Orthogonal arrays of strength 3

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<ul><li><p>- Orthogonal Arrays of Strength 3 </p><p>Donald 1. Kreher Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 4993 7 - 7295 </p><p>ABSTRACT </p><p>A new construction for orthogonal arrays of strength 3 is given. 0 1996 John Wiley &amp; Sons, Inc. </p><p>1 . INTRODUCTION </p><p>An orthogonal array of size N, degree k, order s, and strength t is a k by N array with entries from a set of s 2 2 symbols, having the property that in every t by N subarray, every t by I column array appears the same number A = $ times. We denote such an array by OA,(t, k , s). The parameter A is called the index of the array. </p><p>Existence results for orthogonal arrays of strength greater than or equal to three are few and far between. A summary of these results is given in [2]. For t = 3, the best known upper bound on k for fixed A and s is the Bose-Bush bound [3]: </p><p>A improvement is obtained when A - 1 = b (mod s - I ) and 1 5 b 5 s - 1: </p><p>1. A s 2 - 1 J1 + 4s(s - 1 - b ) - (2s - 2b - 1) k 5 k 1 - 1 2 In [4] orthogonal arrays of strength 2 were constructed by applying a 2-transitive group </p><p>to certain matrices arising from resolvable linear spaces. This construction motivates the use of 3-transitive groups and resolvable 3-designs to form orthogonal arrays of strength 3. Indeed such a construction is possible and in Section 2 we give some new families of strength 3 orthogonal arrays. </p><p>0 1996 John Wiley &amp; Sons, Inc. Journal of Cornbinatorial Designs, Vol. 4, No. 1 (1996) </p><p>CCC 1063-8539/96/010067-03 </p><p>67 </p></li><li><p>68 KREHER </p><p>2. A NEW CONSTRUCTION </p><p>A t - (v , k , A) design is a pair ( X , 3) where X is a v-element set of points and 23 is a collection of k-element subsets of X called blocks such that every t-element subset of points is contained in exactly h blocks. A parallel class in a t-(v, k , A) design ( X , 3) with v = wk is a collection of w disjoint blocks. If 3 can be partitioned into parallel classes we say that the design is resolvable. There are r = A( y?yl)/( ) parallel classes in such a partition. Thus if the t - ( v , k , A) design ( X , 3) is resolvable we can construct a w by r matrix A whose entries are the blocks in 3 and whose columns are parallel classes. We will call A a resolution matrix for ( X , 23). </p><p>It is well known that for i + j 5 t the number b{ of blocks containing a given set of i points and none of a disjoint set of j points is given by </p><p>A permutation group G on R is said to be 3-transitive if for every pair of 3-tuples (a , , a2, CQ) and (PI , / 3 2 , /33) there is g E G that sends ai to Pi, i = 1,2,3. If there is a unique such g E G for every pair of 3-tuples we say that G acts sharply 3-transitively on a. In this case ICl = n3 - n where n + 1 = In]. The only sharply 3-transitive groups have n a prime power and for every such n , PGLZ(n) acts sharply 3-transitive on the projective line GF(n) U {m}. </p><p>Theorem 2.1. Let G act 3-transitively on the ( n + l)-element set R and let m(n3 - n ) be the order of G. I f a resolvable 3-(wk, k , A) design ( X , 3) exists such that n = ( r - h)/b;? - 2 with w - 1 5 n 5 b:/h, then an OAmb;(a-lI(3, wk,n + 1 ) also exists. ProoJ: Let ( X , 3) be a resolvable 3-(wk, k , A) design with resolution matrix A and define M to be the v by r matrix with entries in R = { w l , w2, . . . , w , + I } given by </p><p>M [ i , j ] = W k if and only if x , E A [ k , j ] ; where X = {XI, x2,. . . , x v } . If g E G, then MR denotes the matrix obtained by applying g to each of the entries in M . Lastly let C be a v by (bi(n - 1) - An(n - l ) )m(n + 1) matrix with rn(bi(n - 1) - An(n - 1)) constant columns with value w for each w in R . We claim that the array </p><p>where G = {gl,g2,. . . ,g l~ l } , is an OAmh;(,-,)(3, v , n + 1). To see this consider any 3 rows. On these rows there are five types of columns [i]: (i) x = y = z , (ii) x = y f z , (iii) x = z # y , (iv) y = z # x and (v) x # y # z # x. </p><p>A column of type (i) occurs on these three rows A times in M and rn(bi(n - 1) - hn(n - 1)) times in C. Thus columns of this type occur nfl h + rn(bi(n - 1) - hn(n - 1 ) ) = mbi (n - 1) times in array (*) for each such choice of x , y , and z . </p><p>A column of type (ii) (or of type (iii) or type (iv)) occurs bi times in M and never in C. Thus such a column occurs (n+l)n b: = mb:(n - 1) times in array (*) for each such choice of x, y , and z . </p><p>A column of type (v) occurs r - A - 3bi times in M and never in C. Thus such a IGI 1 1 column occurs ( n + l ) n ( n - l ) ( r - A - 3b2) = m(r - A - 3bi) = mbz(n - 1) times in </p><p>0 </p><p>1 </p><p>[M", C ] = [MR' , MK*, . . . , M g l G I , C ] , ("1 </p><p>IGl </p><p>IGI </p><p>array (*) for each such choice of x , y , and z . </p></li><li><p>ORTHOGONAL ARRAYS STRENGTH 3 69 </p><p>The best results are of course obtained when the group G in Theorem 2.1 is sharply 3-transitive. In this case m = 1 and n is some prime power q. Applying this to resolvable Steiner quadruple systems we obtain Corollary 2.2. </p><p>Corollary 2.2. OAgY+l)(,-1)/2(3,3q + 5 , q + I ) , except possibly for q = 197 or 773. </p><p>Let q be a prime power, q = 1,5,9 (mod 12). Then there exists an </p><p>Prooj Set v = 3q + 5, then v = 8 (mod 12) and there exists a resolvable SQS(v) (see [5] ) , i.e., a resolvable 3-(v, 4, 1) design with w = = 7, r = (v - 1) (v - 2)/6 = (39 + 4)(9 + 1)/2 and b2 = 2 - 7. Thus w - 1 5 q 5 b: holds and </p><p>0 </p><p>3q+5 </p><p>I u-4 - 3q+l </p><p>furthermore q = ( r - l ) / b l - 2. The result now follows from Theorem 2.1. </p><p>The only other block size we know of for which general existence results are known are the resolvable 3-(v, 3, 1 ) designs. These exist according to Baranyais result [ I ] if and only if v = 0 (mod 3). </p><p>Corollary 2.3. Let q be a prime power, q = 1 (mod 3). Then there exists an OA(2q+i)(q-i)(3,2q + 4,q + 1). </p><p>ProoJ: Set v = 2q + 4, then v = 0 (mod 3) and there exists by [ l ] a resolvable 3-(v,3, 1) design with w = 5 = v , r = ( v - l ) (v - 2)/2 = (2q + 3)(q + I), and b: = v - 3 = 2q + 1. Thus w - 1 5 q 5 b: holds and furthermore q = </p><p>0 ( r - l)/bi - 2. The result now follows from Theorem 2.1. </p><p>ACKNOWLEDGMENTS </p><p>The author is grateful to C. J. Colbourn for his careful reading of an earlier version of this article. </p><p>REFERENCES </p><p>[ l ] Z. Baranyai, On the,factorization of the complete uniform hypergraph, COIL Math. SOC. Janos Bolyai 10 (1975), 91-108. </p><p>[2] J. Bierbrauer and C. J. Colbourn, Orthogonal arrays of strength more than two, The CRC handbook of combznatorial designs. C. J. Colbourn and J. H. Dinitz (Editors), CRC Press, Boca Raton, 1996. </p><p>[3] R.C. Bose and K. A. Bush, Orthogonal arrays of strength two and three, Ann. Math. Stat. 23 (1952), 508-524. </p><p>[4j C. J. Colbourn and D. L. Kreher, Concerning difference matrices, Designs, Codes and Cryp- tography, to appear. </p><p>[5] A. Hartman and K. T. Phelps, Steiner quadruple systems, Contemporary design theory: A collection of surveys, J. H. Dinitz and D. R. Stinson (Editors), Wiley, New York, 1992. </p><p>Received April 7, 1995 Accepted June 26, 1995 </p></li></ul>