Moments of orthogonal arrays

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1 Thirteenth International Workshop on Algebraic and Combinatorial Coding Theory Moments of orthogonal arrays Peter Boyvalenkov, Hristina Kulina Pomorie, BULGARIA, June 15-21, 2012

2 Orthogonal arrays H(n, 2) - binary Hamming space of dimension n. an orthogonal array, or equivalently, a τ -design C in H(n, 2) is an M n matrix of a code C such that every M τ submatrix contains all ordered τ-tuples of H(τ, 2), each one exactly C 2 τ times as rows. the maximal τ with this property is called strength of the array. we consider H(n, 2) with the inner product x, y = 1 2d(x, y), n where d(x, y) is the Hamming distance between x and y.

3 Orthogonal arrays Denition 1. A code C H(n, 2) is a τ-design in H(n, 2) if and only if every real polynomial f (t) of degree at most τ and every point y H(n, 2) satisfy f ( x, y ) = f 0 C, (1) x C where f 0 is the rst coecient in the expansion f (t) = n Q (n) i (t) are the normalized Krawtchouk polynomials. i=1 f i Q (n) i (t),

4 Orthogonal arrays The identity C f (1) + x,y C,x y f ( x, y ) = C 2 f 0 + holds for every real polynomial f (t) = n r i = ( ) n i v ij (x) - Boolean functions i=1 n f i r i i=1 r i f i Q (n) i (t). ( v ij (x)) 2 j=1 x C (2)

5 hirteenth International Workshop on Algebraic and Combinatorial Coding Theory, June 2012, Pomorie, BULGARIA Moments of orthogonal arrays Denition 2. The numbers M i = 1 r i ( r i 2 v ij (x)), 1 i n, j=1 x C are called moments of C. C is OA of strength τ M i = 0 for i = 1, 2,..., τ. C is antipodal M i = 0 for every odd i. every moment M i is a rational number whose denominator is a divisor of the LCM of all denominators of the coecients of Q i (t).

6 Basic properties of the moments Main identity Theorem 1. C f (1) + x,y C,x y f ( x, y ) = C 2 f 0 + n f i M i Let C H(n, 2). We have M i = C + x,y C,x y Q i( x, y ), for every i = 1, 2,..., n. Proof. We set f (t) = Q i (t) in main identity and have f i = 1, f j = 0 for j i. Q i (1) = 1. i=1

7 Basic properties of the moments assume that C H(n, 2) is a τ-design t j = 1 + 2j n, j = 0, 1, 2,..., n k j = {(x, y) : x, y = t j }, j = 0, 1, 2,..., n Theorem 2. Let f (t) = n 1 j=0 (t t j) = n i=0 f iq (n) i (t). Then n f i M i = C (f (1) f 0 C ). i=τ+1

8 Basic properties of the moments Theorem 3. Let the polynomial f (t) = k i=0 f iq (n) i (t) of degree k = n 1 or n vanishes at all but one points t 0, t 1, t 2,..., t n 1, say f (t j ) 0. Then k f i M i = C (f (1) f 0 C ) + k j f (t j ). Example i=τ+1 n = 10, τ = 5, C = 192 k 0 A = {144, 146,..., 192} 0 k 9 r, where r = k

9 Orthogonal arrays and spherical codes H(n, 2) S n 1 : 1 1/ n, 0 1/ n in each coordinate. τ-design C H(n, 2) C S n 1 Theorem 4. If τ 3 then C has at least strength 3 as a spherical design. Moreover, all moments M i, i = 4, 5,..., τ, of C as a spherical design can be calculated. Proof. 1 the rst four (up to degree 3) Gegenbauer and Krawtchouk polynomials coincide 2 we set in main identity f (t) = t i for i = 4, 5,..., τ.

10 Orthogonal arrays and spherical codes Example Consider again the case n = 10, τ = 5 and C = it gives a spherical 3 design on S 9 with moments M 4 187, 671, M 5 = 0. 1 for the smallest k 0 = 144 we have unique solution for all k i, i = 1,..., 9 and this implies M 6 389, 366, M 7 55, 4352, M 8 326, 391, etc. 2 for k 0 = 192, we obtain an antipodal spherical code with M i = 0 for all odd i.

11 References I M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover, New York, P. Boyvalenhkov, H. Kulina, Computing distance distributions of orthogonal arrays, Proc. 12th Intern. Workshop ACCT, Novosibirsk, Russia, 2010, G. Fazekas, V. I. Levenshtein, On the upper bounds for code distance and covering radius of designs in polynomial metric spaces, J. Comb. Theory A, 70, 1995, A.S.Hedayat, N. J. A. Sloane, J. Stufken, Orthogonal Arrays: Theory and Applications, Spinger-Verlag, NY, 1999.

12 References II V. I. Levenshtein, Bounds for packings in metric spaces and certain applications, Probl. Kibern. 40, 1983, (in Russian). V. I. Levenshtein, Universal bounds for codes and designs, Chapter 6 ( ) in Handbook of Coding Theory, Eds. V.Pless and W.C.Human, Elsevier Science B.V., 1998.

13 Thank you for your attention!

Nonexistence of (9, 112, 4) and (10, 224, 5) binary orthogonal arrays

Nonexistence of (9, 112, 4) and (10, 224, 5) binary orthogonal arrays Tanya Marinova, Maya Stoyanova Faculty of Mathematics and Informatics, Sofia University St. Kliment Ohridski, Sofia, Bulgaria 15 th