A Tutorial on Orthogonal Arrays: Constructions, Bounds and Links to Error-correcting Codes

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1 A Tutorial on Orthogonal Arrays: onstructions, ounds and Links to Error-correcting odes Douglas R. Stinson David R. heriton School of omputer Science University of Waterloo D.R. Stinson 1

2 Talk Outline ffl OA(k; n) and MOLS ffl OA (k; n), Plackett-urman bound, constructions, affine designs OA (t; k; n), Rao bound, ush bounds, ierbrauer-friedman bound, ffl constructions ffl linear codes and orthogonal arrays, duality nonlinear codes and orthogonal arrays, linear programming bounds, ffl duality ffl orthogonal arrays with a non-prime-power number of symbols D.R. Stinson 2

3 Definition Let k 2 and n 1 be integers. An orthogonal array OA(k; n) is an n 2 k array, A, with entries from a set X of cardinality n such that, within any two columns of A, every ordered pair of symbols from X occurs in exactly one row of A. An OA(s + 2; n) is equivalent to a set of s mutually orthogonal latin squares (MOLS) of order n. Every row of the OA corresponds to a particular cell in each of the latin squares. D.R. Stinson 3

4 Example 2 MOLS of order 3 OA(3; 4) D.R. Stinson 4

5 Example onsider the cells in row 1, column 1: 2 MOLS of order 3 OA(3; 4) D.R. Stinson 5

6 Example onsider the cells in row 3, column 2: 2 MOLS of order 3 OA(3; 4) D.R. Stinson 6

7 Some asic Results Theorem 1 Let n 2. If there is an OA(k; n),thenk» n + 1. Theorem 2 Let n 2. AnOA(n + 1; n) is equivalent to any of the following designs: 1. n 1 MOLS of order n; 2. an affine plane of order n; 3. a projective plane of order n. D.R. Stinson 7

8 First Generalization: Higher Let k 2, n 1 and 1 be integers. An orthogonal array OA (k; n) is a n 2 k array, A, with entries from a set X of cardinality n such that, within any two columns of A, every ordered pair of symbols from X occurs in exactly rows of A. Theorem 3 [Plackett-urman ound (1946)] Let n 2. If there is an OA (k; n),then n2 1 k» 1 : n D.R. Stinson 8

9 a i = k( n 1) a i (a i 1) = k(k 1)( 1) a i 2 overing Arrays Workshop May 14 16, 2006 First Proof: Variance Method Relabel the symbols so the last row of A is Define N = n 2. For 1» i» N 1,leta i denote the number of 1 s in row i of A. Then X N 1 i=1 X N 1 i=1 X N 1 = k(k( 1) + (n 1)): Define i=1 a = 1) k( n 1 : N D.R. Stinson 9

10 (a i a) 2 overing Arrays Workshop May 14 16, 2006 Then First Proof: Variance Method (cont.) X N 1 0» i=1 X N 1 X N 1 = a i + a 2 (N 1) a i 2 2a i=1 i=1 Therefore, = k(k( 1) + (n 1)) k2 ( n 1) 2 n 2 1 1) 2» ( n 2 1)(k( 1) + (n 1)): k( n This simplifies to yield n2 1 k» 1 : n D.R. Stinson 10

11 `ffi 2 `w + `ffi : w overing Arrays Workshop May 14 16, 2006 Second Proof: Applying the Johnson ound Again, relabel the symbols so the last row of A is Delete the last row of A, and replace every symbol x 6= 1 by 0. onsider the code formed by the columns of the resulting array. is a binary code consisting of k vectors of length ` = n 2 1, each of which has constant hamming weight w = n 1, such that the hamming distance between any two distinct codewords is equal to 2ffi = 2 (n 1). Apply the second Johnson bound for constant weight binary codes: jj» Hence, k» 2 1) (n 1) ( n n2 1) 2 ( n 2 1)( n 1) + ( n 2 1) (n 1) = 1 n 1 : ( n D.R. Stinson 11

12 Third Proof: Linear Algebra Approach Assume that A is defined on the symbols in Z n. Let the columns of A be denoted 1 ; : : : ; k.let 0 be the column vector of 0 s. For» m» n 1, construct m j from j by multiplying every entry by m 1 (modulo n). Let! = e 2ßi=n and define ffi : Z n! by the rule ffi(s) =! s. onsider the set of 1 + k(n 1) vectors D = fffi( 0 )g [ fffi(m j ) : 1» m» n 1; 1» j» kg: It can be shown that h; Di = 0 for all ; D 2 D, 6= D,whereh ; i denotes the hermitian inner product of two (complex-valued) vectors. Since D consists of mutually orthogonal vectors, they are linearly independent. Hence, we have a set of 1 + k(n 1) linearly independent vectors in N, and it follows that 1 + k(n 1)» N (= n 2 ). D.R. Stinson 12

13 IDs and Resolvable IDs A (v; k; )-ID (balanced incomplete block design)isapair(x; ), where X is a set of v elements called points and is a collection of k-subsets of X (called blocks) such that every unordered pair of points occurs in exactly blocks. The number of blocks is denoted by b,and each point occurs in exactly r blocks, where (v 1) = r(k 1) and bk = vr: (v; k; )-ID,say(X; A ),isresolvable if can be partitioned r into parallel classes, where each parallel consists v=k of disjoint blocks. D.R. Stinson 13

14 Equality in the Plackett-urman ound A resolvable (v; `; )-ID is affine resolvable if any two blocks from different parallel classes intersect in μ points. ose showed that an affine resolvable ID has v = μn 2, ` = μn and = (μn 1)=(n 1) for integers μ and n. Furthermore, there are n blocks in each parallel class and the number of parallel classes is r = (μn 2 1)=(n 1). Sucha design is denoted AD(n; μ). Theorem 4 An AD(n; μ) is equivalent to An OAμ(k; n) where k = (μn 2 1)=(n 1). Proof. Suppose we have an AD(n; μ). For1» i» r, let the blocks in the ith parallel class be named i;1 ; : : : ; i;n. For all points x 2 i;j,define A[x; i] = j. ThenA is the desired OA. The construction can be reversed, proving the converse. D.R. Stinson 14

15 Example Here is an AD(2; 2) (i.e., an affine resolvable (8; 4; 3)-ID) and the resulting OA 2 (2; 7; 2): ;0 = f0; 1; 2; 3g; 1;1 = f4; 5; 6; 7g 2;0 = f0; 1; 4; 5g; 2;1 = f2; 3; 6; 7g ;0 = f0; 1; 6; 7g; 3;1 = f2; 3; 4; 5g ;0 = f0; 2; 4; 6g; 4;1 = f1; 3; 5; 7g 5;0 = f0; 2; 5; 7g; 5;1 = f1; 3; 4; 6g ;0 = f0; 3; 4; 7g; 6;1 = f1; 2; 5; 6g ;0 = f0; 3; 5; 6g; 7;1 = f1; 2; 4; 7g D.R. Stinson 15

16 Existence of Affine Designs There are only two families of AD(n; μ) known to exist: 1. AD(q; q d ),whereq is a prime power (this design consists of the hyperplanes in an affine geometry of dimension d + 2) and 2. AD(2; s), whenever a Hadamard matrix of order 4s exists. The first family yields OA q d q d+2 1 q 1 ; q and the second family yields OAs(4s 1; 2). An OAs(4s 1; 2) is constructed from a Hadamard matrix of order 4s by standardizing a column of the matrix, and then deleting it. D.R. Stinson 16

17 Example Here is a Hadamard matrix of order 8 and the resulting OA 2 (2; 7; 2): A D.R. Stinson 17

18 Second Generalization: Higher t Let k t 2, n 1 and 1 be integers. An orthogonal array OA (t; k; n) is a n t k array, A, with entries from a set X of cardinality n such that, within any t columns of A, every ordered t-tuple of symbols from X occurs in exactly rows of A. Theorem 5 [Rao ound (1947)] Let n 2. If there is an OA (t; k; n),then n t 8 < 1 + P (t 1)=2 : 1 + P t=2 i=1 k i (n 1) i if t is even k i (t+1)=2 k 1 (t 1)=2 (n 1) t 1) + if is odd. i (n i=1 The Plackett-urman bound is the special case of the Rao bound with t = 2. D.R. Stinson 18

19 The ase = 1: The ush ound Theorem 6 [ush ound (1952)] Let n; t 2. If there is an OA 1 (t; k; n), then 8 >< if n is even and t» n k» + t 1 n + t 2 n t + 1 if t n. if n is odd and 3» t» n Theorem 7 If there exists an OA (t; k; n), then there exists an OA (t 1; k 1; n). >: Applying Theorem 7, we have that OA 1 (t; k; n) ) OA 1 (t 1; k 1; n) ) ) OA 1 (2; k t + 2; n); so k t + 2» n + 1 always holds. This yields the bound k» n + t 1. D.R. Stinson 19

20 A 0 overing Arrays Workshop May 14 16, 2006 Derived OAs Proof of Theorem 7. Let A be an OA (t; k; n) and let x be any symbol. The indicated subarray A 0 is an OA (t 1; k 1; n). x x x x A 0 is formed from A by deleting all rows of A that do not contain x in the first column, and then deleting the first column. D.R. Stinson 20

21 Example Theorem 8 For all t and all n, there exists an OA 1 (t; t + 1; n). Proof. For all t-tuples (x 1 ; : : : ; x t ) 2 (Z n) t,define a row of A consisting of the (t + 1)-tuple x 1 x 2 x t (x 1 + x t ) mod n: That is, we write down all the (t + 1)-tuples that sum to 0 modulo n. Observe that an OA 1 (t; t + 1; n) meets the ush bound with equality if t n. D.R. Stinson 21

22 t n k 1 n 1)k (n n 1)k (n + 1) n(t overing Arrays Workshop May 14 16, 2006 The ase of Large t: The ierbrauer-friedman ound Theorem 9 [ierbrauer-friedman ound (1995)] Let n; t 2. If there is an OA (t; k; n), then : The bound is nontrivial if If = 1 and k = t + 2, then the bound yields t» v 1. Hence, if t v, it must be the case that k» t + 1. This is the third case of the ush bound. t > 1: D.R. Stinson 22

23 Simple and Linear Orthogonal Arrays An orthogonal array A is a simple if all its rows in D are different. An orthogonal array A is linear if the symbol set X = F q for some prime power q and the rows of A form a subspace (of the vector space (F q) k ) having dimension log q jdj. A linear orthogonal array is necessarily simple, and is a power of q in a linear orthogonal array. D.R. Stinson 23

24 onstructing Linear Orthogonal Arrays Theorem 10 [ose (1947)] Let q be a prime power. Suppose M is an ` by k matrix of elements from F q such that every set of t columns of M is linearly independent, and M has rank `. Define A to be the q` by k matrix whose rows consist of all the linear combinations of the rows of M. ThenA is a linear OA (t; k; q) where = q` t. D.R. Stinson 24

25 1 0 1 ff ff 2 ff q 2 overing Arrays Workshop May 14 16, 2006 Example Suppose q is a prime power, and let ff 2 F q be a primitive element. Let M be the following 2 by q + 1 matrix: A Every pair of columns of M is linearly independent, so M generates a linear OA 1 (2; q + 1; q). This orthogonal array is optimal; it is equivalent to a projective or affine plane of order q. D.R. Stinson 25

26 0 0 1 ff ff 2 ff q ff 2 ff 4 ff 2(q 2) overing Arrays Workshop May 14 16, 2006 Example We can generalize the previous example to any t such that 2» t» q + 1. Suppose q is a prime power, and let ff 2 F q be a primitive element. Let M be the following t by q + 1 matrix: ff ff 6 ff 3(q 2) ff t 1 ff (t 1)2 ff (t 1)(q 2) 1 Every set t of columns M of is linearly independent, M so generates a OA linear (t; q + 1; 1 Fort = 3 q). q and odd, this orthogonal array is optimal as it meets the ush bound with A D.R. Stinson 26

27 k 1 overing Arrays Workshop May 14 16, 2006 Gilbert-Varshamov ound Theorem 11 [Gilbert (1952), Varshamov (1957)] Let `, t and k be positive integers such that 2» t» `,andletq be a prime power. Suppose that Xt 1 1) i < q`: (1) (q i Then there exists a OA (t; k; linear where = q),. q` t i=0 Theorem 11 is proven by showing that there is an ` by k matrix satisfying the conditions of Theorem 10. This can be done by an easy counting argument. D.R. Stinson 27

28 odes Let Q be a set of q symbols. Acode is a set of k-tuples (of symbols) called codewords. For x; y 2 Q k,define the Hamming distance between x and y to be y) = jfi : x i 6= y i gj; dist(x; x = (x where ; : : : ; x k ) 1 y = (y and ; : : : ; y k 1 ). The distance of the code, denoted dist(), is the smallest positive integer d such that dist(x; y) d for all x; y 2, x 6= y. is a (k; M;d; q)-code if jj = M and dist() d. D.R. Stinson 28

29 Linear odes Acode is a linear code of dimension m if Q = F q for some prime power q and is an m-dimensional subspace of the vector space (F q) k. The dual code of a linear code is the code?,where? = fy 2 (F q) k : x y = 0 for all x 2 g: (As usual, y denotes the inner product F q over of the two x vectors x and y.) The subspaces and are called orthogonal complements of each other. Observe? that is a linear code of dimension k dim().? D.R. Stinson 29

30 Linear odes and OAs Theorem 12 Suppose that (F q) k is a linear code of dimension m. Then if and only if is a linear OA (d 1; k; q), where dist() d?. k m d+1 q = Theorem 12 says that the theory of linear codes is equivalent to the theory of linear orthogonal arrays. In particular, every bound or construction for linear codes implies a corresponding bound or construction for linear orthogonal arrays, and vice versa. D.R. Stinson 30

31 Hamming and Simplex odes Let q be a prime power and let r 2. TheHamming code is a linear (k; M;d; q)-code where k = (q r 1)=(q 1), M = q k r and d = 3. The Simplex code is the dual of the Hamming code; it is a M;d; q)-code = where 1)=(q 1), = and d = q. r 1 (k; k (q M q It r r q) follows t = q that the Hamming OA (t; code is an where 1, r 1 k; = q and. This is an optimal OA because it k = (q 1)=(q k r t 1) meets the ierbrauer-friedman r bound with equality. The Simplex code is an OA (2; k; q) where k = (q r 1)=(q 1) and =. This is an optimal OA because it meets the Plackett-urman q bound r 2 with equality. D.R. Stinson 31

32 M = A : overing Arrays Workshop May 14 16, 2006 Hamming and Simplex odes (cont.) The Simplex code can be constructed using Theorem 10. From every 1-dimensional subspace of (F q) r, choose a (non-zero) vector v. LetM be the r by (q r 1)=(q 1) matrix whose columns are the chosen vectors. learly no two columns of M are linearly dependent, so M yields an OA with t = 2. (Equivalently, the Hamming code has distance d = 3.) When q = 2, the columns of M comprise all 2 r 1 non-zero vectors in (F 2 ) r. For example, when r = 3, wehave D.R. Stinson 32

33 Rao and Sphere-packing ounds Recall the Rao bound for OA (t; k; q) in the case of even t: Xt=2 q t 1 + k i 1) i : (q i=1 Suppose is a linear (k; q m ; d; q)-code where d is odd. Then? is a k; where = q q),. Applying the Rao bound with OA (d 1; k m d+1 1, we obtain d = t X (d 1)=2 or q k m d+1 q d i=1 k i 1) i ; (q q m» k q + P (d 1)=2 1 k 1) i : i (q This is the Sphere-packing bound for (linear) codes. i=1 D.R. Stinson 33

34 t q k 1 q 1)k (q + 1) q(t k m d+1 q d 1 q k 1 q qd (q 1)k : qd 1)k (q qd overing Arrays Workshop May 14 16, 2006 ierbrauer-friedman and Plotkin ounds Recall the ierbrauer-friedman bound for OA (t; k; q): : Suppose is a linear (k; q m ; d; q)-code. Then? is a OA (d 1; k; q), where = q k m d+1. Applying the ierbrauer-friedman bound with t = d 1, we obtain : Suppose that d > (q 1)k=q; then q m» This is the Plotkin bound for (linear) codes. D.R. Stinson 34

35 0 t 1 j c j 1 A + z; a overing Arrays Workshop May 14 16, 2006 A Family of Nonlinear OAs Theorem 13 [Mudhopadhyay (1981), ierbrauer (1995)] Suppose q is a prime power, ` m and 2» t» q`. Then there exists an OA q (t 1)(` m)(t; q`; q m ). Proof. : q` Let q) be any surjective m F q-linear mapping. The ffi rows F of are indexed by! t-tuples (F : : : ; a t 1 ),wherez 2 (F q) m A 1 (z;a ; and q` (1» i» t i 1), and the columns A of are indexed by the a F 2 elements of A[r; c] q`. An entry is defined F as follows: A[r; c] = ffi where r = (z;a 1 ; : : : ; a t 1 ). j=1 D.R. Stinson 35

36 A Family of Nonlinear OAs (cont.) For t = 2, we obtain OA q` m(2; q`; q m ).If`» 2m, then these OAs have the minimum possible value of permitted by the Plackett-urman bound: q`(qm 1) + 1 2m q q` m q` 1 = 2m q > q` m 1: Since is an integer, it must be the case that q` m. D.R. Stinson 36

37 A i = 1 M overing Arrays Workshop May 14 16, 2006 Distance Distributions of odes Suppose is (k; M;d; 2) an (binary) code. The distance distribution of is defined to be the (A sequence ; A 1 ; : : : ; A k 0 ),where y) : x; y 2 ; dist(x; y) = igj; jf(x; = 0; : : : ; k. The following properties are easily verified: i A 0 = 1; (2) A i 0 for 0» i» k; and (3) Also, A 0 + A 1 + : : : + A k = M: (4) A i = 0 for 1» i» d 1, anda d > 0. D.R. Stinson 37

38 A 0 i = 1 M overing Arrays Workshop May 14 16, 2006 Krawtchouk Polynomials Let ` be a non-negative integer, and let P`(x) be the Krawtchouk polynomial defined as follows: `X ( 1) j x j x k j ` : P`(x) = j=0 The dual distance distribution of is defined to be (A 0 0 ; A0 1 ; : : : ; A0 k ), where kx A j P i (j); j=0 i = 0; : : : ; k. We will express this notationally as (A 0 0; A 0 1; : : : ; A 0 k ) = Kr(A 0; A 1 ; : : : ; A k ): D.R. Stinson 38

39 Krawtchouk Polynomials (cont.) The following properties, analogous to (2), (3) and (4), were proved by Delsarte: = 1; (5) A 0 0 A 0 i 0 for 0» i» k; and (6) (7) 0 A0 1 + : : : + A0 k = 2k A : M 0 Delsarte further showed that Kr is an involutory transformation: ; : : : ; Ak ) = (A 0; A 1 ; : : : ; A k ): (8) A 0; Kr(A D.R. Stinson 39

40 Dual Distance A If for 0 i» d0 1 and 0 A d 0 > 0, thend is called the dual 0 i distance of the code. Suppose we write the codewords = in as rows of 0 1» an array. Delsarte proved the following important k M result. Theorem 14 [Delsarte (1973)] is an OA M=2 d 0 1(d 0 1; k; 2). If is linear, then the dual distance distribution of is the same as the distance distibution of?. Hence, Theorem 14 generalizes Theorem 12. D.R. Stinson 40

41 x 0 = 1 kx overing Arrays Workshop May 14 16, 2006 Linear Programming ounds Let D and k be positive integers such that D» k. We employ the following linear program, L(k; D), which is due to McEliece, Rodemich, Rumsey and Welch. Maximize S = x 0 + x x k subject to i = 0 for 1» i» D 1 x i 0 for D» i» k x x j P i (j) 0 for 0» i» k: j=0 D.R. Stinson 41

42 Linear Programming ounds (cont.) Theorem 15 Suppose that is a (k; M;d; 2) code. Then the following hold: 1. Let S opt be the optimal solution to L(k; d). If has distance d,then M» S opt. 2. Let S opt be the optimal solution to L(k; d 0 ).If has dual distance d 0, then M 2 k =S opt. D.R. Stinson 42

43 Linear Programming ounds (cont.) Proof. Let ; A 1 ; : : : ; A k ) 0 be the distance distribution of (k; M;d; 2) a (A code,, having andlet(a distance d, ; : : : ; Ak ) be the dual distance 0; A distribution of. The first assertion is proved as follows. We claim that (A 0 ; : : : ; A k ) is a feasible solution for L(k; d). The constraints of L(k; d) are satisfied because of (2), (3) and (6), and the fact that is assumed to have distance d. Then, from (4), the resulting value of the objective function is M,so the first assertion follows. To prove the second assertion, we show that (A 0 0 ; : : : ; A0 k ) is a feasible solution for L(k; d 0 ). This follows in a similar way from (3), (5), (6) and (8). Then, from (7), the resulting value of the objective function is 2 k =M, so the second assertion follows, as well. D.R. Stinson 43

44 Duality of ounds for odes and OAs Gopalakrishnan (1994) (see also ierbrauer, Gopalakrishnan and Stinson (1998)) observed that Theorem 15 provides an elementary and transparent explanation of the duality of the Sphere-Packing and Rao bounds; and of the Plotkin and ierbrauer bounds. This follows immediately from Theorem 15 once it is proven that the bounds in question are consequences of the more general linear programming bound. For example, it is known that the Sphere-Packing bound is a consequence of the Linear Programming bound. Therefore the Rao bound holds. Some comments: 1. Other examples of dual pairs of bounds also exist. 2. The theory can be generalized to non-binary codes and OAs. 3. A similar observation was made by Levenshtein (1995) using a much more complicated approach. D.R. Stinson 44

45 OAs where the Number of Symbols is not a Prime Power There is an extensive theory of MOLS, which yields many constructions for OAs with t = 2 on an arbitrary number of symbols. As well, recall from Theorem 8 that an OA 1 (t; t + 1; n) exists for all t and n. In contrast, there are very few constructions for OAs with 3» t» k 2, in which the number of symbols is not a prime power. One classical construction is a direct product construction. Theorem 16 [ush (1952)] If there exist OA 1 (t; k; n 1 ) and OA 2 (t; k; n 1 ), then there exists an OA 1 2 (t; k; n 1 n 2 ). D.R. Stinson 45

46 OAs where the Number of Symbols is not a Prime Power (cont.) A recent method of Kreher has been used to produce new infinite classes of OA (3; k; n) with k 5. This technique employs resolvable 3-wise balanced designs and ordered designs of strength 3. Here are two examples of results obtained using applications of this method. Theorem 17 [olbourn, Kreher, McSorley, Stinson (2002)] q Suppose is an odd prime power. Then there exists an q + 3; q + 1). OAq 1(3; Theorem 18 [olbourn, Kreher, McSorley, Stinson (2002)] q Suppose is an odd prime power. Then there exists an OA q 2 1(3; 2(q + 3); q + 1). D.R. Stinson 46

47 Applications of Orthogonal Arrays in omputer Science Secrecy and authentication codes ffl Threshold schemes ffl Perfect local randomizers ffl Derandomization ffl Resilient and correlation-immune functions ffl lock cipher and stream cipher design ffl D.R. Stinson 47

48 References For more information on orthogonal arrays, including applications, see: J. ierbrauer, Introduction to oding Theory, R Press, 2005, ffl hapter 15.. J. olbourn and J. H. Dinitz. The R Handbook of ffl ombinatorial Designs, R Press, 1996, Sections II.4 and II.5. A. S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays: ffl Theory and Applications, Springer, D. R. Stinson, ombinatorial Designs: onstructions and Analysis, ffl Springer, 2004, hapters 10 and 11. D.R. Stinson 48