Hadamard matrices of order 36 and double-even self-dual [72,36,12] codes

A balanced incomplete block design (BIBD) [1] with parameters 2-(v, b, r, k, λ) (short 2-(v, k, λ)) is a pair (V,B) where V is a v-set (elements are called points) and B is a collection of b k-subsets (elements are called blocks) of V such that each point is contained in exactly r blocks and any pair of points is contained in exactly λ blocks. A Hadamard matrix of order n is an n × n (1,−1)-matrix satisfying HH = nI . Each Hadamard matrix can be normalized, i.e. replaced by an equivalent Hadamard matrix whose first row and column are ones. When deleting the first row and column of a normalized Hadamard matrix of order 4m, a symmetric 2-(4m− 1, 2m− 1,m− 1) design is obtained which is called a Hadamard design. Hadamard matrices have been classified up to order 28. For higher orders, only partial classifications are known. Lin, Wallis and Zhu [3] found 66104 inequivalent Hadamard matrices of order 32. Extensive results on order 32 appear in [4] and [5]. Before this work, at least 762 inequivalent Hadamard matrices of order 36 were known, see [2], [6] and [7]. We found 7238 Hadamard matrices of order 36, which are obtained from all 2-(35, 17, 8) designs with an automorphism of order 3 and 2 fixed points and blocks. In order to be sure about our computer results, we made two independent implementations. A linear code with block length n, dimension k, and minimum distance d is referred to as an [n, k, d]code. Hadamard designs are related to self-dual codes, see [9] and [10]. Let A be the incidence matrix