There is no 2‐(22, 8, 4) block design

In this article, we show that a 2‐(22,8,4) design does not exist. This result was obtained by a computer search. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 262–267, 2007     

There exists no (15,5,4) RBIBD†*

An (n, M, d)_q code is a q‐ary code of length n, cardinality M, and minimum distance d. We show that there exists no (15,5,4) resolvable balanced incomplete block design (RBIBD) by showing that there exists no (equidistant) (14,15,10)₃ code. This is accomplished by an exhaustive computer search using an orderly algorithm combined with a maximum clique algorithm. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 357–362, 2001     

There exists no (15,5,4) RBIBD

An (n, M, d)_q code is a q‐ary code of length n, cardinality M, and minimum distance d. We show that there exists no (15,5,4) resolvable balanced incomplete block design (RBIBD) by showing that there exists no (equidistant) (14,15,10)₃ code. This is accomplished by an exhaustive computer search using an orderly algorithm combined with a maximum clique algorithm. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 227–232, 2001     

There is no (95, 40, 12, 20) strongly regular graph

We show that there is no (95, 40, 12, 20) strongly regular graph and, consequently, there is no (96, 45, 24, 18) strongly regular graph, no nontrivial regular two‐graph on 96 vertices, and no partial geometry pg(4, 9, 2). The main idea of the result is based on the star complement technique and requires a moderate amount of computation.     

区传递的$2-(v, k, 1)$ 设计与单群$E_6(q)$

This article is a contribution to the study of block-transitive automorphism groups of 2-（v,k,1） block designs. Let D be a 2-（v,k,1） design admitting a block-transitive, pointprimitive but not flag-transitive automorphism group G. Let kr = （k,v-1） and q = pf for prime p. In this paper we prove that if G and D are as above and q （3（krk-kr ＋ 1）f）1/3, then G does not admit a simple group E6（q） as its socle.     

There Are Integral Heptagons, no Three Points on a Line, no Four on a Circle

We give two configurations of seven points in the plane, no three points in a line, no four points on a circle with pairwise integral distances. This answers a famous question of Paul Erdős.     

New type of block design

We are already familiar with (υ, k, λ)‐difference sets and (υ, k, λ)‐designs. In this paper, we will introduce a new class of difference sets and designs: (υ, k, [λ₁, λ₂, … , λ_m])‐difference sets and (υ, k, [λ₁,λ₂, … , λ_m])‐designs. We will mainly study designs with a relationship we call λ‐equivalence and use them to produce other designs. Some existence or nonexistence theorems will be given. © 2002 Wiley Periodicals, Inc. J Combin Designs 11: 1–23, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10031     

Existence of (q,k, 1) difference families with q a prime power and k = 4, 5

The existence of a (q, k, 1) difference family in GF(q) has been completely solved for k = 3. For k = 4, 5 partial results have been given by Bose, Wilson, and Buratti. In this article, we continue the investigation and show that the necessary condition for the existence of a (q, k, 1) difference family in GF(q), i.e., q ≡ 1 (mod k(k − 1)) is also sufficient for k = 4, 5. For general k, Wilson's bound shows that a (q, k, 1) difference family in GF(q) exists whenever q ≡ 1 (mod k(k − 1)) and q > [k(k − 1)/2]^k(k−1). An improved bound on q is also presented. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 21–30, 1999     

Self‐dual codes and the (22,8,4) balanced incomplete block design

An Erratum has been published for this article in Journal of Combinatorial Designs 14: 83–83, 2006 . We enumerate a list of 594 inequivalent binary (33,16) doubly‐even self‐orthogonal codes that have no all‐zero coordinates along with their automorphism groups. It is proven that if a (22,8,4) Balanced Incomplete Block Design were to exist then the 22 rows of its incident matrix will be contained in at least one of the 594 codes. Without using computers, we eliminate this possibility for 116 of these codes. © 2005 Wiley Periodicals, Inc. J Combin Designs.     

Efficiency of connected binary block designs when a single observation is unavailable

In this paper the problem of finding the design efficiency is considered when a single observation is unavailable in a connected binary block design. The explicit expression of efficiency is found for the resulting design when the original design is a balanced incomplete block design or a group divisible, singular or semiregular or regular with ₁>0, design. The efficiency does not depend on the position of the unavailable observation. For a regular group divisible design with ₁>0, the efficiency depends on the position of the unavailable observation. The bounds, both lower and upper, on the efficiency are given in this situation. The efficiencies of designs resulting from a balanced incomplete block design and a group divisible design are in fact high when a single observation is unavailable.The work of the first author is sponsored by the Air Force Office of Scientific Research under Grant AFOSR-90-0092.On leave from Indian Statistical Institute, Calcutta, India. The work of the third author was supported by a grant from the CMDS, Indian Institute of Management, Calcutta.     

There are 1239 Steiner Triple Systems STS(31) of 2-rank 27

A computer search over the words of weight 3 in the code of blocks of a classical Steiner triple system (STS) on 31 points is carried out to classify all STS(31) whose incidence matrix has 2-rank equal to 27, one more than the possible minimum of 26. There is a total of 1239 nonisomorphic STS(31) of 2-rank 27.     

Decomposable super‐simple NRBIBDs with block size 4 and index 6

Necessary conditions for the existence of a super‐simple, decomposable, near‐resolvable ‐balanced incomplete block design (BIBD) whose 2‐component subdesigns are both near‐resolvable ‐BIBDs are (mod ) and . In this paper, we show that these necessary conditions are sufficient. Using these designs, we also establish that the necessary conditions for the existence of a super‐simple near‐resolvable ‐RBIBD, namely (mod ) and , are sufficient. A few new pairwise balanced designs are also given.     

Optimum covariate designs in a binary proper equi-replicate block design set-up

The choice of covariates values for a given block design attaining minimum variance for estimation of each of the regression parameters of the model has attracted attention in recent times. In this article, we consider the problem of finding the optimum covariate design (OCD) for the estimation of covariate parameters in a binary proper equi-replicate block (BPEB) design model with covariates, which cover a large class of designs in common use. The construction of optimum designs is based mainly on Hadamard matrices.     

Enumeration of resolvable 2‐(10, 5, 16) and 3‐(10, 5, 6) designs

A backtracking over parallel classes with a partial isomorph rejection (PIR) is carried out to enumerate the resolvable 2‐(10,5,16) designs. Computational results show that the inclusion of PIR reduce substantially the CPU time for the enumeration of all designs. We prove first some results, which enable us to restrict the search space. Since every resolvable 2‐(10,5,16) design is also a resolvable 3‐(10,5,6) design and vice versa, the latter designs are also enumerated. There are 27, 121, 734 such designs with automorphism groups whose order range from 1 to 1,440. From these, designs 2,006,690 are simple. © 2004 Wiley Periodicals, Inc.     

Some (17q, 17, 2) and (25q, 25, 3) BIBD Constructions

We give the explicit construction of a regular (17q, 17, 2)-BIBD for any prime power q 17 (mod 32) such that 2 is not a 4th power in GF(q) and the explicit construction of a regular (25q, 25, 3)-BIBD for any prime power q 25 (mod 48) such that and +3 are non-squares in GF(q).     

There is no triangulation of the torus with vertex degrees 5, 6, ... , 6, 7 and related results: geometric proofs for combinatorial theorems

There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be bicolored. Similar statements hold for 4,8-triangulations and 2,6-quadrangulations. We prove these results, of which the first two are known and the others seem to be new, as corollaries of a theorem on the holonomy group of a euclidean cone metric on the torus with just two cone points. We provide two proofs of this theorem: One argument is metric in nature, the other relies on the induced conformal structure and proceeds by invoking the residue theorem. Similar methods can be used to prove a theorem of Dress on infinite triangulations of the plane with exactly two irregular vertices. The non-existence results for torus decompositions provide infinite families of graphs which cannot be embedded in the torus.     

Existence of (q, 7, 1) difference families with q a prime power

The existence of a (q,k, 1) difference family in GF(q) has been completely solved for k = 3,4,5,6. For k = 7 only partial results have been given. In this article, we continue the investigation and use Weil's theorem on character sums to show that the necessary condition for the existence of a (q,7,1) difference family in GF(q), i.e. q ≡ 1; (mod 42) is also sufficient except for q = 43 and possibly except for q = 127, q = 211, q = 31⁶ and primes q∈ [261239791, 1.236597 × 10¹³] such that in GF(q). © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 126–138, 2002; DOI 10.1002/jcd.998     

New coverings of t-sets with (t + 1)-sets

The minimum number of k-subsets out of a v-set such that each t-set is contained in at least one k-set is denoted by C(v, k, t). In this article, a computer search for finding good such covering designs, leading to new upper bounds on C(v, k, t), is considered. The search is facilitated by predetermining automorphisms of desired covering designs. A stochastic heuristic search (embedded in the general framework of tabu search) is then used to find appropriate sets of orbits. A table of upper bounds on C(v, t + 1, t) for v 28 and t 8 is given, and the new covering designs are listed. © 1999 John Wiley & Sons, Inc. J. Combin Designs 7: 217–226, 1999