Institute of Statistics Mimeo Series No April 1965

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1 _ - ON THE CONSTRUCTW OF DFFERENCE SETS AND THER USE N THE SEARCH FOR ORTHOGONAL ratn SQUARES AND ERROR CORRECTnm CODES by. M. Chakravarti University of North Carolina nstitute of Statistics Mimeo Series No. 427 April 1965 n this paper, a relationship is established between orthogonal mappings of a group and difference sets for construction of orthogonal arrays. A difference set D(lO,lO,5) has been constructed and is given as an illustration. Use of difference sets for construction of orthogona:l Latin squares and error correcting codes is discussed. This research was supported by the U. S. krrrry Research Office Grant No. DA-3l-J24-ARO-D-254. DEPARTMENT OF S'MTSTCS UNVERSTY OF NORTH CAROLmA Chapel Hill, N. C.

2 . _ On the Construction of Difference Sets and their Use in the Search for Orthogonal Latin Squares and Error Correcting Codes by. M. Chakravarti University of North Carolina 1. ntroduction. The concept of orthogonal mapping was developed in [2]. t has been shown there that the existence of s orthogonal mappings of a group of order 4t inqjlies the existence of s mutually orthogonal Latin squares of order 4t. t has been shown in [5], that a single-error correcting code of length 4 over an n-symbol alphabet with n 2 code words corresponds to a pair of orthogonal Latin squares of order n. n general, a set of t pair-wise orthogonal Latin squares of order n is equivalent to a code of n 2 words from an n symbol alphabet, with block length (t+2) and distance (t+l). Difference sets which provide orthogonal arrays were defined in [1]. Difference sets of order 2p for p = 3,5 and 7 were first constructed in [6]. The existence of difference sets of order 2p for p = 2t+1 an odd prime and t(t+l)/2 an odd integer, was also demonstrated in [6]. n [3], a general method of construction of difference sets of order 2p for p an odd prime (without the restriction as in [6]) has been given. n this paper, a relationship is established between orthpgonaj. mappings and difference sets for construction of orthogonal arrays. A difference set D(lO, 10, 5) has been constructed and is given as an illustration. Use of difference sets for construction of orthogonal Latin squares and associated error correcting codes is discussed.

3 2. Definitions Latin squares. A Latin square of order n is defined as an n x n square, the n 2 cells of lmich are occupied by n distinct symbols (which may be Latin or Greek letters or just integers) such that each symbol occurs once in each row and once in each column. is called the cell (i,.1). The cell in the ith row and the jth column Two Latin squares are said to be orthogonal if on superposition each sy:mbol of the first square occurs exactly once with each symbol of the second square Orthogonal arrays. Suppose A = «a ij» is a matrix, i = 1,2,..., m, j = 1,2,..., N and the elements of the matrix are symbols 0,1,2,..., s-l t Consider the s 1 x t matrices x' = (x l 'J!2', x t ) that can be formed by giving different values to xi's, Xi = 0,1,, s-l, i = 1,2,, t. Suppose associated with each t x 1 matrix as, there is a positive integer A (:;"J!2', x t ) which is invariant under permutations of (xl'j!2'..., x t ). t f for every submatrix having t rows of A, the s t x 1 matrices as occur as columns A(xl'~' balanced array of strength t s sy:mbols (or levels) and the specified..., x t ) times then the matrix A is called a partially in N assemblies, m constraints (or factors), r.. (:;"J!2'..., x t ) parameters. \fuen A(:;"~'..., x t ) = A for all x', the array is called an orthogonal arrb -A(N,m, s,t) of strength t, index r.., N assemblies and m constraints. A detailed account of methods of construction of orthogonal arrays is given in [4J._ An orthogonal array having s2 assemblies, t+2 constraints, strength 2 and index 1, corresponds to a set of t mutually orthogonal Latin squares of order s. 2.3 Difference sets. A difference set for constructing an orthogonal array is defined in 2 _,

4 . _ [1], as follows. Let E be a 1Ddule (additive group) consisting of s elements eo' e l,..., es-1. Suppose it is possible to find a scheme S of r rows and n columns, with elements belonging to E, sll s12 sln s2l s22 s.2n (2.1) S. srl sr2 s rn such that 8.lOOng the differences of the corresponding elements of any two rows, each element of E occurs exactly" times (n = "M). Then S is called a difference set generating an orthogonal arr8\y A(" a2, r, s, 2) Given S, the orthogonal array A is formed as follows: Write down the addition table of E. Then replace each element in the scheme S by the row in the addition table, corresponding to the element (use only the suffixes 0,1,2,..., s-l). 3. Generalized Hadamard Matrices. A square matrix H of order h all of whose elements are p th roots of unity, is called [3] a generalized Hadamard matrix (H(p,h) matrix), if HH ct = h, where H ct is the conjugate transpose of H. A generalized Hadamard matrix has been called an equimodular square matrix in [6]. When p is a prime, an H(p, h) matrix can exist only for values h = pt, Where t is a positive integer. A method of constructing H(p, 2 i pj) matrices for any prime p and any non-negative integers i < j has been given in [3]. Let H = «hij»i,j = 0,1,, pt-l, be a Ha.d.a.mard matrix of order pt, where p is an odd prime. Let ex be a primitive p th root of Sij unity and h ij = ex, i,j =0,1,, pt-l, where Sij GF(p).3

5 Then it is easily seen that the matrix S = «Sij» i,j = ~,1,, pt-l, be a Hada.ma.rd matrix of order pt, where p is an odd prime. Let ex be a Sij primitive. p th root of unity and h ij = ex, i,j = 0,1,..., pt-1, where s.. GF(p). Then it is easily seen that the matrix S = «s.j» ~. 1 i,j = 0,1,, pt-l, is a difference set as defined in Section 2.3. generate an orthogonal array Suppose p = 2q+1 is a prime and let n be the smallest quadratic Then it is easy to prove that _knq;_..t, y y- k = 0,1,, p-1 matrix (3.1) K = A(tp2,pt,p,2). non-residue of p. Let. ~ be the Van der Mende matrix defined by V = «v ij», v ij = ~j, i,j = 0,1,, p-l, where ex is a primitive p th root of unity. Then V is a symmetric H(p,p) matrix. Denote by U that permutation matrix for which W = VU has elements w ij = cfij, i,j = 0,1,..., p-1. Let the 2 nntrix Q = «%j» be such that %j = 0 for i f j and C1:i.i = ofl1, i = 0,1,2,..., p-1. Then C = QVQ and B = Q~n are H(p,p) matrices. Let D = «d. j» be the matrix defined by d.. = 0 for i f j, and J d ii = ~ for i = 0,1,, p-1 and T = «t ij» be the permutation ma~rix defined by t.+ 1. = 1, for i = 0,1,, p-1 and t.. = 0, otherwise. 1,1. 1J. Further, let Y = (1,1, 1) and Z = (0,0,, 0) be both of length p. D'kv =V'Jf and Dnkw = ~ t is also easily seen that the k th column of B can be written in the form ~Q'DyT. Let CP be the matrix whose k th column is given by Then it has been proved in [3], that the 4 t 1nll, _

6 . _ is an H(p, 2p) matrix. We give here an example of an S matrix which corresponds to an H(p, 2p) matrix for p = o (3.2) S = o This matrix is a difference set which can be used to generate an orthogonal array A(50,10,5,2), in the manner described in section 2.3. t has been soown in [6], that given an H(p,2p) matrix Where p is a prime, one can construct an orthogonal array A(2p2, 2p+1, p, 2). 4. Orthogonal mappings and construction of mutually orthogonal Latin squares. (4.1) Consider a finite group G of order n. Let a be a one-one mapping of G into itself, the image of the element x in G being denoted by a x Thus, the equation ax = c determines a unique x in G given c in G and conversely. Let denote the identity mapping which maps every x in G on x itself. such mappings define the permutation group ping a then corresponds a unique inverse 5 The set of all M, of order nl. To each map -1 a such that if ax = a, then a- 1 a = x. Let ax. denote the mapping for which the i.ma.ge of x is ox.x, where a and X; are in M t is clear that a X. is in M and that the associative law Jl( ax) = (JU)X holds.

7 Consider a n x n square. Make an (1,1) correspondence between the rows of the square and the elements of G. Similarly, between the columns of the square and the elements of G. ~e cell of the square corresponding to the row x and the column y is said to be the cell (x,y). We quote some results from [2] without proof. Theorem 4.1: f in each cell (x,y) of an n x n square we put the element (<Tx)y of G (er is a mapping belonging to M) we get a Latin square L(er). Let.er and X. be two mappings belonging to M. The> e are said to be orthogonal if the equation (4.2) has a unique solution x in G for every c in G. f the group operation is addition instead of multiplication, the equation (4.2) becomes (4.;) (erx) - (xx) = c Theorem 4.2: The necessary and sufficient condition for the Latin squares L(er) and L(x.) to be orthogonal is that the ma:p]';lings er and X. are orthogonal. Theorem 4.;: The necessary and sufficient condition for a finite mdule (additive Abelian group) to admit two orthogonal mappings is that the sum of the elements of the mdule vanish. Let G(~,~) denote a mdule whose elements are vectors (a1'a ), where 2 ~ is a residue class (mod ~) and a is a residue class (OOd ~), 2 the addition being defined by (4.4) (~,a2) + (b,b l 2 ) = (c l,c 2 ), where ~ + b l = c l (OOd n l ), a 2 + b 2 = c 2 (mod ~). The module has nl~ elements. From theorem 4,;, it follows that for the existence of a pair of orthogonal mappings for G(~,n2)' it is necessary and sufficient that n l and ~ have the same parity. 6 _ i

8 . _,. 5. Difference sets and orthogonal mappings Consider the difference set S of (;.2). We note every element of the group (0,1,2,;,4) occurs exactly twice in every row (column) except in the first row (c01~). And further amng the differences of pairs of corresponding elements in any 00 rows (columns) every element of the group (0,1,2,;,4) occurs exactly twice. This may be described as one to many mapping of the e1en:ents of the group of residue classes md 5 into itself and the additional property enjoyed by the mapping may be caj.1ed orthogonality in a wider sense. Consider a difference set of order ~~ with ~ symbols such that (i) every symbol occurs ~ t:imes in each row(each column) (i1) 8JOOng the differences of pairs of corresponding elements in any two rows (columns) every symbol occurs ~ times. Further, consider a roodule of ~ symbols. The problem is then of adjoining to each one of the symbols in the difference set, a symbol from this new roodu1e as a second co-ordinate, so that the whole scheme will define a difference set in n n elements (eac~ element is now 1 2 composed of two symbols taken from two different rodules). For instance, it vdll be of interest to construct a difference set of order 15 in five symbols (0,1,2,;,4) and then adjoin symbols from the module of the residue classes mod ;, so that the whole scheme becomes a difference set in symbols, (0,0),(0,1),(0,2),(1,0),(1,1),,(4,0),(4,1),(4,2). 6. Computer search for difference sets. The principles outlined in sections 4 and 5, provide a technique for computer search for difference sets Which will provide orthogonal Latin squares or orthogonal arrays. A similar technique was used in [2] in a search for orthogonal Latin squares of order 12. starting with a Hadamard 7

9 second coordinates to its columns (except the first Which consists of zeros only) so that the final array became a difference set with elements from the nodule ~dth elements (a,b) rthere a = 0,1, (mod 2) and b ~ 0,1,2,3,4,5 (mod 6). Using a comput er, an array consisting of five such columns only could be obtained. an illustration. This difference set (first given in [2]) is reproduced here as ~ ~ '\ (6.1) Error correcting codes associated with orthogonal Latin squares. ~le relationship betrreen a n-symbol block code of block length k and a set of (k-2) mutually orthogonal Latin squares of order n, described here is on the lines of [5]. Consider the set V(n,k) of all k-tuples from an n-symbol alphabet. Any subset of V(n,k) may be called a block code of blocl{ length k. The elements of the subset are the code words. The H.a.r.1ming distance between two code words is the number of coordinates in which they differ. A block code in ~Thich any pair of code words are at 8 _ matrix H(2,12), elements 0,1,2,3,4,5 (md 6) were to be adjoined as

10 . _ least a Hamming distance of r apart is called a distance r code. A distance r code can correct (r-l)/2 errors. t is easy to show that a set of t mutually orthogonal Latin squares of order n is equivalent to a set of n 2 (t+2)-tuples in n symbols such that the distance between any two (t+2)-tuples is at least (t+l). Hence this provides us a distance (t+l) code in n symbols of block length (t+2). Thus we have shown how difference sets might be used to construct orthogonal Latin squares, orthogonal arrays and error-correcting codes. 9

11 REFERENCES Bose, R. C. and Bush, K. A. (1952). Orthogonal arrays of strength two and three. Ann. Math. Statist.,~, Bose, R. C., Chakravarti,. M. and Knuth, D. E. (1960). On methods of constructing sets of mutually orthogonal Latin squares using a computer,. Technometrics,!, Butson, A. T. (1962). Generalized Hadamard matrices. Proc. Amer. Math. Soc., ~, Chakravarti,. M. (1963). Orthogonal and partially balanced arrays and their application in design of experiments. Metrika,!, Golomb, S. W. and Posner, E. C. (1964). Rook domains, Latin squares, affine planes and error-distributing codes. EEE Trans. on nform. Theory, T-10, Masuyama, M. (1957). On difference sets for constructing orthogonal arrays of index two and of strength two. Rep. Statist. Appl. Res., JUSE, 5, _ 1 1 1

On Construction of a Class of. Orthogonal Arrays

On Construction of a Class of. Orthogonal Arrays On Construction of a Class of Orthogonal Arrays arxiv:1210.6923v1 [cs.dm] 25 Oct 2012 by Ankit Pat under the esteemed guidance of Professor Somesh Kumar A Dissertation Submitted for the Partial Fulfillment