The construction given in Kreher, J Combin Des 4 (1996) 67 is extended to obtain new infinite families of orthogonal arrays of strength 3. Regular 3-wise balanced designs play a central role in this construction

Colbourn, Charles J.

Kreher, D. L.

McSorley, John P.

Stinson, D. R.

OpenSIUC

Southern Illinois University CarbondaleOpenSIUCArticles and Preprints Department of Mathematics2-2002Orthogonal Arrays of Strength Three from Regular3-Wise Balanced DesignsCharles J. ColbournUniversity of VermontD. L. KreherMichigan Technological UniversityJohn P. McSorleySouthern Illinois University Carbondale, mcsorley60@hotmail.comD. R. StinsonUniversity of WaterlooFollow this and additional works at: http://opensiuc.lib.siu.edu/math_articlesPart of the Discrete Mathematics and Combinatorics Commons, and the Statistics andProbability CommonsPublished in Journal of Statistical Planning and Inference, 100(2), 191-195.This Article is brought to you for free and open access by the Department of Mathematics at OpenSIUC. It has been accepted for inclusion in Articlesand Preprints by an authorized administrator of OpenSIUC. For more information, please contact opensiuc@lib.siu.edu.Recommended CitationColbourn, Charles J., Kreher, D. L., McSorley, John P. and Stinson, D. R. "Orthogonal Arrays of Strength Three from Regular 3-WiseBalanced Designs." (Feb 2002).Orthogonal arrays of strength three fromregular 3-wise balanced designsCharles J. ColbournComputer ScienceUniversity of VermontBurlington, Vermont 05405D. L. KreherMathematical SciencesMichigan Technological UniversityHoughton, Michigan 49931-1295J. P. McSorleyDept. of MathematicsSouthern Illinois UniversityCarbondale, IllinoisD. R. StinsonCombinatorics and OptimizationUniversity of WaterlooWaterloo, OntarioCANADA N2L 3G1AbstractThe construction given in [4] is extended to obtain new infinitefamilies of orthogonal arrays of strength 3. Regular 3-wise balanceddesigns play a central role in this construction.11 IntroductionAn orthogonal array of size N , with k constraints (or of degree k), s levels(or of order s), and strength t, denoted OA(N, k, s, t), is a k ×N array withentries from a set of s ≥ 2 symbols, having the property that in every t×Nsubmatrix, every t×1 column vector appears the same number λ = Nsttimes.The parameter λ is the index of the orthogonal array. An OA(N, k, s, t) isalso denoted by OAλ(t, k, s); in this notation, if t is omitted it is understoodto be 2, and if λ is omitted it is understood to be 1. A parallel class in anOAλ(t, k, s) is a set of s columns so that each row contains all s symbolswithin these s columns. A resolution of the orthogonal array is a partition ofits columns into parallel classes, and an OA with such a resolution is termedresolvable. An OAλ(t, k, n) is class-regular or regular if some group Γ of ordern acts regularly on the symbols of the array. A class-regular OAλ(t, k, n) isresolvable. See [1] for a brief survey on orthogonal arrays of strength at least3.In [4] a construction for orthogonal arrays of strength 3 is given thatstarts from resolvable 3-(v, k, λ) designs and uses 3-transitive groups. Theconditions on the resolvable 3-design ingredient can be relaxed and a moregeneral theorem can be stated using a resolvable set system (X,B) such that:1. the number of blocks containing three points x, y, z ∈ X, x 6= y 6= z 6=x, is a constant λ3 that does not depend on the choice of x, y, z;2. the number of blocks containing two points x, y ∈ X but disjoint froma third point z ∈ X, x 6= y 6= z 6= x, is a constant b12 that does notdepend on the choice of x, y, z.We allow (X,B) to contain blocks of any size, including 1, 2, 3 and |X|.If x, y ∈ X, x 6= y, then the number of blocks containing x and y isλ2 = b12 + λ3 independent of the choice of x and y. These set systems neednot be balanced for points. For example, the set system{{1, 2, 3, 4}, {1, 2,∞}, {1, 3,∞}, {1, 4,∞}, {2, 3,∞}, {2, 4,∞},{3, 4,∞}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}has λ3 = 1, λ2 = 3, points 1, 2, 3, 4 each occur in 7 blocks, but∞ in 6 blocks.If resolvability is required, then every point must occur in the same number2λ1 = r of blocks. Kageyama [3] called a t-wise balanced design that is alsoi-balanced for each i < t a regular t-wise balanced design.Theorem 1.1 (Kageyama [3]) Let (X,B) be a regular 3-wise balanced designwith at most two distinct block sizes, k1, k2. Then the subdesignsBki = {B ∈ B : |B| = ki}are each 2-designs, i = 1, 2.If λ3 6= 0, and the block size is constant, then such a design is a 3-design. But these conditions are not necessary. For example, the edges ofthe complete graph Kv when v is even have λ3 = 0, λ2 = 1, and λ1 = v − 1.Furthermore Kv has a 1-factorization and so this set system is resolvable.A 3-(v,K,Λ) design of width w is a pair (X,B) where X is a v-elementset of points and B is a collection of subsets of X called blocks satisfying:1. the size of every block is in K;2. Λ = [λ1, λ2, λ3] and every i-element subset is in λi blocks, i = 1, 2, 3and3. the blocks can be partitioned into λ1 resolution classes using no morethan w blocks in any one class.The revised theorem is then:Theorem 1.2 Let G act 3-transitively on the (n+ 1)-element set Ω and letm(n3 − n) be the order of G. If a 3-(v,K,Λ) design of width w exists suchthat n = (λ1 − λ3)/(λ2 − λ3)− 2 with w ≤ n+ 1 and λ3(n+ 1) ≤ λ2, then aresolvable OAm(n−1)(λ2−λ3)(3, v, n+ 1) also exists.Proof: This is exactly the same as in the proof of Theorem 2.1 in [4].Resolvability of the OA follows from the transitivity of the action of G (thiswas not pointed out in [4]). 232 Applications of the ConstructionAs in [4] we apply Theorem 1.2 with sharply 3-transitive groups, so thatn = q is a power of a prime, and m = 1.Lemma 2.1 Let q be an odd prime power. Then there exists an OAq−1(3, q+3, q + 1).Proof: Set v = q+3. Then a 1-factorization of Kv is a 3-(v, {2}, [v−1, 1, 0])design of width w = v/2 = (q + 3)/2. Then w − 1 ≤ q and (λ1 − λ3)/(λ2 −λ3) − 2 = (v − 1 − 0)/(1 − 0) − 2 = v − 3 = q, and the result follows fromTheorem 1.2. 2Theorem 2.2 For all x ≥ 2 there exists a 3-(4x, {2, 4, 2x}, [1+2(x−1)(2x−1), 2x− 1, 1]) design of width 2(x− 1).Proof: The construction is essentially the doubling construction for Steinerquadruple systems (see [2], for example). Let A and B be two disjoint sets ofsize 2x and let {a1, a2, . . . , a2x−1} and {b1, b2, . . . , b2x−1} be one factorizationsof A and B respectively. Take as blocks1. the sets A and B each of size 2x;2. the x2 4-element subsets in each of the 2x − 1 families: {α ∪ β : α ∈ai, β ∈ bi}, for all i = 1, 2, . . . , 2x− 1; and3. all the 2(2x2)pairs that are either in A or in B each repeated x − 2times.Arrange the blocks into resolution classes with at most 2(x− 1) blocks each,to produce the required design. 2If we use PGL2(q) and the 3-(v,K,Λ) designs constructed in Theorem 2.2as ingredients to Theorem 1.2 thenq =(λ1 − λ3)(λ2 − λ3) − 2=(1 + 2(x− 1)(2x − 1)− 1)2(x− 1) − 2= 2x− 3.Consequently, the following arrays are obtained.4Lemma 2.3 An OAq2−1(3, 2(q + 3), q + 1) exists for every odd prime powerq.Another way to construct 3-(v,K,Λ) designs is given next.Theorem 2.4 If there exists an OAµ(3, n, yw), then there exists a 3-(n,K,Λ)design of width w withΛ = [µy3w3, µy3w2, µy3w].Proof: Let A be an OAµ(3, n, v). We think of A as an n×µv3 array definedon symbol set X, |X| = yw. Partition X into subsets Yi, i = 1, 2, . . . , wwith each |Yi| = y. We define a 3-(n,K,Λ) design of width w with wµv3blocks, as follows: for i = 1, 2, . . . , w and each column j of A, define a blockBi,j = {h : A[h, j] ∈ Yi}. Then ({1, . . . , n}, {Bij}) is a 3-(n,K,Λ) design ofwidth w with Λ = [µy3w3, µy3w2, µy3w]. 2If Di is a 3-(v,Ki,Λi) design of width wi, for i = 1, 2, . . . , n then fornatural numbers αi, the union with repeated blocks ∪ni=1αiDi of αi copiesof Di, 1, 2, . . . , n is a 3-(v,∪iKi,∑i αiΛi) design of width w = maxiwi. Weillustrate this idea next.Theorem 2.5 Let q be a prime power and choose integers a, b,m ≥ 1 suchthat1. q + 3 = m(a+ b);2. ma ≥ 4;3. m(a+ 2b) ≡ 0 (mod 4); and4. (m(a+ 2b)− 4)/4 ≡ 0 (mod b).Then an OA(a+ b)4b(q − 1 +mb)(q − 1)(3,(a+ 2ba+ b)(q + 3), q + 1) exists.5Proof: Let x = m(a + 2b)/4. Then x ≥ 2 is a positive integer and byTheorem 2.2 there is 3-(4x, {2, 4, 2x}, [1 + 2(x− 1)(2x− 1), 2x− 1, 1]) designD1 of width 2(x − 1). Also the edges of the complete graph K4x (see theproof of Corollary 2.1) form a 3-(4x, {2}, [4x − 1, 1, 0]) design D2 of widthw = 2x. Take one copy of D1 andab(x− 1) copies of D2 to form a3-(4x, {2, 4, 2x},[1 +(x− 1)(4(a+ b)x− (a+ 2b))b,(x− 1)(a+ 2b)b, 1])design D of width w = 2x. The conditions of Theorem 1.2 are satisfied. 2The main applications of Theorem 1.2 rest on finding suitable regular3-wise balanced designs. We have illustrated in this section the applicationsof some easily constructed designs of this type, but expect that further con-structions can lead to more existence results for orthogonal arrays.AcknowledgmentsThe authors’ research is supported as follows: ARO grant DAAG55-98-1-0272 (Colbourn), NSA grant MDA904-97-1-0072 (Kreher) and NSA grantMDA904-96-1-0084 (Stinson).References[1] J. Bierbrauer and C.J. Colbourn, Orthogonal arrays of strength morethan two, in: CRC Handbook of Combinatorial Designs, CRC Press,1996, 179-184.[2] A. Hartman and K.T. Phelps, Steiner quadruple systems, in: Contempo-rary Design Theory: A Collection of Surveys (J.H. Dinitz, D.R. Stinson;eds.) John Wiley & Sons, New York, 1992, 205-240.[3] S. Kageyama, A property of t-wise balanced designs, Ars Combinatoria,31, (1991), 237-238.[4] D.L. Kreher, Orthogonal Arrays of Strength 3, Journal of CombinatorialDesigns 4 (1996), 67-69.6

Orthogonal Arrays of Strength Three from Regular 3-Wise Balanced Designs

The construction given in Kreher, J Combin Des 4 (1996) 67 is extended to obtain new infinite families of orthogonal arrays of strength 3. Regular 3-wise balanced designs play a central role in this construction. © 2002 Elsevier Science B.V. All rights reserved

Colbourn, C. J.

Mcsorley, J. P.

Michigan Technological University

Orthogonal arrays of strength three from regular 3-wise balanced designs

http://opensiuc.lib.siu.edu/cgi/viewcontent.cgi?article=1039&context=math_articles

Orthogonal Arrays of Strength Three from Regular 3-Wise Balanced Designs

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