Skew Hadamard Matrices Research Articles

A skew-Hadamard matrix of order 92

There is a skew-Hadamard matrix of order 92.Previously the smallest order for which a skew-Hadamard matrix was not known was 92. We construct such a matrix below. The orders < 200 which are now undecided are 100, 116, 148, 156, 172, 188, 196; see [2], [3]. The existence of any Hadamard matrix of order 92 was unknown until 1962 [1].

An infinite family of skew Hadamard matrices

It has been conjectured that iϊ-matrices and even skew iJ-matrices always exist for n divisible by 4. Constructions of both types of matrices have been given for particular values of n and also for various infinite classes of values (see [1] for the pertinent references). In [1] D. Blatt and G. Szekeres constructed for the first time a skew ίf-matrix of order 52. Their construction is summarized in Theorems 1 and 2 of this paper. Given an additive abelian group G of order 2m + 1, two subsets A c G , BczG, each of order m, are called complementary difference sets in G if ( i ) a G A ==> —a g A, and (ii) for each de (?, d Φ 0, the total number of solutions (αx, α2) e A x A, (&!, b2) e B x B of the equations

Some (1, −1) matrices

We define an n-type (1, −1) matrix N = I + R of order n ≡ 2 (mod 4) to have R symmetric and R 2 = ( n − 1) I n . These matrices are analogous to skew-type matrices M = I + W which have W skew-symmetric. If n is the order of an n-type matrix, h 1 and h 2 the orders of Hadamard matrices, h the order of a skew-Hadamard matrix, and p r ≡ 1 (mod 4) is a prime power then we show there are: n-type matrices of orders p r + 1,( h − 1) 2 + 1,( n −1) 3 + 1; symmetric Hadamard matrices of orders2 n,2 n(n −1),2 p r ( p r + 1); Hadamard matrices of orders h 1 n, h 1 h 2 n(n − 1), h 1 h 2 n(n − 3) (this latter with n + 4 also the order of an n-type matrix); Hadamard matrices of orders 452, 612, 2452 and 3044, all “new”. We also give existence conditions for many other classes of Hadamard matrices and another formulation for Goldberg's skew-Hadamard matrix of order( h − 1) 3 + 1.

A skew Hadamard matrix of order 36

Hadamard matrices exist for infinitely many orders 4m, m ≧ 1, m integer, including all 4m < 100, cf. [3], [2]. They are conjectured to exist for all such orders. Skew Hadamard matrices have been constructed for all orders 4m < 100 except for 36, 52, 76, 92, cf. the table in [4]. Recently Szekeres [6] found skew Hadamard matrices of the order 2(pt +1)≡ 12 (mod 16), p prime, thus covering the case 76. In addition, Blatt and Szekeres [1] constructed one of order 52. The present note contains a skew Hadamard matrix of order 36 (and one of order 52), thus leaving 92 as the smallest open case.

(v, k, λ) Configurations and Hadamard matrices

Using the terminology in 2 (where the expression m-type is also explained) we will prove the following theorems: Theorem 1. If there exist (i) a skew-Hadamard matrix H = U+I of order h, (ii)m-type matrices M = W+I and N = NT of order m, (iii) three matrices X, Y, Z of order x = 3 (mod 4) satisfying (a) XYT, YZT and ZXT all symmetric, and (b) XXT = aIx+bJxthen is an Hadamard matrix of order mxh.

A Skew Hadamard Matrix of Order 52

1. A Hadamard (H-) matrix H = (hij) of order n is an n × n square matrix satisfying the conditionsfor all i, j ≦ n. A skew H-matrix is an H-matrix of the formwhere I is the identity matrix and S’ the transpose of 5. In particular,Skew H-matrices have applications in the theory of finite projective planes (2) and tournaments (4), also in the construction of H-matrices of certain orders. For example, if there is a skew H-matrix of order n, then there is an H-matrix of order n(n – 1) (Williamson, see (1, p. 213)).