained by polishing a nice group code. For example, A(25,6,11) ≥ 8125^g, A(25,6,12) ≥ 9375^g, A(26,6,9) ≥ 5700^g, A(26,6,11) ≥ 11660^g, A(26,6,13) ≥ 15625^g, A(28,6,10) ≥ 17738^g, A(29,6,10) ≥ 23520^g. The above A(22,6,10) ≥ 2183 was obtained by polishing the A(22,6,10) ≥ 2180^t from [BSSS].

Values of A(n,6,4)

Values of A(n,6,5)

A(63,6,5) = 3906 follows from A(64,6,6) = 41664, from Preparata.

Several of the above ugly codes were obtained by polishing a nice group code. For example, A(34,6,5) ≥ 495^g, A(42,6,5) ≥ 1010^g, A(43,6,5) ≥ 1050^g, A(51,6,5) ≥ 1734^g, A(55,6,5) ≥ 2211^g, A(56,6,5) ≥ 2352^g, A(59,6,5) ≥ 2842^g.

The group codes A(41,6,5) ≥ 943^g, A(47,6,5) ≥ 1363^g, A(49,6,5) ≥ 1617^g, A(53,6,5) ≥ 2067^g were found by [MS2].

Values of A(n,6,6)

Pr: from extended Preparata code.

Several of the above ugly codes were obtained by polishing a nice group code. For example, A(34,6,6) ≥ 1922^g, A(35,6,6) ≥ 2100^g, A(36,6,6) ≥ 2387^g, A(45,6,6) ≥ 6810^g, A(49,6,6) ≥ 9947^g, A(50,6,6) ≥ 10810^g, A(51,6,6) ≥ 11010^g.

The group codes A(32,6,6) ≥ 1643^g, A(33,6,6) ≥ 1829^g, A(37,6,6) ≥ 3108^g, A(42,6,6) ≥ 5002^g, A(43,6,6) ≥ 5719^g, A(45,6,6) ≥ 6810^g, A(53,6,6) ≥ 13780^g were found by [MS2].

Preparata codes

x^64+41664*x^58*y^6+2118168*x^56*y^8+74203584*x^54*y^10+1602647424*x^52*y^12+23369897088*x^50*y^14+
238532662620*x^48*y^16+1758643689600*x^46*y^18+9579950593920*x^44*y^20+39232098538560*x^42*y^22+
122387418032040*x^40*y^24+293729091759936*x^38*y^26+546275088069376*x^36*y^28+791155554970368*x^34*y^30+
894836772921798*x^32*y^32+791155554970368*x^30*y^34+546275088069376*x^28*y^36+293729091759936*x^26*y^38+
122387418032040*x^24*y^40+39232098538560*x^22*y^42+9579950593920*x^20*y^44+1758643689600*x^18*y^46+
238532662620*x^16*y^48+23369897088*x^14*y^50+1602647424*x^12*y^52+74203584*x^10*y^54+2118168*x^8*y^56+
41664*x^6*y^58+y^64

CXY find by averaging various subsets of the Preparata code with (n,d,M) = (63,5,2⁵²) that A(63,6,7) ≥ 270468, A(63,6,8) ≥ 1893276, A(63,6,11) ≥ 300700062, A(63,6,12) ≥ 1302990507, and A(64,6,7) ≥ 303354, A(64,6,8) ≥ 2163744, A(64,6,9) ≥ 13447707, A(64,6,11) ≥ 363105666, A(64,6,12) ≥ 1603680624, A(64,6,13) ≥ 6414487191.

Jump to d=4, d=6, d=8, d=10, d=12, d=14, d=16, d=18.

Bounds on A(n,8,w)

Comments

G: From Golay code.

Some of the above ugly codes were obtained by polishing a nice group code. For example, A(28,8,8) ≥ 1029^g yielded the entries for (n,w)=(28,8), and 27 ≤ n ≤ 32, w = 7. and A(28,8,9) ≥ 1303^g yielded the entries for (n,w)=(28,9), (27,9), and A(28,8,11) ≥ 2688^g yielded the entries for (n,w)=(28,11), (27,11).

The values A(26,8,11) ≥ 2037 and A(27,8,10) ≥ 1721 and A(28,8,10) ≥ 2067 were obtained by polishing the 2030 and 1704 and 2028 found by [R].

The values A(28,8,12) ≥ 4927 and A(28,8,14) ≥ 6218 were obtained by polishing the group part of from the A(28,8,12) ≥ 4914+2 and A(28,8,14) ≥ 5920+170 found by [Nu].

Values of A(n,8,5)

S: S(2,5,41), S(2,5,45), S(2,5,61), S(2,5,65).

A(37,8,5) = 65 from a PBD(37,{5,9*}) (completion of RB(4,1; 28)). A(36,8,5) = 57 by deleting one point.

A(48,8,5) ≥ 102 by replacing the six groups of GD(5,1,8; 48) by 5-lines.

A(53,8,5) ≥ 133 from a PBD(53,{5,13*}) (completion of RB(4,1; 40)). A(52,8,5) ≥ 123 by deleting one point.

[MS] claims A(29,8,5) ≥ 39 and refers to [SWY], but there is no such statement there. Replaced this lower bound by the 36 given in [Bl].

This problem is that of packing pairs by quintuples. See [SWY].

Values of A(n,8,6)

A(30,8,6) ≥ 155: take a 3-OA(6,5) (six groups of size 5, every triple across covered once, 125 blocks) and add 30 blocks that meet two groups, both in 3 points, where no triple is used twice. (Some detail left out, see the explicit code given.)

A(42,8,6) > 343: take a 3-OA(6,7) (six groups of size 7, every triple across covered once, 343 blocks) and add some blocks that meet at most two groups. Optimum not determined, the code given was found by adding words greedily.

A(48,8,6) > 512: similar.

A(54,8,6) > 729: similar.

A(66,8,6) > 1331.

The codes for n=38, 39, 40 were obtained by polishing A(38,8,6) ≥ 285^g.

The group codes A(55,8,6) ≥ 1045^g and A(61,8,6) ≥ 1464^g were found by [MS2]. The codes for 52≤n≤59 were obtained by polishing the former.

Values of A(n,8,7)

A(49,8,7) > 2401: use a 4-OA(7,7) and add words. Adding words in lexmin order gives 3240, further polishing yields 3270.

4-OA(7,8) and 4-OA(7,9) give A(56,8,7) > 4096 and A(63,8,7) > 6561.

Let AGL(m) be the group of maps x → ax+b (mod m) for gcd(a,m)=1. Using AGL(55) one finds A(55,8,7) > 3850. Adding words in lexmin order gives A(55,8,7) ≥ 4206, polishing A(55,8,7) ≥ 4337. Shortening yields A(54,8,7) ≥ 3795, adding words and polishing gives A(54,8,7) ≥ 3828.

Using AGL(61) one finds A(61,8,7) > 5490. Adding words in lexmin order gives A(61,8,7) ≥ 6425.

The code for n=42 was obtained by polishing A(38,8,7) ≥ 912^g.

The group code A(42,8,7) ≥ 1394^g was found by [MS2].

Lower bounds for A(n,8,8) and kissing numbers

gA: Group code using AGL(37) (by Yves Edel).
gL: Group code using PSL(2,43).
Tap: from János Tapolcai: A(48,8,8) ≥ 12642=16*759+498, from A(24,8,8)=759, A(8,4)=16, and A(24,4,4)=498.

The codes for 29 ≤ n ≤ 34 were obtained by lengthening and polishing A(28,8,8) ≥ 1029^g. The code for n = 36 was obtained by adding 3 words to A(36,8,8) ≥ 2739^g.

A(56,8,8) > 7⁵ = 16807.

The bounds given above yield improved lower bounds for the kissing numbers τ_n.

Cf.

Yves Edel, E. M. Rains & N. J. A. Sloane,
On kissing numbers in dimensions 32 to 128,
Electr. J. Comb. 5 (1998) R22. (PDF)

Comparison between lexmin and reverse lexmin

Thus, for n > 26 reverse lexmin is better than lexmin (for d=w=8). After polishing the opposite is true for n ≤ 37, but for n ≥ 42 plain reverse lexmin is even better than polished lexmin. Note that the lexmin bounds increase monotically with n, but the reverse lexmin bounds do not.

Goethals codes

Jump to d=4, d=6, d=8, d=10, d=12, d=14, d=16, d=18.

Bounds on A(n,10,w)

[BSSS] showed A(21,10,8) ≥ 21, but the period indicating equality may have been a mistake. However, equality follows from [AVZ] (26).

Some of the above ugly codes were obtained by polishing a nice group code. For example, A(29,10,12) ≥ 539^g, A(30,10,9) ≥ 196^g, A(30,10,12) ≥ 750^g, A(31,10,15) ≥ 1860^g, A(32,10,8) ≥ 144^g, A(32,10,16) ≥ 2976^g.

A(25,10,8) ≥ 51 was obtained by polishing the 50 found by [R].

Values of A(n,10,6)

For n ≤ 37, see here.

A(31,10,6) ≥ 31 from PG(2,5).

A(42,10,6) ≥ 55 from TD(6,7) and adding six 6-lines contained in the groups. This can be shortened twice in such a way that only 13 blocks are lost, so that A(40,10,6) ≥ 42.

A(46,10,6) ≤ 67 since no S(2,6,46) exists: S.K. Houghten, L.H. Thiel, J. Janssen and C.W.H. Lam, There is no (46,6,1) Block Design, J. Comb. Designs 9 (2001) 60-71.

The code showing that A(48,10,6) = 72 is a GD(6,1,3; 48). Maybe such a design was unknown.

A(54,10,6) ≥ 87 from TD(6,9) and adding six 6-lines contained in the groups.

A(60,10,6) ≥ 104 from TD(6,10)–TD(6,2) (with 96 blocks) and adding six 6-lines in the groups and two 6-lines in the hole. There is a point in the hole contained in 8 blocks only, and shortening there yields A(59,10,6) ≥ 96.

A(61,10,6) ≥ 110 from TD(6,10)–TD(6,2) with a point ∞ added to all groups, and adding twelve 6-lines in the extended groups and two 6-lines in the hole.

A(66,10,6) = 143 from S(2,6,66). A(65,10,6) = 130.

A(76,10,6) = 190 from S(2,6,76).

Values of A(n,10,7)

One has A(n–2,d–2,w–1) ≥ A(n,d,w) (Honkala et al., cf. [BSSS], Thm 21), and this yields A(48,10,7) ≥ 350. Shortening yields codes for 34 ≤ n ≤ 48.

A 3-OA(7,7) yields A(49,10,7) > 343. Polishing gives A(49,10,7) ≥ 385. Lengthening and polishing yields the above bounds for 49 ≤ n ≤ 53.

A 3-OA(7,8) yields A(56,10,7) > 512. Polishing gives A(56,10,7) ≥ 583. Shortening and polishing yields codes for n = 54, 55. Lengthening and polishing gives 57 ≤ n ≤ 60.

A 3-OA(7,9) yields A(63,10,7) > 729. Polishing gives A(63,10,7) ≥ 831. Shortening and polishing yields codes for n = 61, 62.

Values of A(n,10,8)

A(56,10,8) > 2401 from a 4-OA(8,7) and A(64,10,8) > 4096 from a 4-OA(8,8). All codes nearby are polished versions of shortenings or lengthenings of these.

Jump to d=4, d=6, d=8, d=10, d=12, d=14, d=16, d=18.

Bounds on A(n,12,w)

A(30,12,9) ≥ 44 was obtained by polishing the 43 found by [R].

BCH code

x^32 + 496x^12 + 1054x^16 + 496x^20 + x^32.

A coset has weight enumerator

20x^8+12x^432+1144x^16+432x^20+20x^24,

Two cosets of the non-extended code have weight enumerators

x^5+x^6+35x^9+87x^10+400x^13+500x^14+500x^17+400x^18+87x^21+35x^22+x^25+x^26
2x^6+40x^9+82x^10+390x^13+510x^14+510x^17+390x^18+82x^21+40x^22+xx^25

Values of A(n,12,7)

(The code words can be viewed as the points of a partial linear space, the n positions as the lines. The condition is that each point is in precisely 7 lines, and two lines have at most one common point.
A(28,12,7) = 8 follows by taking the complete graph K₈;
A(30,12,7) = 9 follows by taking the linear space with lines {0,3,6}, {0,1}, {0,2}, {0,4} (mod 9);
A(32,12,7) = 10 follows by taking the partial linear space on 10 points with 3-lines {0,1,2}, {3,4,5}, {6,7,8}, {0',3,6}, {1,4,7}, {2,5,8} and otherwise 2-lines, except that the pair 00' is not covered;
A(33,12,7) = 11 follows by taking the linear space with lines {0,1,3}, {0,4}, {0,5} (mod 11);
A(34,12,7) = 12 follows by taking the linear space with lines {0,1,3}, {0,4,8}, {0,5}, {0,6} (mod 12);
A(35,12,7) = 15 follows by taking STS(15);
A(36,12,7) = 16 follows by taking the linear space with lines {0,4,8,12}, {0,1,7}, {0,2,5} (mod 16);
A(37,12,7) = 17 follows by taking the linear space on 17 points A,B,C0,C1,C2,0-5,0'-5' with the 5-line {A,B,C0,C1,C2}, and the six 4-lines {0,1,2',4'} (mod 6), and the thirty 3-lines {A,0,0'} (mod 6), {B,0,5'} (mod 6), {C0,0,3}, {C0,1,2}, {C0,4,5}, {C0,0',3'}, {C0,2',4'}, {C0,1',5'} (mod 6) with C3=C0;
A(38,12,7) = 19 follows by taking the linear space with lines {0,1,7,11}, {0,2,5} (mod 19);
A(39,12,7) ≥ 21 follows by taking the partial linear space on 21 points obtained from a PBD(22,{4,7*}) by replacing the 7-line by a Fano plane and removing one point and the three incident Fano lines from it;
A(45,12,7) = 45 follows by taking a GD(7,1,3;45), Baker's elliptic semiplane.
A(49,12,7) = 56 follows by taking AG(2,7), with the points as coordinate positions;
A(56,12,7) ≥ 71 and A(63,12,7) ≥ 88 follow by taking a TD(7,8) or TD(7,9) and adding seven 7-lines contained in the groups.)

Note on the upper bounds: A(41,12,7) < 30 and A(42,12,7) < 36 and A(43,12,7) < 43 since equality would give two MOLS(6), resp. AG(2,6), resp. PG(2,6) and these do not exist. In fact A(41,12,7) ≤ 29 implies A(42,12,7) ≤ 34 and A(43,12,7) ≤ 40. János Tapolcai observed that A(36,10,6) ≤ 37 implies A(43,12,7) ≤ 38.

Somewhat related is the result in

J.I. Hall, A.J.E.M. Jansen, A.W.J. Kolen & J.H. van Lint,
Equidistant codes with distance 12,
Discr. Math. 17 (1977) 71-83.

D. McCarthy & S.A. Vanstone,
Embedding (r,1)-designs in finite projective planes,
Discr. Math. 19 (1977) 67-76.

K. Metsch,
Linear spaces with few lines,
Springer LNM 1490, 1991.

Values of A(n,12,8)

A(31,12,8) ≤ 31 follows by [AVZ], (26).

A(50,12,8) = 350 from S(3,8,50), a Möbius plane. Now A(50+m,12,8) ≥ 350 + A(m,8,6) for m ≤ 25. Removing 8 points in a block shows A(42,12,8) ≥ 105.

A(65,14,9) = 520 from S(3,9,65), a Möbius plane. It follows that A(63,12,8) ≥ 520, and shortening yields the above bounds for 57 ≤ n ≤ 63.

Stronger results follow by using a 3-OA(8,7) or 3-OA(8,8). One finds A(56,12,8) > 343 and A(64,12,8) > 512.

Lower bounds for A(n,12,12)

Jump to d=4, d=6, d=8, d=10, d=12, d=14, d=16, d=18.

Bounds on A(n,14,w)

bibd: 2-(31,10,3) design.
Similarly, a 2-(37,9,2) design shows A(37,14,9) = 37, A(36,14,9) = 28.

A Steiner system S(3,9,65) shows A(65,14,9) = 520.

Values of A(n,14,8)

(The code words can be viewed as the points of a partial linear space, the n positions as the lines. The condition is that each point is in precisely 8 lines, and two lines have at most one common point.
A(36,14,8) = 9 follows by taking K₉;
A(39,14,8) = 10 follows by taking the partial linear space with two disjoint 3-lines and otherwise 2-lines, with two noncollinear pairs AB, CD;
A(41,14,8) = 11 follows by taking the partial linear space on 11 points 0-8,0',8' with 3-lines 012,345,678,0'36,147,258' and otherwise 2-lines, where the pairs 00' and 88' are not covered;
A(42,14,8) = 12 follows by taking the linear space on 12 points with lines {0,1,3}, {0,4}, {0,5}, {0,6} (mod 12);
A(44,14,8) = 13 follows by taking the partial linear space on 13 points A,0-11 with 3-lines 013 (mod 12), A06,A17,A28,A39 and otherwise 2-lines, where the pairs {4,10} and {5,11} are not covered;
A(45,14,8) = 15 follows by taking the linear space on 15 points with lines {0,1,3}, {0,4,9}, {0,7} mod 15;
A(46,14,8) = 17 follows by taking the partial linear space obtained from a PBD(17,{3,5*}) by replacing the 5-line by two 3-lines and two 2-lines;
A(47,14,8) = 18 follows by taking the partial linear space on 18 points a-f,0-11 with lines {0,3,6,9} (mod 12), {0,4,8} (mod 12), a01,b12,a23,b34,a45,..., c05,d16,c27,d38,..., e02,e13,f24,f35,e46,e57,f68,..., ace,adf,bcf,bde, where the pairs ab, cd, ef are not covered;
A(48,14,8) = 19 is given in [MS];
A(49,14,8) = 21 follows by taking the linear space on 21 points with lines {0,2,8,11}, {0,4,5}, {0,7,14} mod 21;
A(50,14,8) = 25 follows by taking S(2,4,25);
Lower bounds A(52,14,8) ≥ 27, A(53,14,8) ≥ 31 are given in [SHP];
A(57,14,8) = 57 follows by taking PG(2,7);
A(62,14,8) ≥ 58 was earlier claimed without proof by [SHP].
A(64,14,8) follows by taking AG(2,8), this time with points as coordinate positions.
We have 89 ≤ A(72,14,8) ≤ 90, where the lower bound follows by deleting the points of one line from AG(2,9), and removing a point from the eight lines parallel to it.)

Lower bounds for A(n,14,14)

Jump to d=4, d=6, d=8, d=10, d=12, d=14, d=16, d=18.

Bounds on A(n,16,w)

A(32,16,14), A(32,16,15) ≤ 32 follows by [AVZ], (26).

A(33,16,11) = 12.

Values of A(n,16,9)

The values for n = 59, 60, 61 are due to I. Gashkov & D. Taub, New optimal constant weight codes, Electr. J. Comb. 14 (2007) #N13. (That reference also claims that A(45,14,8) = 14, but in fact A(45,14,8) = 15.)

(The code words can be viewed as the points of a partial linear space, the n positions as the lines. The condition is that each point is in precisely 9 lines, and two lines have at most one common point. For example, A(48,16,9) = 11 follows from a partial linear space with 11 points, the union of three 3-lines and two points A, B, with AB not joined, but 2-lines otherwise; A(50,16,9) = 12 follows from a linear space with 12 points and 8 triples, two on each point (and otherwise 2-lines); A(52,16,9) = 13 follows from the linear space with 13 points and lines {0,1,3}, {0,4}, {0,5}, {0,6} (mod 13); A(54,16,9) = 14 follows from a partial linear space with points 0,1,...,11,0',1' and lines {0,1,3}, {0,4}, {0,5}, {0,6,0'}, {0,1'} (mod 12), where 2' = 0'; A(55,16,9) = 15 follows from the linear space with 15 points and 25 3-lines formed by a TD(3,5); A(57,16,9) = 19 follows from the existence of STS(19); A(63,16,9) = 63 from S(2,4,28); A(73,16,9) = 73 from PG(2,8); A(78,16,9) = 78 from PG(2,9) minus a Baer subplane; A(81,16,9) = 90 from AG(2,9), the last two with points as coordinate positions.)

Two further exact values are A(99,16,9) = 132 and A(135,22,12) = 135, obtained from Mathon's elliptic semiplane.

A(40,16,10) = 16. A(42,16,10) = 21. A(45,16,10) < 36.

Jump to d=4, d=6, d=8, d=10, d=12, d=14, d=16, d=18.

Bounds on A(n,18,w)

A(36,18,15) = 36 (bibd), A(36,18,18) = 70 (Had), A(40,18,13) = 40 (PG(3,3)).

Values of A(n,18,10)

A(82,18,10) = 41, A(90,18,10) = 81, A(91,18,10) = 91.

(The code words can be viewed as the points of a partial linear space, the n positions as the lines. The condition is that each point is in precisely 10 lines, and two lines have at most one common point. For example, A(61,18,10) = 13 follows from the linear space that is the union of a 4-line, and a 3x3 grid (6 3-lines on 9 points), where all remaining lines are 2-lines; A(63,18,10) = 14 follows from the linear space with lines {0,1,3}, {0,4}, {0,5}, {0,6}, {0,7} (mod 14); A(65,18,10) = 15 follows from the linear space with lines {0,1,3}, {0,5,10}, {0,4}, {0,6}, {0,7} (mod 15); A(67,18,10) = 16 follows from the linear space with points 0,...,11,0',...,3' and lines {0,1,3} mod 12, {0',0,5} mod 12 (with 4'=0'), {0',1',2',3'} and otherwise 2-lines; A(68,18,10) = 17 follows from the linear space with lines {0,1,3}, {0,4,9}, {0,6}, {0,7}; A(70,18,10) = 21 follows by taking an STS(21); A(72,18,10) = 22 follows from a GDD with blocksize 3 and groups 4*4+6 (with the 6-group split into 3*2); A(73,18,10) = 23 follows from a (10,1)-design PBD(23,{3,4,5*}); A(74,18,10) = 24 follows by deleting the 7-block in an PBD(31,{4,7*}). A(75,18,10) = 25 follows from the linear space with lines {0,1,4,12},{0,2,7},{0,6,15} (mod 25).)

Jump to d=4, d=6, d=8, d=10, d=12, d=14, d=16, d=18.

Acknowledgements