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1.
Evidence is presented to suggest that, in three dimensions, spherical 6-designs withN points exist forN=24, 26,≥28; 7-designs forN=24, 30, 32, 34,≥36; 8-designs forN=36, 40, 42,≥44; 9-designs forN=48, 50, 52,≥54; 10-designs forN=60, 62, ≥64; 11-designs forN=70, 72,≥74; and 12-designs forN=84,≥86. The existence of some of these designs is established analytically, while others are given by very accurate numerical
coordinates. The 24-point 7-design was first found by McLaren in 1963, and—although not identified as such by McLaren—consists
of the vertices of an “improved” snub cube, obtained from Archimedes' regular snub cube (which is only a 3-design) by slightly
shrinking each square face and expanding each triangular face. 5-designs with 23 and 25 points are presented which, taken
together with earlier work of Reznick, show that 5 designs exist forN=12, 16, 18, 20,≥22. It is conjectured, albeit with decreasing confidence fort≥9, that these lists oft-designs are complete and that no other exist. One of the constructions gives a sequence of putative sphericalt-designs withN=12m points (m≥2) whereN=1/2t
2(1+o(1)) ast→∞. 相似文献
2.
Shiro Iwasaki 《组合设计杂志》1997,5(2):95-110
Let p be an odd prime number such that p − 1 = 2em for some odd m and e ≥ 2. In this article, by using the special linear fractional group PSL(2, p), for each i, 1 ≤ i ≤ e, except particular cases, we construct a 2-design with parameters v = p + 1, k = (p − 1)/2i + 1 and λ = ((p − 1)/2i+1)(p − 1)/2 = k(p − 1)/2, and in the case i = e we show that some of these 2-designs are 3-designs. Likewise, by using the linear fractional group PGL(2,p) we construct an infinite family of 3-designs with the same v k and λ = k(k − 2). These supplement a part of [4], in which we gave an infinite family of 3-designs with parameters v = q + 1, k = (q + 1)/2 = (q − 1)/2 + 1 and λ = (q + 1)(q − 3)/8 = k(k − 2)/2, where q is a prime power such that q − 1 = 2m for some odd m and q > 7. Some of the designs given in this article and in [4] fill in a few blanks in the table of Chee, Colbourn, and Kreher [2]. © 1997 John Wiley & Sons, Inc. 相似文献
3.
Jürgen Bierbrauer 《Graphs and Combinatorics》1992,8(3):207-224
We construct simple 3-designs and 4-designs of block-size 6 in the classical projective planesPG(2,q),q a power of 2. All of our designs are invariant under the projective groupPGL(3,q). Aside from several infinite series of 3-designs we get some relatively small designs of independent interest, e.g. designs
with parameters 4-(21, 6, 16) and 4-(73, 6, 330) defined in the planes of orders 4 and 8, respectively. 相似文献
4.
John Arhin 《Designs, Codes and Cryptography》2007,43(2-3):103-114
We introduce the notion of an unrefinable decomposition of a 1-design with at most two block intersection numbers, which is
a certain decomposition of the 1-designs collection of blocks into other 1-designs. We discover an infinite family of 1-designs
with at most two block intersection numbers that each have a unique unrefinable decomposition, and we give a polynomial-time
algorithm to compute an unrefinable decomposition for each such design from the family. Combinatorial designs from this family
include: finite projective planes of order n; SOMAs, and more generally, partial linear spaces of order (s, t) on (s + 1)2 points; as well as affine designs, and more generally, strongly resolvable designs with no repeated blocks.
相似文献
5.
Optimal constant weight covering codes and nonuniform group divisible 3-designs with block size four
Let K
q
(n, w, t, d) be the minimum size of a code over Z
q
of length n, constant weight w, such that every word with weight t is within Hamming distance d of at least one codeword. In this article, we determine K
q
(n, 4, 3, 1) for all n ≥ 4, q = 3, 4 or q = 2
m
+ 1 with m ≥ 2, leaving the only case (q, n) = (3, 5) in doubt. Our construction method is mainly based on the auxiliary designs, H-frames, which play a crucial role
in the recursive constructions of group divisible 3-designs similar to that of candelabra systems in the constructions of
3-wise balanced designs. As an application of this approach, several new infinite classes of nonuniform group divisible 3-designs
with block size four are also constructed. 相似文献
6.
Antonino Giorgio Spera 《组合设计杂志》1995,3(3):203-212
We study the action of the group PGL(m,A) on the projective space PG(m − 1,A) over a finite commutative local algebra A in order to construct a class of divisible designs, denoted by Dm(d,A), which is the classical one of 2-designs (of points and of flats of fixed projective dimension) in the case where A is a field. We also study the constructed divisible designs with particular care for the case where d = m − 1. © 1995 John Wiley & Sons, Inc. 相似文献
7.
In this work we give upper bounds for the Coulomb energy of a sequence of well separated spherical n-designs, where a spherical n-design is a set of m points on the unit sphere S
2 ⊂ ℝ3 that gives an equal weight cubature rule (or equal weight numerical integration rule) on S
2 which is exact for spherical polynomials of degree ⩽ n. (A sequence Ξ of m-point spherical n-designs X on S
2 is said to be well separated if there exists a constant λ > 0 such that for each m-point spherical n-design X ∈ Ξ the minimum spherical distance between points is bounded from below by .) In particular, if the sequence of well separated spherical designs is such that m and n are related by m = O(n
2), then the Coulomb energy of each m-point spherical n-design has an upper bound with the same first term and a second term of the same order as the bounds for the minimum energy
of point sets on S
2.
Dedicated to Edward B. Saff on the occasion of his 60th birthday. 相似文献
8.
Hiroyuki Nakasora 《Journal of Combinatorial Theory, Series A》2010,117(8):1289-1294
We give a construction of a 2-(mn2+1,mn,(n+1)(mn−1)) design starting from a Steiner system S(2,m+1,mn2+1) and an affine plane of order n. This construction is applied to known classes of Steiner systems arising from affine and projective geometries, Denniston designs, and unitals. We also consider the extendability of these designs to 3-designs. 相似文献
9.
10.
Reinhard Laue 《Designs, Codes and Cryptography》2004,32(1-3):277-301
A general group theoretic approach is used to find resolvable designs. Infinitely many resolvable 3-designs are obtained where each is block transitive under some PSL(2, p f ) or PGL(2, p f ). Some known Steiner 5-designs are assembled from such resolvable 3-designs such that they are also resolvable. We give some visualizations of Steiner systems which make resolvability obvious. 相似文献
11.
Cunsheng Ding 《Designs, Codes and Cryptography》2018,86(3):703-719
It has been known for a long time that t-designs can be employed to construct both linear and nonlinear codes and that the codewords of a fixed weight in a code may hold a t-design. While a lot of progress in the direction of constructing codes from t-designs has been made, only a small amount of work on the construction of t-designs from codes has been done. The objective of this paper is to construct infinite families of 2-designs and 3-designs from a type of binary linear codes with five weights. The total number of 2-designs and 3-designs obtained in this paper are exponential in any odd m and the block size of the designs varies in a huge range. 相似文献
12.
Bela Bajnok 《Geometriae Dedicata》1992,43(2):167-179
Spherical t-designs are Chebyshev-type averaging sets on the d-dimensional unit sphere S
d–1, that are exact for polynomials of degree at most t. The concept of such designs was introduced by Delsarte, Goethals and Seidel in 1977. The existence of spherical t-designs for every t and d was proved by Seymour and Zaslavsky in 1984. Although some sporadic examples are known, no general construction has been given. In this paper we give an explicit construction of spherical t-designs on S
d–1 containing N points, for every t,d and N,NN
0, where N
0 = C(d)t
O(d
3). 相似文献
13.
Michael Huber 《Journal of Algebraic Combinatorics》2007,26(2):183-207
Among the properties of homogeneity of incidence structures flag-transitivity obviously is a particularly important and natural
one. Consequently, in the last decades flag-transitive Steinert-designs (i.e. flag-transitive t-(v,k,1) designs) have been investigated, whereas only by the use of the classification of the finite simple groups has it been
possible in recent years to essentially characterize all flag-transitive Steiner 2-designs. However, despite the finite simple
group classification, for Steiner t-designs with parameters t > 2 such characterizations have remained challenging open problems for about 40 years (cf. [11, p. 147] and [12 p. 273],
but presumably dating back to around 1965). The object of the present paper is to give a complete classification of all flag-transitive
Steiner 4-designs. Our result relies on the classification of the finite doubly transitive permutation groups and is a continuation
of the author's work [20, 21] on the classification of all flag-transitive Steiner 3-designs.
2000 Mathematics Subject Classification. Primary 51E10 . Secondary 05B05 . 20B25 相似文献
14.
Béla Bajnok 《Graphs and Combinatorics》1998,14(2):97-107
Spherical t-designs are Chebyshev-type averaging sets on the d-sphere which are exact for polynomials of degree at most t. This concept was introduced in 1977 by Delsarte, Goethals, and Seidel, who also found the minimum possible size of such
designs, in particular, that the number of points in a 3-design on S
d must be at least . In this paper we give explicit constructions for spherical 3-designs on S
d consisting of n points for d=1 and ; d=2 and ; d=3 and ; d=4 and ; and odd or even. We also provide some evidence that 3-designs of other sizes do not exist. We will introduce and apply a concept from
additive number theory generalizing the classical Sidon-sequences. Namely, we study sets of integers S for which the congruence mod n, where and
, only holds in the trivial cases. We call such sets Sidon-type sets of strength t, and denote their maximum cardinality by s(n, t). We find a lower bound for s(n, 3), and show how Sidon-type sets of strength 3 can be used to construct spherical 3-designs. We also conjecture that our
lower bound gives the true value of s(n, 3) (this has been verified for n≤125).
Received: June 19, 1996 相似文献
15.
In this article, the existence of additive BIB designs is discussed with direct and recursive constructions, together with investigation of a property of resolvability. Such designs can be used to construct infinite families of BIB designs. In particular, we obtain a series of B(sn, tsm, λt (tsm ? 1) (sn‐m ? 1)/[2(sm ? 1)]) for any positive integer λ, such that sn (sn ? 1) λ ≡ 0 (mod sm (sm ? 1) and for any positive integer t with 2 ≤ t ≤ sn‐m, where s is an odd prime power. Connections between additive BIB designs and other combinatorial objects such as multiply nested designs and perpendicular arrays are discussed. A construction of resolvable BIB designs with v = 4k is also proposed. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 235–254, 2007 相似文献
16.
Etsuko Bannai 《Journal of Algebraic Combinatorics》2006,24(4):391-414
Neumaier and Seidel (1988) generalized the concept of spherical designs and defined Euclidean designs in ℝ
n
. For an integer t, a finite subset X of ℝ
n
given together with a weight function w is a Euclidean t-design if holds for any polynomial f(x) of deg(f)≤ t, where {S
i
, 1≤ i ≤ p} is the set of all the concentric spheres centered at the origin that intersect with X, X
i
= X∩ S
i
, and w:X→ ℝ> 0. (The case of X⊂ S
n−1 with w≡ 1 on X corresponds to a spherical t-design.) In this paper we study antipodal Euclidean (2e+1)-designs. We give some new examples of antipodal Euclidean tight 5-designs. We also give the classification of all antipodal Euclidean tight 3-designs, the classification of antipodal Euclidean tight 5-designs supported by 2 concentric spheres. 相似文献
17.
Recently, active research has been performed on constructing t-designs from linear codes over Z
4. In this paper, we will construct a new simple 3 – (2
m
, 7, 14/3 (2
m
– 8)) design from codewords of Hamming weight 7 in the Z
4-Goethals code for odd m 5. For 3 arbitrary positions, we will count the number of codewords of Hamming weight 7 whose support includes those 3 positions. This counting can be simplified by using the double-transitivity of the Goethals code and divided into small cases. It turns out interestingly that, in almost all cases, this count is related to the value of a Kloosterman sum. As a result, we can also prove a new Kloosterman sum identity while deriving the 3-design. 相似文献
18.
Michael Huber 《Journal of Algebraic Combinatorics》2007,26(4):453-476
As a consequence of the classification of the finite simple groups, it has been possible in recent years to characterize Steiner
t-designs, that is t-(v,k,1) designs, mainly for t=2, admitting groups of automorphisms with sufficiently strong symmetry properties. However, despite the finite simple group
classification, for Steiner t-designs with t>2 most of these characterizations have remained long-standing challenging problems. Especially, the determination of all
flag-transitive Steiner t-designs with 3≤t≤6 is of particular interest and has been open for about 40 years (cf. Delandtsheer (Geom. Dedicata 41, p. 147, 1992 and Handbook of Incidence Geometry, Elsevier Science, Amsterdam, 1995, p. 273), but presumably dating back to 1965).
The present paper continues the author’s work (see Huber (J. Comb. Theory Ser. A 94, 180–190, 2001; Adv. Geom. 5, 195–221, 2005; J. Algebr. Comb., 2007, to appear)) of classifying all flag-transitive Steiner 3-designs and 4-designs. We give a complete classification of all
flag-transitive Steiner 5-designs and prove furthermore that there are no non-trivial flag-transitive Steiner 6-designs. Both
results rely on the classification of the finite 3-homogeneous permutation groups. Moreover, we survey some of the most general
results on highly symmetric Steiner t-designs.
相似文献
19.
William J. Martin 《Designs, Codes and Cryptography》1999,16(3):271-289
A number of important families of association schemes—such as the Hamming and Johnson schemes—enjoy the property that, in
each member of the family, Delsarte t-designs can be characterised combinatorially as designs in a certain partially ordered set attached to the scheme. In this
paper, we extend this characterisation to designs in a product association scheme each of whose components admits a characterisation
of the above type. As a consequence of our main result, we immediately obtain linear programming bounds for a wide variety
of combinatorial objects as well as bounds on the size and degree of such designs analogous to Delsarte's bounds for t-designs in Q-polynomial association schemes. 相似文献
20.
Vladimir D. Tonchev 《组合设计杂志》1996,4(3):203-204
The minimum weight codewords in the Preparata code of length n = 4m are utilized for the construction of an infinite family of Steiner S(4, {5, 6}, 4m + 1) designs for any m ≥ 2. © 1996 John Wiley & Sons, Inc. 相似文献